constant - a value that always assumes the same value independent of how its parameters are varied

$1,100 per month for 10 years, if the account earns 2% per year

$1,100 per month for 10 years, if the account earns 2% per year
What the student or parent is asking is: If they deposit $1,100 per month in a savings/investment account every month for 10 years, and they earn 2% per year, how much will the account be worth after 10 years?
Deposits are monthly. But interest crediting is annual. What we want is to match the two based on interest crediting time, which is annual or yearly.
1100 per month. * 12 months in a year = 13,200 per year in deposit
Since we matched interest crediting period with deposits, we now want to know:
If they deposit $13,200 per year in a savings/investment account every year for 10 years, and they earn 2% per year, how much will the account be worth after 10 years?
This is an annuity, which is a constant stream of payments with interest crediting at a certain period.
[SIZE=5][B]Calculate AV given i = 0.02, n = 10[/B]
[B]AV = Payment * ((1 + i)^n - 1)/i[/B][/SIZE]
[B]AV =[/B]13200 * ((1 + 0.02)^10 - 1)/0.02
[B]AV =[/B]13200 * (1.02^10 - 1)/0.02
[B]AV =[/B]13200 * (1.2189944199948 - 1)/0.02
[B]AV =[/B]13200 * 0.21899441999476/0.02
[B]AV = [/B]2890.7263439308/0.02
[B]AV = 144,536.32[/B]

-5n - 5n - 5 = 5

-5n - 5n - 5 = 5
Solve for [I]n[/I] in the equation -5n - 5n - 5 = 5
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(-5 - 5)n = -10n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
-10n - 5 = + 5
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -5 and 5. To do that, we add 5 to both sides
-10n - 5 + 5 = 5 + 5
[SIZE=5][B]Step 4: Cancel 5 on the left side:[/B][/SIZE]
-10n = 10
[SIZE=5][B]Step 5: Divide each side of the equation by -10[/B][/SIZE]
-10n/-10 = 10/-10
n = [B]- 1
[URL='https://www.mathcelebrity.com/1unk.php?num=-5n-5n-5%3D5&pl=Solve']Source[/URL][/B]

1 - n = n - 1

1 - n = n - 1
Solve for [I]n[/I] in the equation 1 - n = n - 1
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables -n and n. To do that, we subtract n from both sides
-n + 1 - n = n - 1 - n
[SIZE=5][B]Step 2: Cancel n on the right side:[/B][/SIZE]
-2n + 1 = -1
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 1 and -1. To do that, we subtract 1 from both sides
-2n + 1 - 1 = -1 - 1
[SIZE=5][B]Step 4: Cancel 1 on the left side:[/B][/SIZE]
-2n = -2
[SIZE=5][B]Step 5: Divide each side of the equation by -2[/B][/SIZE]
-2n/-2 = -2/-2
n = [B]1
[URL='https://www.mathcelebrity.com/1unk.php?num=1-n%3Dn-1&pl=Solve']Source[/URL][/B]

10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6

10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6
Solve for [I]n[/I] in the equation 10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(10 - 9 + 8 - 7 + 6)n = 8n
[SIZE=5][B]Step 2: Group the constant terms on the right hand side:[/B][/SIZE]
10 - 9 + 8 - 7 + 6 = 8
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
8n = + 8
[SIZE=5][B]Step 4: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 8/8
n = [B]1[/B]

2n + 1 = n + 10

2n + 1 = n + 10
Solve for [I]n[/I] in the equation 2n + 1 = n + 10
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 2n and n. To do that, we subtract n from both sides
2n + 1 - n = n + 10 - n
[SIZE=5][B]Step 2: Cancel n on the right side:[/B][/SIZE]
n + 1 = 10
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 1 and 10. To do that, we subtract 1 from both sides
n + 1 - 1 = 10 - 1
[SIZE=5][B]Step 4: Cancel 1 on the left side:[/B][/SIZE]
n = [B]9[/B]

2n + 10 = 3n + 5

2n + 10 = 3n + 5
Solve for [I]n[/I] in the equation 2n + 10 = 3n + 5
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 2n and 3n. To do that, we subtract 3n from both sides
2n + 10 - 3n = 3n + 5 - 3n
[SIZE=5][B]Step 2: Cancel 3n on the right side:[/B][/SIZE]
-n + 10 = 5
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 10 and 5. To do that, we subtract 10 from both sides
-n + 10 - 10 = 5 - 10
[SIZE=5][B]Step 4: Cancel 10 on the left side:[/B][/SIZE]
-n = -5
[SIZE=5][B]Step 5: Divide each side of the equation by -1[/B][/SIZE]
-1n/-1 = -5/-1
n = [B]5[/B]

2n + 8 - n = 20

2n + 8 - n = 20
Solve for [I]n[/I] in the equation 2n + 8 - n = 20
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(2 - 1)n = n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
n + 8 = + 20
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 8 and 20. To do that, we subtract 8 from both sides
n + 8 - 8 = 20 - 8
[SIZE=5][B]Step 4: Cancel 8 on the left side:[/B][/SIZE]
n = [B]12[/B]

2n + 8 = 24

2n + 8 = 24
Solve for [I]n[/I] in the equation 2n + 8 = 24
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 8 and 24. To do that, we subtract 8 from both sides
2n + 8 - 8 = 24 - 8
[SIZE=5][B]Step 2: Cancel 8 on the left side:[/B][/SIZE]
2n = 16
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2n/2 = 16/2
n = [B]8[/B]

2n - 7 = 0

2n - 7 = 0
Solve for [I]n[/I] in the equation 2n - 7 = 0
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -7 and 0. To do that, we add 7 to both sides
2n - 7 + 7 = 0 + 7
[SIZE=5][B]Step 2: Cancel 7 on the left side:[/B][/SIZE]
2n = 7
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2n/2 = 7/2
n = [B]3.5[/B]

4n - 8 = n + 1

4n - 8 = n + 1
Solve for [I]n[/I] in the equation 4n - 8 = n + 1
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 4n and n. To do that, we subtract n from both sides
4n - 8 - n = n + 1 - n
[SIZE=5][B]Step 2: Cancel n on the right side:[/B][/SIZE]
3n - 8 = 1
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -8 and 1. To do that, we add 8 to both sides
3n - 8 + 8 = 1 + 8
[SIZE=5][B]Step 4: Cancel 8 on the left side:[/B][/SIZE]
3n = 9
[SIZE=5][B]Step 5: Divide each side of the equation by 3[/B][/SIZE]
3n/3 = 9/3
n = [B]3[/B]

5n - 5 = 85

5n - 5 = 85
Solve for [I]n[/I] in the equation 5n - 5 = 85
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -5 and 85. To do that, we add 5 to both sides
5n - 5 + 5 = 85 + 5
[SIZE=5][B]Step 2: Cancel 5 on the left side:[/B][/SIZE]
5n = 90
[SIZE=5][B]Step 3: Divide each side of the equation by 5[/B][/SIZE]
5n/5 = 90/5
n = [B]18[/B]

7 and 105 are successive terms in a geometric sequence. what is the term following 105?

7 and 105 are successive terms in a geometric sequence. what is the term following 105?
Geometric sequences are set up such that the next term in the sequence equals the prior term multiplied by a constant. Therefore, we express the relationship in the following equation:
7k = 105 where k is the constant
[URL='https://www.mathcelebrity.com/1unk.php?num=7k%3D105&pl=Solve']Type this equation into our search engine[/URL] and we get:
k = 15
The next term in the geometric sequence after 105 is found as follows:
105*15 = [B]1,575[/B]

7n + 4 + n - 5 = 63

7n + 4 + n - 5 = 63
Solve for [I]n[/I] in the equation 7n + 4 + n - 5 = 63
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(7 + 1)n = 8n
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
4 - 5 = -1
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
8n - 1 = + 63
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants -1 and 63. To do that, we add 1 to both sides
8n - 1 + 1 = 63 + 1
[SIZE=5][B]Step 5: Cancel 1 on the left side:[/B][/SIZE]
8n = 64
[SIZE=5][B]Step 6: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 64/8
n = [B]8[/B]

A 98-inch piece of wire must be cut into two pieces. One piece must be 10 inches shorter than the ot

A 98-inch piece of wire must be cut into two pieces. One piece must be 10 inches shorter than the other. How long should the pieces be?
The key phrase in this problem is [B]two pieces[/B].
Declare Variables:
[LIST]
[*]Let the short piece length be s
[*]Let the long piece length be l
[/LIST]
We're given the following
[LIST=1]
[*]s = l - 10
[*]s + l = 98 (Because the two pieces add up to 98)
[/LIST]
Substitute equation (1) into equation (2) for s:
l - 10+ l = 98
Group like terms:
2l - 10 = 98
Solve for [I]l[/I] in the equation 2l - 10 = 98
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -10 and 98. To do that, we add 10 to both sides
2l - 10 + 10 = 98 + 10
[SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE]
2l = 108
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2l/2 = 108/2
l = [B]54[/B]
To solve for s, we substitute l = 54 into equation (1):
s = 54 - 10
s = [B]44[/B]
Check our work:
The shorter piece is 10 inches shorter than the longer piece since 54 - 44 = 10
Second check: Do both pieces add up to 98
54 + 44 ? 98
98 = 98

A barn contains cows, ducks, and a 3-legged dog named Tripod. There are twice as many cows as ducks

A barn contains cows, ducks, and a 3-legged dog named Tripod. There are twice as many cows as ducks in the barn and a total of 313 legs. How many ducks are there in the barn?
[LIST]
[*]Let the number of ducks be d. Duck legs = 2 * d = 2d
[*]Number of cows = 2d. Cow legs = 4 * 2d = 8d
[*]1 dog Tripod has 3 legs
[/LIST]
Total legs:
2d + 8d + 3 = 313
Solve for [I]d[/I] in the equation 2d + 8d + 3 = 313
[SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE]
(2 + 8)d = 10d
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
10d + 3 = + 313
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 3 and 313. To do that, we subtract 3 from both sides
10d + 3 - 3 = 313 - 3
[SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE]
10d = 310
[SIZE=5][B]Step 5: Divide each side of the equation by 10[/B][/SIZE]
10d/10 = 310/10
d = [B]31[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=2d%2B8d%2B3%3D313&pl=Solve']Source[/URL]

a baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many m

a baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many more tickets would they need to sell to earn at least $441?
Let the number of tickets above 42 be t.
A few things to note on this question:
[LIST]
[*]The phrase [I]at least[/I] means greater than or equal to, so we have an inequality.
[*]Earnings = Price * Quantity
[/LIST]
We're given:
Earnings = 4.50 * 42 + 4.5t >= 441
Earnings = 189 + 4.5t >= 441
We want to solve this inequality for t:
Solve for [I]t[/I] in the inequality 189 + 4.5t ? 441
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 189 and 441. To do that, we subtract 189 from both sides
4.5t + 189 - 189 ? 441 - 189
[SIZE=5][B]Step 2: Cancel 189 on the left side:[/B][/SIZE]
4.5t ? 252
[SIZE=5][B]Step 3: Divide each side of the inequality by 4.5[/B][/SIZE]
4.5t/4.5 ? 252.4.5
[B]t ? 56[/B]

A boat traveled at a constant speed for 32 hours, covering a total distance of 597 kilometers. To th

A boat traveled at a constant speed for 32 hours, covering a total distance of 597 kilometers. To the nearest hundredth of a kilometer per hour, how fast was it going?
Distance = Rate * Time
We're given t = 32, and d = 597. Using our [URL='https://www.mathcelebrity.com/drt.php?d=+597&r=+&t=32&pl=Calculate+the+missing+Item+from+D%3DRT']distance, rate, and time calculator[/URL], we get:
r = [B]18.656 km/hr[/B]

A cell phone provider is offering an unlimited data plan for $70 per month or a 5 GB plan for $55 pe

A cell phone provider is offering an unlimited data plan for $70 per month or a 5 GB plan for $55 per month. However, if you go over your 5 GB of data in a month, you have to pay an extra $10 for each GB. How many GB would be used to make both plans cost the same?
Let g be the number of GB.
The limited plan has a cost as follows:
C = 10(g - 5) + 55
C = 10g - 50 + 55
C = 10g + 5
We want to set the limited plan equal to the unlimited plan and solve for g:
10g + 5 = 70
Solve for [I]g[/I] in the equation 10g + 5 = 70
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 5 and 70. To do that, we subtract 5 from both sides
10g + 5 - 5 = 70 - 5
[SIZE=5][B]Step 2: Cancel 5 on the left side:[/B][/SIZE]
10g = 65
[SIZE=5][B]Step 3: Divide each side of the equation by 10[/B][/SIZE]
10g/10 = 65/10
g = [B]6.5[/B]
Check our work for g = 6.5:
10(6.5) + 5
65 + 5
70

A group of scientists studied the effect of a chemical on various strains of bacteria. Strain A star

A group of scientists studied the effect of a chemical on various strains of bacteria. Strain A started with 6000 cells and decreased at a constant rate of 2000 cells per hour after the chemical was applied. Strain B started with 2000 cells and decreased at a constant rate of 1000 cells per hour after the chemical was applied. When will the strains have the same number of cells? Explain.
Set up strain equations where h is the number of hours since time 0:
[LIST]
[*]Strain A: 6000 - 2000h
[*]Strain B: 2000 - 1000h
[/LIST]
Set them equal to each other
6000 - 2000h = 2000 - 1000h
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=6000-2000h%3D2000-1000h&pl=Solve']equation solver[/URL], we see that [B]h = 4[/B]

A man purchased 20 tickets for a total of $225. The tickets cost $15 for adults and $10 for children

A man purchased 20 tickets for a total of $225. The tickets cost $15 for adults and $10 for children. What was the cost of each ticket?
Declare variables:
[LIST]
[*]Let a be the number of adult's tickets
[*]Let c be the number of children's tickets
[/LIST]
Cost = Price * Quantity
We're given two equations:
[LIST=1]
[*]a + c = 20
[*]15a + 10c = 225
[/LIST]
Rearrange equation (1) in terms of a:
[LIST=1]
[*]a = 20 - c
[*]15a + 10c = 225
[/LIST]
Now that I have equation (1) in terms of a, we can substitute into equation (2) for a:
15(20 - c) + 10c = 225
Solve for [I]c[/I] in the equation 15(20 - c) + 10c = 225
We first need to simplify the expression removing parentheses
Simplify 15(20 - c): Distribute the 15 to each term in (20-c)
15 * 20 = (15 * 20) = 300
15 * -c = (15 * -1)c = -15c
Our Total expanded term is 300-15c
Our updated term to work with is 300 - 15c + 10c = 225
We first need to simplify the expression removing parentheses
Our updated term to work with is 300 - 15c + 10c = 225
[SIZE=5][B]Step 1: Group the c terms on the left hand side:[/B][/SIZE]
(-15 + 10)c = -5c
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
-5c + 300 = + 225
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 300 and 225. To do that, we subtract 300 from both sides
-5c + 300 - 300 = 225 - 300
[SIZE=5][B]Step 4: Cancel 300 on the left side:[/B][/SIZE]
-5c = -75
[SIZE=5][B]Step 5: Divide each side of the equation by -5[/B][/SIZE]
-5c/-5 = -75/-5
c = [B]15[/B]
Recall from equation (1) that a = 20 - c. So we substitute c = 15 into this equation to solve for a:
a = 20 - 15
a = [B]5[/B]

A man's age (a) 10 years ago is 43.

A man's age (a) 10 years ago is 43.
Years ago means we subtract
[B]a - 10 = 43
[/B]
If the problem asks you to solve for a, we type this equation into our math engine and we get:
Solve for [I]a[/I] in the equation a - 10 = 43
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -10 and 43. To do that, we add 10 to both sides
a - 10 + 10 = 43 + 10
[SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE]
a = [B]53[/B]

A school conducts 27 tests in 36 weeks. Assume the school conducts tests at a constant rate. What is

A school conducts 27 tests in 36 weeks. Assume the school conducts tests at a constant rate. What is the slope of the line that represents the number of tests on the y-axis and the time in weeks on the x-axis?
Slope is y/x,so we have 27/36.
[URL='https://www.mathcelebrity.com/fraction.php?frac1=27%2F36&frac2=3%2F8&pl=Simplify']Using our fraction simplifier[/URL], we can reduce 27/36 to 3/4. So this is our slope.
[B]3/4[/B]

A school conducts 27 tests in 36 weeks. Assume the school conducts tests at a constant rate. What is

A school conducts 27 tests in 36 weeks. Assume the school conducts tests at a constant rate. What is the slope of the line that represents the number of tests on the y-axis and the time in weeks on the x-axis?
Slope = Rise/Run or y/x
Since tests are on the y-axis and time is on the x-axis, we have:
Slope = 27/36
We can simplify this, so we [URL='https://www.mathcelebrity.com/fraction.php?frac1=27%2F36&frac2=3%2F8&pl=Simplify']type in 27/36 into our search engine[/URL], and get:
[B]Slope = 3/4[/B]

A tow truck charges a service fee of $50 and an additional fee of $1.75 per mile. What distance was

A tow truck charges a service fee of $50 and an additional fee of $1.75 per mile. What distance was Marcos car towed if he received a bill for $71
Set up a cost equation C(m) where m is the number of miles:
C(m) = Cost per mile * m + Service Fee
Plugging in the service fee of 50 and cost per mile of 1.75, we get:
C(m) = 1.75m + 50
The question asks for what m is C(m) = 71. So we set C(m) = 71 and solve for m:
1.75m + 50 = 71
Solve for [I]m[/I] in the equation 1.75m + 50 = 71
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 50 and 71. To do that, we subtract 50 from both sides
1.75m + 50 - 50 = 71 - 50
[SIZE=5][B]Step 2: Cancel 50 on the left side:[/B][/SIZE]
1.75m = 21
[SIZE=5][B]Step 3: Divide each side of the equation by 1.75[/B][/SIZE]
1.75m/1.75 = 21/1.75
m = [B]12[/B]

A turtle and rabbit are in a race to see who is the first to reach a point 100 feet away. The turtle

A turtle and rabbit are in a race to see who is the first to reach a point 100 feet away. The turtle travels at a constant speed of 20 feet per minute for the entire 100 feet. The rabbit travels at a constant speed of 40 feet per minute for the first 50 feet, stops for 3 minutes, and then continuous at a constant speed of 40 feet per minute for the last 50 feet.
(i) Determine which animal won the race.
(ii). By how much time the animal won the race.
(iii) Explain one life lesson from the race.
We know the distance formula is:
d = rt
For the turtle, he has a rate (r) of 20 feet / minute and distance (d) of 100. We want to solve for time:
[URL='https://www.mathcelebrity.com/drt.php?d=+100&r=+20&t=&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 5
The rabbit has 3 parts of the race:
Rabbit Part 1: Distance (d) = 50 and rate (r) = 40
[URL='https://www.mathcelebrity.com/drt.php?d=50&r=40&t=+&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 1.25
Rabbit Part 2: The rabbit stops for 3 minutes (t = 3)
Rabbit Part 3: Distance (d) = 50 and rate (r) = 40
[URL='https://www.mathcelebrity.com/drt.php?d=50&r=40&t=+&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 1.25
Total time for the rabbit from the 3 parts is (t) = 1.25 + 3 + 1.25
Total time for the rabbit from the 3 parts is (t) = 5.5
[LIST]
[*](i) The [B]turtle won[/B] the race because he took more time to finish and they both started at the same time
[*](ii) We subtract the turtles time from the rabbit's time: 5.5 - 5 = [B]0.5 minutes which is also 30 seconds[/B]
[*](iii) [B]Slow and Steady wins the race[/B]
[/LIST]

A varies directly as B and inversely as C.

A varies directly as B and inversely as C.
There exists a constant k such that:
[B]a = kb/c
[/B]
Inversely means we divide by and directly means we multiply by

a varies directly with b and inversely with c

a varies directly with b and inversely with c
Direct variation means we multiply.
Inverse variation means we divide.
There exists a constant k such that:
[B]a = kb/c[/B]

a varies inversely with b, c and d

a varies inversely with b, c and d
Varies inversely means we divide. Given a constant, k, we have:
[B]a = k/bcd[/B]

A vehicle purchased for $25,000 depreciates at a constant rate of 5%. Determine the approximate valu

A vehicle purchased for $25,000 depreciates at a constant rate of 5%. Determine the approximate value of the vehicle 11 years after purchase. Round to the nearest whole dollar.
Depreciation at 5% means it retains 95% of the value. Set up the depreciation equation to get Book Value B(t) at time t.
B(t) = $25,000 * (1 - 0.05)^t
Simplifying, this is:
B(t) = $25,000 * (0.95)^t
The problem asks for B(11)
B(11) = $25,000 * (0.95)^11
B(11) = $25,000 * 0.5688
B(11) = [B]$14,220[/B]

Algebra Master (Polynomials)

Given 2 polynomials this does the following:

1) Polynomial Addition

2) Polynomial Subtraction

Also generates binomial theorem expansions and polynomial expansions with or without an outside constant multiplier.

1) Polynomial Addition

2) Polynomial Subtraction

Also generates binomial theorem expansions and polynomial expansions with or without an outside constant multiplier.

Approximations of Interest Rate

Interest Rate Approximations: Approximates a yield rate of interest based on 4 methods:

1) Max Yield denoted as i_{max}

2) Min Yield denoted as i_{min}

3) Constant Ratio denoted as i_{cr}

4) Direct Ratio denoted as i_{dr}

1) Max Yield denoted as i

2) Min Yield denoted as i

3) Constant Ratio denoted as i

4) Direct Ratio denoted as i

b varies directly as the sum of x and y

b varies directly as the sum of x and y
This is a direct variation problem.
Direct variation means there exists a constant k such that:
[B]b = k(x + y)[/B]

B varies jointly as u, v, and w

B varies jointly as u, v, and w
Varies jointly means we have a constant k such that:
[B]B = kuvw[/B]

Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will

Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will it take for them to have the same amount of money?
Let w be the number of weeks that go by for saving/spending.
Set up Barney's balance equation, B(w). Spending means we [U]subtract[/U]
B(w) = Initial Amount - spend per week * w weeks
B(w) = 450 - 3w
Set up Betty's balance equation, B(w). Saving means we [U]add[/U]
B(w) = Initial Amount + savings per week * w weeks
B(w) = 120 + 8w
The same amount of money means both of their balance equations B(w) are equal. So we set Barney's balance equal to Betty's balance and solve for w:
450 - 3w = 120 + 8w
Add 3w to each side to isolate w:
450 - 3w + 3w = 120 + 8w + 3w
Cancelling the 3w on the left side, we get:
450 = 120 + 11w
Rewrite to have constant on the right side:
11w + 120 = 450
Subtract 120 from each side:
11w + 120 - 120 = 450 - 120
Cancelling the 120's on the left side, we get:
11w = 330
To solve for w, we divide each side by 11
11w/11 = 330/11
Cancelling the 11's on the left side, we get:
w = [B]30
[MEDIA=youtube]ifG_q-utgJI[/MEDIA][/B]

Basic m x n Matrix Operations

Given 2 matrices |A| and |B|, this performs the following basic matrix operations

* Matrix Addition |A| + |B|

* Matrix Subtraction |A| - |B|

* Matrix Multiplication |A| x |B|

* Scalar multiplication rA where r is a constant.

* Matrix Addition |A| + |B|

* Matrix Subtraction |A| - |B|

* Matrix Multiplication |A| x |B|

* Scalar multiplication rA where r is a constant.

Bawi solves a problem that has an answer of x = -4. He first added 7 to both sides of the equal sign

Bawi solves a problem that has an answer of x = -4. He first added 7 to both sides of the equal sign, then divided by 3. What was the original equation
[LIST=1]
[*]If we added 7 to both sides, that means we had a minus 7 (-7) to start with as a constant. Since subtraction undoes addition.
[*]If we divided by 3, this means we multiplied x by 3 to begin with. Since division undoes multiplication
[/LIST]
So we have the start equation:
3x - 7
If the answer was x = -4, then we plug this in to get our number on the right side of the equation:
3(-4) - 7
-12 - 7
-19
This means our original equation was:
[B]3x - 7 = -19[/B]
And if we want to solve this to prove our answer, we [URL='https://www.mathcelebrity.com/1unk.php?num=3x-7%3D-19&pl=Solve']type the equation into our search engine [/URL]and we get:
x = -4

C varies directly as d use k as the constant of variation

C varies directly as d use k as the constant of variation
Direct variation means we multiply below:
[B]C = kd[/B]

C varies directly as the cube of a and inversely as the 4th power of B

C varies directly as the cube of a and inversely as the 4th power of B
The cube of a means we raise a to the 3rd power:
a^3
The 4th power of B means we raise b to the 4th power:
b^4
Varies directly means there exists a constant k such that:
C = ka^3
Also, varies inversely means we divide by the 4th power of B
C = [B]ka^3/b^4[/B]
Varies [I]directly [/I]as means we multiply by the constant k.
Varies [I]inversely [/I]means we divide k by the term which has inverse variation.
[MEDIA=youtube]fSsG1OB3qdk[/MEDIA]

c varies jointly as the square of q and cube of p

c varies jointly as the square of q and cube of p
The square of q means we raise q to the 2nd power:
q^2
The cube of p means we raise p to the rdd power:
p^3
The phrase [I]varies jointly[/I] means there exists a constant k such that:
[B]c = kp^3q^2[/B]

Caleb earns points on his credit card that he can use towards future purchases.

Let f = dollars spent on flights, h dollars spent on hotels, and p dollars spent on all other purchases.
[U]Set up our equations:[/U]
(1) 4f + 2h + p = 14660
(2) f + h + p = 9480
(3) f = 2h + 140
[U]First, substitute (3) into (2)[/U]
(2h + 140) + h + p = 9480
3h + p + 140 = 9480
3h + p = 9340
[U]Subtract 3h to isolate p to form equation (4)[/U]
(4) p = 9340 - 3h
[U]Take (3) and (4), and substitute into (1)[/U]
4(2h + 140) + 2h + (9340 - h) = 14660
[U]Multiply through[/U]
8h + 560 + 2h + 9340 - 3h = 14660
[U]Combine h terms and constants[/U]
(8 + 2 - 3)h + (560 + 9340) = 14660
7h + 9900 = 14660
[U]Subtract 9900 from both sides:[/U]
7h = 4760
[U]Divide each side by 7[/U]
[B]h = 680[/B]
[U]Substitute h = 680 into equation (3)[/U]
f = 2(680) + 140
f = 1360 + 140
[B]f = 1,500[/B]
[U]
Substitute h = 680 and f = 1500 into equation (2)[/U]
1500 + 680 + p = 9480
p + 2180 = 9480
[U]Subtract 2180 from each side:[/U]
[B]p = 7,300[/B]

Casey is 26 years old. Her daughter Chloe is 4 years old. In how many years will Casey be double her

Casey is 26 years old. Her daughter Chloe is 4 years old. In how many years will Casey be double her daughter's age
Declare variables for each age:
[LIST]
[*]Let Casey's age be c
[*]Let her daughter's age be d
[*]Let n be the number of years from now where Casey will be double her daughter's age
[/LIST]
We're told that:
26 + n = 2(4 + n)
26 + n = 8 + 2n
Solve for [I]n[/I] in the equation 26 + n = 8 + 2n
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables n and 2n. To do that, we subtract 2n from both sides
n + 26 - 2n = 2n + 8 - 2n
[SIZE=5][B]Step 2: Cancel 2n on the right side:[/B][/SIZE]
-n + 26 = 8
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 26 and 8. To do that, we subtract 26 from both sides
-n + 26 - 26 = 8 - 26
[SIZE=5][B]Step 4: Cancel 26 on the left side:[/B][/SIZE]
-n = -18
[SIZE=5][B]Step 5: Divide each side of the equation by -1[/B][/SIZE]
-1n/-1 = -18/-1
n = [B]18[/B]
Check our work for n = 18:
26 + 18 ? 8 + 2(18)
44 ? 8 + 36
44 = 44

Charles Law

This will solve for any of the 4 items in Charles Law assuming constant pressure

V1 ÷ T1 = V2 ÷ T2

V1 ÷ T1 = V2 ÷ T2

David roller skates for 3 1/3 hours with a constant speed of 24 km/h and then for another 1 hour 10

David roller skates for 3 1/3 hours with a constant speed of 24 km/h and then for another 1 hour 10 minutes with constant speed of 12 km/h. What distance did he go?
Distance = Rate x Time
[U]Part 1 of his trip:[/U]
D1 = R1 x T1
D1 = 3 & 1/3 hours * 24 km/h
D1 = 80 km
[U]Part 2 of his trip:[/U]
D2 = R2 x T2
D2 = 1 & 1/6 hours * 12 km/h (Note, 10 minutes = 1/6 of an hour)
D2 = 14 km
[U]Calculate Total Distance (D)[/U]
D = D1 + D2
D = 80 + 14
D = [B]94 km[/B]

Dennis was getting in shape for a marathon. The first day of the week he ran n miles. Dennis then ad

Dennis was getting in shape for a marathon. The first day of the week he ran n miles. Dennis then added a mile to his run each day. By the end of the week (7 days), he had run a total of 70 miles. How many miles did Dennis run the first day?
Setup distance ran for the 7 days:
[LIST=1]
[*]n
[*]n + 1
[*]n + 2
[*]n + 3
[*]n + 4
[*]n + 5
[*]n + 6
[/LIST]
Add them all up:
7n + 21 = 70
Solve for [I]n[/I] in the equation 7n + 21 = 70
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 21 and 70. To do that, we subtract 21 from both sides
7n + 21 - 21 = 70 - 21
[SIZE=5][B]Step 2: Cancel 21 on the left side:[/B][/SIZE]
7n = 49
[SIZE=5][B]Step 3: Divide each side of the equation by 7[/B][/SIZE]
7n/7 = 49/7
n =[B] 7
[URL='https://www.mathcelebrity.com/1unk.php?num=7n%2B21%3D70&pl=Solve']Source[/URL][/B]

Deon opened his account starting with $650 and he is going to take out $40 per month. Mai opened up

Deon opened his account starting with $650 and he is going to take out $40 per month. Mai opened up her account with a starting amount of $850 and is going to take out $65 per month. When would the two accounts have the same amount of money?
We set up a balance equation B(m) where m is the number of months.
[U]Set up Deon's Balance equation:[/U]
Withdrawals mean we subtract from our current balance
B(m) = Starting Balance - Withdrawal Amount * m
B(m) = 650 - 40m
[U]Set up Mai's Balance equation:[/U]
Withdrawals mean we subtract from our current balance
B(m) = Starting Balance - Withdrawal Amount * m
B(m) = 850 - 65m
When the two accounts have the same amount of money, we can set both balance equations equal to each other and solve for m:
650 - 40m = 850 - 65m
Solve for [I]m[/I] in the equation 650 - 40m = 850 - 65m
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables -40m and -65m. To do that, we add 65m to both sides
-40m + 650 + 65m = -65m + 850 + 65m
[SIZE=5][B]Step 2: Cancel -65m on the right side:[/B][/SIZE]
25m + 650 = 850
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 650 and 850. To do that, we subtract 650 from both sides
25m + 650 - 650 = 850 - 650
[SIZE=5][B]Step 4: Cancel 650 on the left side:[/B][/SIZE]
25m = 200
[SIZE=5][B]Step 5: Divide each side of the equation by 25[/B][/SIZE]
25m/25 = 200/25
m = [B]8[/B]

Derivatives

This lesson walks you through the derivative definition, rules, and examples including the power rule, derivative of a constant, chain rule

Determine whether the statement is true or false. If y = e^2, then y’ = 2e

Determine whether the statement is true or false. If y = e^2, then y’ = 2e
e^2 is a constant, and the derivative of a constant is 0. So y' = 0
So this is [B]FALSE[/B]

Dunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offers

Dunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offers business cards for $0.15 each with a setup charge of $10. What numbers of business cards cost the same from either company
Declare variables:
[LIST]
[*]Let b be the number of business cards.
[/LIST]
[U]Set up the cost function C(b) for Dunder Mifflin:[/U]
C(b) = Cost to print each business card * b + Setup Charge
C(b) = 0.1b + 15
[U]Set up the cost function C(b) for Werham Hogg:[/U]
C(b) = Cost to print each business card * b + Setup Charge
C(b) = 0.15b + 10
The phrase [I]cost the same[/I] means we set both C(b)'s equal to each other and solve for b:
0.1b + 15 = 0.15b + 10
Solve for [I]b[/I] in the equation 0.1b + 15 = 0.15b + 10
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 0.1b and 0.15b. To do that, we subtract 0.15b from both sides
0.1b + 15 - 0.15b = 0.15b + 10 - 0.15b
[SIZE=5][B]Step 2: Cancel 0.15b on the right side:[/B][/SIZE]
-0.05b + 15 = 10
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 15 and 10. To do that, we subtract 15 from both sides
-0.05b + 15 - 15 = 10 - 15
[SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE]
-0.05b = -5
[SIZE=5][B]Step 5: Divide each side of the equation by -0.05[/B][/SIZE]
-0.05b/-0.05 = -5/-0.05
b = [B]100[/B]

Equation and Inequalities

Solves an equation or inequality with 1 unknown variable and no exponents as well as certain absolute value equations and inequalities such as |x|=c and |ax| = c where a and c are constants. Solves square root, cube root, and other root equations in the form ax^2=c, ax^2 + b = c. Also solves radical equations in the form asqrt(bx) = c. Also solves open sentences and it will solve one step problems and two step equations. 2 step equations and one step equations and multi step equations

Erin has 72 stamps in her stamp drawer along with a quarter, two dimes and seven pennies. She has 3

Erin has 72 stamps in her stamp drawer along with a quarter, two dimes and seven pennies. She has 3 times as many 3-cent stamps as 37-cent stamps and half the number of 5-cent stamps as 37-cent stamps. The value of the stamps and coins is $8.28. How many 37-cent stamps does Erin have?
Number of stamps:
[LIST]
[*]Number of 37 cent stamps = s
[*]Number of 3-cent stamps = 3s
[*]Number of 5-cent stamps = 0.5s
[/LIST]
Value of stamps and coins:
[LIST]
[*]37 cent stamps = 0.37s
[*]3-cent stamps = 3 * 0.03 = 0.09s
[*]5-cent stamps = 0.5 * 0.05s = 0.025s
[*]Quarter, 2 dime, 7 pennies = 0.52
[/LIST]
Add them up:
0.37s + 0.09s + 0.025s + 0.52 = 8.28
Solve for [I]s[/I] in the equation 0.37s + 0.09s + 0.025s + 0.52 = 8.28
[SIZE=5][B]Step 1: Group the s terms on the left hand side:[/B][/SIZE]
(0.37 + 0.09 + 0.025)s = 0.485s
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
0.485s + 0.52 = + 8.28
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 0.52 and 8.28. To do that, we subtract 0.52 from both sides
0.485s + 0.52 - 0.52 = 8.28 - 0.52
[SIZE=5][B]Step 4: Cancel 0.52 on the left side:[/B][/SIZE]
0.485s = 7.76
[SIZE=5][B]Step 5: Divide each side of the equation by 0.485[/B][/SIZE]
0.485s/0.485 = 7.76/0.485
s = [B]16[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.37s%2B0.09s%2B0.025s%2B0.52%3D8.28&pl=Solve']Source[/URL]

f - g = 1/4b for b

f - g = 1/4b for b
Multiply each side of the equation by 4 to remove the 1/4 and isolate b:
4(f - g) = 4/4b
4/4 = 1, so we have:
b = [B]4(f - g)[/B]
[I]the key to this problem was multiplying by the reciprocal of the constant[/I]

F varies directly as g and inversely as r^2

F varies directly as g and inversely as r^2
[U]Givens and assumptions[/U]
[LIST]
[*]We take a constant of variation called k.
[*][I]Varies directly means we multiply our variable term by k[/I]
[*][I]Varies inversely means we divide k by our variable term[/I]
[/LIST]
The phrase varies directly or varies inversely means we have a constant k such that:
[B]F = kg/r^2[/B]

f varies jointly with u and h and inversely with the square of y.

f varies jointly with u and h and inversely with the square of y.
Variation means we have a constant k.
Varies jointly with u and h means we multiply k by hu
Varies inversely with the square of y means we divide by y^2
[B]f = khu/y^2[/B]

for the function, h(x) = bx - 22, b is a constant and h(-5) = -7. Find the value of h(5)

for the function, h(x) = bx - 22, b is a constant and h(-5) = -7. Find the value of h(5)
h(-5) = -5b - 22
Since we're given h(-5) = -7, we have:
-5b - 22 = -7
[URL='https://www.mathcelebrity.com/1unk.php?num=-5b-22%3D-7&pl=Solve']Typing this equation into our search engine[/URL], we get:
b = -3
So our h(x) equation is now:
h(x) = -3x - 22
The problem asks for h(5):
h(5) = -3(5) - 22
h(5) = 15 - 22
h(5) = [B]-37[/B]

Frequency and Wavelength and Photon Energy

Provides the following 3 items using the speed of light and Plancks constant (h):

- Given a frequency of centimeters, feet, meters, or miles the calculator will determine wavelength in Hz, KHz, MHz, GHz

- Given a wavelength of Hz, KHz, MHz, GHz, the calculator will determine frequency in centimeters, feet, meters, or miles

- Calculates photon energy

- Given a frequency of centimeters, feet, meters, or miles the calculator will determine wavelength in Hz, KHz, MHz, GHz

- Given a wavelength of Hz, KHz, MHz, GHz, the calculator will determine frequency in centimeters, feet, meters, or miles

- Calculates photon energy

Function

Takes various functions (exponential, logarithmic, signum (sign), polynomial, linear with constant of proportionality, constant, absolute value), and classifies them, builds ordered pairs, and finds the y-intercept and x-intercept and domain and range if they exist.

Gamma Constant γ

This calculator generates 5000 iterations for the development of the gamma constant γ

Given that E[Y]=2 and Var [Y] =3, find E[(2Y + 1)^2]

Given that E[Y]=2 and Var [Y] =3, find E[(2Y + 1)^2]
Multiply through
E[(2Y + 1)^2] = E[4y^2 + 4y + 1]
We can take the expected value of each term
E[4y^2] + E[4y] + E[1]
For the first term, we have:
4E[Y^2]
We define the Var[Y] = E[Y^2] - (E[Y])^2
Rearrange this term, we have E[Y^2] = Var[Y] + (E[Y])^2
E[Y^2] = 3+ 2^2
E[Y^2] = 3+ 4
E[Y^2] = 7
So our first term is 4(7) = 28
For the second term using expected value rules of separating out a constant, we have
4E[Y] = 4(2) = 8
For the third term, we have:
E[1] = 1
Adding up our three terms, we have:
E[4y^2] + E[4y] + E[1] = 28 + 8 + 1
E[4y^2] + E[4y] + E[1] = [B]37[/B]

Given y= 4/3x what is the constant of proportionality

Given y= 4/3x what is the constant of proportionality
Direct variation means the constant of proportionality is y/x.
Cross multiplying, we get:
y/x = [B]4/3[/B]

Given: 9 - 4x = -19 Prove: x = 7

Given: 9 - 4x = -19 Prove: x = 7
Solve for [I]x[/I] in the equation 9 - 4x = - 19
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 9 and -19. To do that, we subtract 9 from both sides
-4x + 9 - 9 = -19 - 9
[SIZE=5][B]Step 2: Cancel 9 on the left side:[/B][/SIZE]
-4x = -28
[SIZE=5][B]Step 3: Divide each side of the equation by -4[/B][/SIZE]
-4x/-4 = -28/-4
x = [B]7[/B]

Gravitational Force

Using Sir Isaac Newtons Law of Gravitational Force, this calculator determines the force between two objects with mass in kilograms at a distance apart in meters using the constant of gravity.

If 2 is added to the numerator and denominator it becomes 9/10 and if 3 is subtracted from the numer

If 2 is added to the numerator and denominator it becomes 9/10 and if 3 is subtracted from the numerator and denominator it become 4/5. Find the fractions.
Convert 2 to a fraction with a denominator of 10:
20/2 = 10, so we multiply 2 by 10/10:
2*10/10 = 20/10
Add 2 to the numerator and denominator:
(n + 2)/(d + 2) = 9/10
Cross multiply and simplify:
10(n + 2) = 9(d + 2)
10n + 20 = 9d + 18
Move constants to right side by subtracting 20 from each side and subtracting 9d:
10n - 9d = -2
Subtract 3 from the numerator and denominator:
(n - 3)/(d - 3) = 4/5
Cross multiply and simplify:
5(n - 3) = 4(d - 3)
5n - 15 = 4d - 12
Move constants to right side by adding 15 to each side and subtracting 4d:
5n - 4d = 3
Build our system of equations:
[LIST=1]
[*]10n - 9d = -2
[*]5n - 4d = 3
[/LIST]
Multiply equation (2) by -2:
[LIST=1]
[*]10n - 9d = -2
[*]-10n + 8d = -6
[/LIST]
Now add equation (1) to equation (2)
(10 -10)n (-9 + 8)d = -2 - 6
The n's cancel, so we have:
-d = -8
Multiply through by -1:
d = 8
Now bring back our first equation from before, and plug in d = 8 into it to solve for n:
10n - 9d = -2
10n - 9(8) = -2
10n - 72 = -2
To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=10n-72%3D-2&pl=Solve']plug this equation into our search engine[/URL] and we get:
n = 7
So our fraction, n/d = [B]7/8[/B]

If EF = 9x - 17, FG = 17x - 14, and EG = 20x + 17, what is FG?

If EF = 9x - 17, FG = 17x - 14, and EG = 20x + 17, what is FG?
By segment addition, we know that:
EF + FG = EG
Substituting in our values for the 3 segments, we get:
9x - 17 + 17x - 14 = 20x + 17
Group like terms and simplify:
(9 + 17)x + (-17 - 14) = 20x - 17
26x - 31 = 20x - 17
Solve for [I]x[/I] in the equation 26x - 31 = 20x - 17
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 26x and 20x. To do that, we subtract 20x from both sides
26x - 31 - 20x = 20x - 17 - 20x
[SIZE=5][B]Step 2: Cancel 20x on the right side:[/B][/SIZE]
6x - 31 = -17
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -31 and -17. To do that, we add 31 to both sides
6x - 31 + 31 = -17 + 31
[SIZE=5][B]Step 4: Cancel 31 on the left side:[/B][/SIZE]
6x = 14
[SIZE=5][B]Step 5: Divide each side of the equation by 6[/B][/SIZE]
6x/6 = 14/6
x = [B]2.3333333333333[/B]

If p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equ

If p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equal to 2.
We set up the variation equation with a constant k such that:
p = k/q^2 [I](inversely proportional means we divide)
[/I]
When q is 4 and p is 2, we have:
2 = k/4^2
2 = k/16
Cross multiply:
k = 2 * 16
k = 32
Now, the problem asks for p when q = 2:
p = 32/2^2
p = 32/4
p = [B]8[/B]

If two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two n

If two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two numbers?
Let the smaller number be n.
The next consecutive even number is n + 2.
Add them together to equal 226:
n + n + 2 = 226
Solve for [I]n[/I] in the equation n + n + 2 = 226
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 1)n = 2n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2n + 2 = + 226
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 2 and 226. To do that, we subtract 2 from both sides
2n + 2 - 2 = 226 - 2
[SIZE=5][B]Step 4: Cancel 2 on the left side:[/B][/SIZE]
2n = 224
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2n/2 = 224/2
n = [B]112
[URL='https://www.mathcelebrity.com/1unk.php?num=n%2Bn%2B2%3D226&pl=Solve']Source[/URL][/B]

If x varies directly with y and x = -3 when y = 12, what is the constant of variation?

If x varies directly with y and x = -3 when y = 12, what is the constant of variation?
Using our [URL='https://www.mathcelebrity.com/community/forums/calculator-requests.7/create-thread']variation calculator[/URL], we see the constant of variation (k) is:
k =[B] -1/4 or -0.25[/B]

If y varies directly as x and inversely as z, then which equation describes the relation?

If y varies directly as x and inversely as z, then which equation describes the relation?
Directly means we multiply, inversely means we divide, so we have a constant k such that:
[B]y = kx/z[/B]

In the year 1980, Rick was twice as old as Nancy who was twice as old as Michael. In the year 1992 R

In the year 1980, Rick was twice as old as Nancy who was twice as old as Michael. In the year 1992 Ric, Nancy, and Michael ages added up to 78 years. How old was Ric in 1980?
Age in 1980:
[LIST]
[*]Let Michael's age be m
[*]Nancy's age is 2m
[*]Rick's age is 2 * 2m = 4m
[/LIST]
Age in 1992:
[LIST]
[*]Michael's age = m + 12
[*]Nancy's age is 2m + 12
[*]Rick's age is 2 * 2m = 4m + 12
[/LIST]
Total them up:
m + 12 + 2m + 12 + 4m + 12 = 78
Solve for [I]m[/I] in the equation m + 12 + 2m + 12 + 4m + 12 = 78
[SIZE=5][B]Step 1: Group the m terms on the left hand side:[/B][/SIZE]
(1 + 2 + 4)m = 7m
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
12 + 12 + 12 = 36
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
7m + 36 = + 78
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants 36 and 78. To do that, we subtract 36 from both sides
7m + 36 - 36 = 78 - 36
[SIZE=5][B]Step 5: Cancel 36 on the left side:[/B][/SIZE]
7m = 42
[SIZE=5][B]Step 6: Divide each side of the equation by 7[/B][/SIZE]
7m/7 = 42/7
m = 6
Rick's age = 6 * 4 = [B]24
[URL='https://www.mathcelebrity.com/1unk.php?num=m%2B12%2B2m%2B12%2B4m%2B12%3D78&pl=Solve']Source[/URL]
[/B]

Interpolation

Given a set of data, this interpolates using the following methods:

* Linear Interpolation

* Nearest Neighbor (Piecewise Constant)

* Polynomial Interpolation

* Linear Interpolation

* Nearest Neighbor (Piecewise Constant)

* Polynomial Interpolation

is 6x a monomial?

[B]Yes[/B]. It's an algebraic expression consisting of one term.
The constant is 6, and the variable is x.

jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an ine

jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an inequality to show how many sodas he can buy.
Let s be the number of sodas.
Cost for the day is:
Price per soda * s + Admission Price
4.25s + 42
We're told that Jane has 55, which means Jane cannot spend more than 55. Jane can spend up to or less than 55. We write this as an inequality using <= 55
[B]4.25s + 42 <= 55[/B]
[B][/B]
If the problems asks you to solve for s, we type it in our math engine and we get:
Solve for [I]s[/I] in the inequality 4.25s + 42 ? 55
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 42 and 55. To do that, we subtract 42 from both sides
4.25s + 42 - 42 ? 55 - 42
[SIZE=5][B]Step 2: Cancel 42 on the left side:[/B][/SIZE]
4.25s ? 13
[SIZE=5][B]Step 3: Divide each side of the inequality by 4.25[/B][/SIZE]
4.25s/4.25 ? 13/4.25
[B]s ? 3.06[/B]

Johnny Rocket can run 300 meters in 90 seconds. If his speed remains constant, how far could he ru

Johnny Rocket can run 300 meters in 90 seconds. If his speed remains constant, how far could he run in 500 seconds? Round to one decimal place.
Set up the distance equation:
Distance = Rate * Time
300 = 90r
Solving this equation for r, we [URL='https://www.mathcelebrity.com/1unk.php?num=300%3D90r&pl=Solve']type it in our search engine[/URL] and we get:
r = 3.333
For 500 seconds, we set up our distance equation again:
Distance = 500 * 3.333333
Distance = [B]1666.7 meters[/B]

K varies inversely with square root of m and directly with the cube of n.

K varies inversely with square root of m and directly with the cube of n.
[LIST]
[*]We take a constant c as our constant of proportionality.
[*]The word inversely means we divide
[*]The word directly means we multiply
[/LIST]
[B]k = cn^3/sqrt(m)[/B]

k varies jointly with m,n, p

k varies jointly with m,n, p
The phrase [I]varies jointly[/I] means we have a constant c such that:
[B]k= cmnp[/B]

larger of 2 numbers is 12 more than the smaller number. if the sum of the 2 numbers is 74 find the 2

larger of 2 numbers is 12 more than the smaller number. if the sum of the 2 numbers is 74 find the 2 numbers
Declare Variables for each number:
[LIST]
[*]Let l be the larger number
[*]Let s be the smaller number
[/LIST]
We're given two equations:
[LIST=1]
[*]l = s + 12
[*]l + s = 74
[/LIST]
Equation (1) already has l solved for. Substitute equation (1) into equation (2) for l:
s + 12 + s = 74
Solve for [I]s[/I] in the equation s + 12 + s = 74
[SIZE=5][B]Step 1: Group the s terms on the left hand side:[/B][/SIZE]
(1 + 1)s = 2s
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2s + 12 = + 74
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 12 and 74. To do that, we subtract 12 from both sides
2s + 12 - 12 = 74 - 12
[SIZE=5][B]Step 4: Cancel 12 on the left side:[/B][/SIZE]
2s = 62
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2s/2 = 62/2
s = [B]31[/B]
To solve for l, we substitute in s = 31 into equation (1):
l = 31 + 12
l = [B]43[/B]

Laura weighs 45 pounds more than her pet dog. When they are on the scale together, they weigh 85 pou

Laura weighs 45 pounds more than her pet dog. When they are on the scale together, they weigh 85 pounds. How much does Laura weigh?
Let Laura weigh l and her dog weigh d. WE have:
[LIST=1]
[*]l = d + 45
[*]d + l = 85
[/LIST]
Substitute equation (1) into Equation (2) for l:
d + d + 45 = 85
Solve for [I]d[/I] in the equation d + d + 45 = 85
[SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE]
(1 + 1)d = 2d
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2d + 45 = + 85
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 45 and 85. To do that, we subtract 45 from both sides
2d + 45 - 45 = 85 - 45
[SIZE=5][B]Step 4: Cancel 45 on the left side:[/B][/SIZE]
2d = 40
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2d/2 = 40/2
d = 20
From equation (1), we substitute d = 20:
l = d + 45
l = 20 + 45
l = [B]65 pounds
[URL='https://www.mathcelebrity.com/1unk.php?num=d%2Bd%2B45%3D85&pl=Solve']Source[/URL][/B]

Lebron James scored 288 points in 9 games this season. Assuming he continues to score at this consta

Lebron James scored 288 points in 9 games this season. Assuming he continues to score at this constant rate, write a linear equation that represents the scenario.
288 points / 9 games = 32 points per game
Let g be the number of games Lebron plays. We build an equation for his season score:
Lebron's Season Score = Points per game * number of games
Lebron's Season Score = [B]32g[/B]

Leilani can read 20 pages in 2 minutes. if she can maintain this page, how many pages can she read i

Leilani can read 20 pages in 2 minutes. if she can maintain this page, how many pages can she read in an hour?
We know that 1 hour is 60 minutes.
Let p be the number of pages Leilani can read in 1 hour (60 minutes)
The read rate is constant, so we can build a proportion.
20 pages /2 minutes = p/60
We can cross multiply:
Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2
[SIZE=5][B]Solving for Numerator 2 we get:[/B][/SIZE]
Numerator 2 = Numerator 1 * Denominator 2/Denominator 1
[SIZE=5][B]Evaluating and simplifying using your input values we get:[/B][/SIZE]
p = 20 * 60/ 2
p = 1200/2
p = [B]600[/B]

Logarithms and Natural Logarithms and Eulers Constant (e)

This calculator does the following:

* Takes the Natural Log base e of a number x Ln(x) → log_{e}x

* Raises e to a power of y, e^{y}

* Performs the change of base rule on log_{b}(x)

* Solves equations in the form b^{cx} = d where b, c, and d are constants and x is any variable a-z

* Solves equations in the form ce^{dx}=b where b, c, and d are constants, e is Eulers Constant = 2.71828182846, and x is any variable a-z

* Exponential form to logarithmic form for expressions such as 5^{3} = 125 to logarithmic form

* Logarithmic form to exponential form for expressions such as Log_{5}125 = 3

* Takes the Natural Log base e of a number x Ln(x) → log

* Raises e to a power of y, e

* Performs the change of base rule on log

* Solves equations in the form b

* Solves equations in the form ce

* Exponential form to logarithmic form for expressions such as 5

* Logarithmic form to exponential form for expressions such as Log

m is inversely proportional to the square of p-1 when p=4 m=5 find m when p=6

m is inversely proportional to the square of p-1 when p=4 and m=5. find m when p=6
Inversely proportional means there is a constant k such that:
m = k/(p - 1)^2
When p = 4 and m = 5, we have:
5 = k/(4 - 1)^2
5 = k/3^2
5 = k/9
[U]Cross multiply:[/U]
k = 45
[U]The problems asks for m when p = 6. And we also now know that k = 45. So plug in the numbers:[/U]
m = k/(p - 1)^2
m = 45/(6 - 1)^2
m = 45/5^2
m = 45/25
m = [B]1.8[/B]

Mathematical Constants and Identities

Calculates and explains various mathematical constants such as:

* Gelfonds (Gelfond's) Constant

* Eulers Constant

* Gelfonds (Gelfond's) Constant

* Eulers Constant

Matthew's cat weighs 10 pounds more than his pet hamster. His dog weighs the same as his cat. If the

Matthew's cat weighs 10 pounds more than his pet hamster. His dog weighs the same as his cat. If the weight of all three pets is 35 pounds, ow much does his hamster weigh?
Setup weights and relations:
[LIST]
[*]Hamster weight: w
[*]Cat weight: w + 10
[*]Dog weight:w + 10
[/LIST]
Add all the weights up:
w + w + 10 + w + 10 = 35
Solve for [I]w[/I] in the equation w + w + 10 + w + 10 = 35
[SIZE=5][B]Step 1: Group the w terms on the left hand side:[/B][/SIZE]
(1 + 1 + 1)w = 3w
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
10 + 10 = 20
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
3w + 20 = + 35
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants 20 and 35. To do that, we subtract 20 from both sides
3w + 20 - 20 = 35 - 20
[SIZE=5][B]Step 5: Cancel 20 on the left side:[/B][/SIZE]
3w = 15
[SIZE=5][B]Step 6: Divide each side of the equation by 3[/B][/SIZE]
3w/3 = 15/3
w =[B] 5[/B]
[B]
[URL='https://www.mathcelebrity.com/1unk.php?num=w%2Bw%2B10%2Bw%2B10%3D35&pl=Solve']Source[/URL][/B]

n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6

n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6
Solve for [I]n[/I] in the equation n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 2 + 3 + 4)n = 10n
[SIZE=5][B]Step 2: Group the constant terms on the right hand side:[/B][/SIZE]
2 + 3 + 4 + 5 + 6 = 20
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
10n = + 20
[SIZE=5][B]Step 4: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = 20/10
n = [B]2[/B]

n + 9n - 8 - 5 = 2n + 3

n + 9n - 8 - 5 = 2n + 3
Solve for [I]n[/I] in the equation n + 9n - 8 - 5 = 2n + 3
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 9)n = 10n
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
-8 - 5 = -13
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
10n - 13 = 2n + 3
[SIZE=5][B]Step 4: Group variables:[/B][/SIZE]
We need to group our variables 10n and 2n. To do that, we subtract 2n from both sides
10n - 13 - 2n = 2n + 3 - 2n
[SIZE=5][B]Step 5: Cancel 2n on the right side:[/B][/SIZE]
8n - 13 = 3
[SIZE=5][B]Step 6: Group constants:[/B][/SIZE]
We need to group our constants -13 and 3. To do that, we add 13 to both sides
8n - 13 + 13 = 3 + 13
[SIZE=5][B]Step 7: Cancel 13 on the left side:[/B][/SIZE]
8n = 16
[SIZE=5][B]Step 8: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 16/8
n = [B]2[/B]

n + 9n - 90 = 0

n + 9n - 90 = 0
Solve for [I]n[/I] in the equation n + 9n - 90 = 0
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 9)n = 10n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
10n - 90 =
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -90 and 0. To do that, we add 90 to both sides
10n - 90 + 90 = 0 + 90
[SIZE=5][B]Step 4: Cancel 90 on the left side:[/B][/SIZE]
10n = 90
[SIZE=5][B]Step 5: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = 90/10
n = [B]9[/B]

Nancy is 10 years less than 3 times her daughters age. If Nancy is 41 years old, how old is her daug

Nancy is 10 years less than 3 times her daughters age. If Nancy is 41 years old, how old is her daughter?
Declare variables for each age:
[LIST]
[*]Let Nancy's age be n
[*]Let her daughter's age be d
[/LIST]
We're given two equations:
[LIST=1]
[*]n = 3d - 10
[*]n = 41
[/LIST]
We set 3d - 10 = 41 and solve for d:
Solve for [I]d[/I] in the equation 3d - 10 = 41
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -10 and 41. To do that, we add 10 to both sides
3d - 10 + 10 = 41 + 10
[SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE]
3d = 51
[SIZE=5][B]Step 3: Divide each side of the equation by 3[/B][/SIZE]
3d/3 = 51/3
d = [B]17[/B]

P varies directly as q and the square of r and inversely as s

P varies directly as q and the square of r and inversely as s
There exists a constant k such that:
p = kqr^2/s
[I]Note: Direct variations multiply and inverse variations divide[/I]

p varies directly as the square of r and inversely as q and s and p = 40 when q = 5, r = 4 and s = 6

p varies directly as the square of r and inversely as q and s and p = 40 when q = 5, r = 4 and s = 6, what is the equation of variation?
Two rules of variation:
[LIST=1]
[*]Varies directly means we multiply
[*]Varies inversely means we divide
[/LIST]
There exists a constant k such that our initial equation of variation is:
p = kr^2/qs
[B][/B]
With p = 40 when q = 5, r = 4 and s = 6, we have:
4^2k / 5 * 6 = 40
16k/30 = 40
Cross multiply:
16k = 40 * 30
16k = 1200
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=16k%3D1200&pl=Solve']equation calculator[/URL], we get:
k = [B]75[/B]
So our final equation of variation is:
[B]p = 75r^2/qs[/B]

Perimeter of a rectangle is 372 yards. If the length is 99 yards, what is the width?

Perimeter of a rectangle is 372 yards. If the length is 99 yards, what is the width?
The perimeter P of a rectangle with length l and width w is:
2l + 2w = P
We're given P = 372 and l = 99, so we have:
2(99) + 2w = 372
2w + 198 = 372
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 198 and 372. To do that, we subtract 198 from both sides
2w + 198 - 198 = 372 - 198
[SIZE=5][B]Step 2: Cancel 198 on the left side:[/B][/SIZE]
2w = 174
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2w/2 = 174/2
w = [B]87[/B]

Polynomial

This calculator will take an expression without division signs and combine like terms.

It will also analyze an polynomial that you enter to identify constant, variables, and exponents. It determines the degree as well.

It will also analyze an polynomial that you enter to identify constant, variables, and exponents. It determines the degree as well.

Pressure Law

This will solve for any of the 4 items in the Pressure Law equation, also known as Gay-Lussacs Law assuming constant volume

P1 ÷ T1 = P2 ÷ T2

P1 ÷ T1 = P2 ÷ T2

r varies directly with s and inversely with the square root of t

r varies directly with s and inversely with the square root of t
Varies directly means we multiply
Varies inversely means we divide
There exists a constant k such that:
[B]r = ks/sqrt(t)[/B]

Rachel buys some scarves that cost $10 each and 2 purses that cost $16 each. The cost of Rachel's to

Rachel buys some scarves that cost $10 each and 2 purses that cost $16 each. The cost of Rachel's total purchase is $62. What equation can be used to find n, the number of scarves that Rebecca buys
Scarves Cost + Purses Cost = Total Cost
[U]Calculate Scarves Cost[/U]
Scarves cost = Cost per scarf * number of scarves
Scarves cost = 10n
[U]Calculate Purses Cost[/U]
Purses cost = Cost per purse * number of purses
Purses cost = 16 * 2
Purses cost = 32
Total Cost = 62. Plug in our numbers and values to the Total Cost equation :
10n + 32 = 62
Solve for [I]n[/I] in the equation 10n + 32 = 62
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 32 and 62. To do that, we subtract 32 from both sides
10n + 32 - 32 = 62 - 32
[SIZE=5][B]Step 2: Cancel 32 on the left side:[/B][/SIZE]
10n = 30
[SIZE=5][B]Step 3: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = 30/10
n = [B]3[/B]

S varies jointly with t cubed and v

S varies jointly with t cubed and v
Varied jointly means there exists a constant k such that:
[B]s = kt^3v[/B]

Seth is constantly forgetting the combination to his lock. He has a lock with four dials. (Each ha 1

Seth is constantly forgetting the combination to his lock. He has a lock with four dials. (Each has 10 numbers 0-9). If Seth can try one lock combination per second, how many seconds will it take him to try every possible lock combination?
Start with 0001, 0002, all the way to 9999
[URL='https://www.mathcelebrity.com/inclusnumwp.php?num1=0&num2=9999&pl=Count']When you do this[/URL], you get 10,000 combinations. One per second = 10,000 seconds

Sonia visited a park in California that had redwood trees. When Sonia asked how tall a certain large

Sonia visited a park in California that had redwood trees. When Sonia asked how tall a certain large redwood tree was, the ranger said that he wouldn't tell its height, but would give Sonia a clue. How tall is the redwood tree Sonia asked about?
Sonia said the tree is 64 times my height. The tree is also 112 feet taller than the tree next to it. The two trees plus my height total 597.5 feet.
[LIST]
[*]Rangers's height = n
[*]Tree height = 64n
[*]Smaller tree height = 64n - 112
[*]Total height = 64n - 112 + 64n = 597.5
[/LIST]
Solve for [I]n[/I] in the equation 64n - 112 + 64n = 597.5
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(64 + 64)n = 128n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
128n - 112 = + 597.5
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -112 and 597.5. To do that, we add 112 to both sides
128n - 112 + 112 = 597.5 + 112
[SIZE=5][B]Step 4: Cancel 112 on the left side:[/B][/SIZE]
128n = 709.5
[SIZE=5][B]Step 5: Divide each side of the equation by 128[/B][/SIZE]
128n/128 = 709.5/128
n = 5.54296875
Tree height = 64 * ranger height
Tree height = 64 * 5.54296875
Tree height = [B]354.75 feet[/B]

Stanley bought a ruler and a yardstick for $1.25. If the yardstick cost 45 cents more than the ruler

Stanley bought a ruler and a yardstick for $1.25. If the yardstick cost 45 cents more than the ruler, what was the cost of the yardstick?
Let r be the cost of the ruler
Let y be the cost of the yardstick
We're given 2 equations:
[LIST=1]
[*]r + y = 1.25
[*]y = r + 0.45
[/LIST]
Substitute equation (2) into equation (1) for y
r + r + 0.45 = 1.25
Solve for [I]r[/I] in the equation r + r + 0.45 = 1.25
[SIZE=5][B]Step 1: Group the r terms on the left hand side:[/B][/SIZE]
(1 + 1)r = 2r
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2r + 0.45 = + 1.25
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 0.45 and 1.25. To do that, we subtract 0.45 from both sides
2r + 0.45 - 0.45 = 1.25 - 0.45
[SIZE=5][B]Step 4: Cancel 0.45 on the left side:[/B][/SIZE]
2r = 0.8
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2r/2 = 0.8/2
r = 0.4
Substitute r = 0.4 into equation (2) above:
y = r + 0.45
y = 0.4 + 0.45
r = [B]0.85
[URL='https://www.mathcelebrity.com/1unk.php?num=r%2Br%2B0.45%3D1.25&pl=Solve']Source[/URL][/B]

sum of 3 consecutive odd integers equals 1 hundred 17

sum of 3 consecutive odd integers equals 1 hundred 17
The sum of 3 consecutive odd numbers equals 117. What are the 3 odd numbers?
1) Set up an equation where our [I]odd numbers[/I] are n, n + 2, n + 4
2) We increment by 2 for each number since we have [I]odd numbers[/I].
3) We set this sum of consecutive [I]odd numbers[/I] equal to 117
n + (n + 2) + (n + 4) = 117
[SIZE=5][B]Simplify this equation by grouping variables and constants together:[/B][/SIZE]
(n + n + n) + 2 + 4 = 117
3n + 6 = 117
[SIZE=5][B]Subtract 6 from each side to isolate 3n:[/B][/SIZE]
3n + 6 - 6 = 117 - 6
[SIZE=5][B]Cancel the 6 on the left side and we get:[/B][/SIZE]
3n + [S]6[/S] - [S]6[/S] = 117 - 6
3n = 111
[SIZE=5][B]Divide each side of the equation by 3 to isolate n:[/B][/SIZE]
3n/3 = 111/3
[SIZE=5][B]Cancel the 3 on the left side:[/B][/SIZE]
[S]3[/S]n/[S]3 [/S]= 111/3
n = 37
Call this n1, so we find our other 2 numbers
n2 = n1 + 2
n2 = 37 + 2
n2 = 39
n3 = n2 + 2
n3 = 39 + 2
n3 = 41
[SIZE=5][B]List out the 3 consecutive odd numbers[/B][/SIZE]
([B]37, 39, 41[/B])
37 ? 1st number, or the Smallest, Minimum, Least Value
39 ? 2nd number
41 ? 3rd or the Largest, Maximum, Highest Value

Suppose that h(x) varies directly with x and h(x)=44 when x = 2. What is h(x) when x = 1.5?

Suppose that h(x) varies directly with x and h(x)=44 when x = 2. What is h(x) when x = 1.5?
Direct variation means we set up an equation:
h(x) = kx where k is the constant of variation.
For h(x) = 44 when x = 2, we have:
2k = 44
[URL='https://www.mathcelebrity.com/1unk.php?num=2k%3D44&pl=Solve']Type this equation into our search engine[/URL], we get:
k = 22
The question asks for h(x) when x = 1.5. So we set up our variation equation, knowing that k = 22.
kx = h(x)
With k = 22 and x = 1.5, we get:
22(1.5) = h(x)
h(x) = [B]33[/B]

Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in ga

Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 20 gallons of fuel, the airplane weighs 2012 pounds. When carrying 55 gallons of fuel, it weighs 2208 pounds. How much does the airplane weigh if it is carrying 65 gallons of fuel?
Linear functions are written in the form of one dependent variable and one independent variable. Using g as the number of gallons and W(g) as the weight, we have:
W(g) = gx + c where c is a constant
We are given:
[LIST]
[*]W(20) = 2012
[*]W(55) = 2208
[/LIST]
We want to know W(65)
Using our givens, we have:
W(20) = 20x + c = 2012
W(55) = 55x + c = 2208
Rearranging both equations, we have:
c = 2012 - 20x
c = 2208 - 55x
Set them both equal to each other:
2012 - 20x = 2208 - 55x
Add 55x to each side:
35x + 2012 = 2208
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=35x%2B2012%3D2208&pl=Solve']equation solver[/URL], we see that x is 5.6
Plugging x = 5.6 back into the first equation, we get:
c = 2012 - 20(5.6)
c = 2012 - 112
c = 2900
Now that we have all our pieces, find W(65)
W(65) = 65(5.6) + 2900
W(65) = 264 + 2900
W(65) = [B]3264[/B]

Synthetic Division

Using Ruffinis Rule, this performs synthetic division by dividing a polynomial with a maximum degree of 6 by a term (x ± c) where c is a constant root using the factor theorem. The calculator returns a quotient answer that includes a remainder if applicable. Also known as the Rational Zero Theorem

t varies directly with the square of r and inversely with w

t varies directly with the square of r and inversely with w
There exists a constant k such that:
[B]t = kr^2/w[/B]
[I]Directly means multiply and inversely means divide[/I]

The average cost of printing a book in a publishing company is c(x) = 5.5x+kx , where x is the numbe

The average cost of printing a book in a publishing company is c(x) = 5.5x+kx , where x is the number of books printed that day and k is a constant. Find k, if on the day when 200 were printed the average cost was $9 per book.
We are given: c(200) = 9, so we have:
9 = 5.5(200) + k(200)
200k + 1100 = 9
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=200k%2B1100%3D9&pl=Solve']equation solver[/URL], we get:
[B]k = -5.455[/B]

the fuel tank of a jet used gas at a constant rate of 300 gallons for each hour of flight. the tank

the fuel tank of a jet used gas at a constant rate of 300 gallons for each hour of flight. the tank can hold a maximum of 2400 gallons of gas. write an equation representing the amount of fuel left in the tank as a function of the number of hours spent flying.
We have an equation F(h) where h is the number of hours since the flight took off:
[B]F(h) = 2400 - 300h[/B]

The income i is directly proportional to working hours h

The income i is directly proportional to working hours h
The phrase [I]directly proportional[/I] means there exists a constant k such that:
[B]I = kh[/B]

The phone company charges Rachel 12 cents per minute for her long distance calls. A discount company

The phone company charges Rachel 12 cents per minute for her long distance calls. A discount company called Rachel and offered her long distance service for 1/2 cent per minute, but will charge a $46 monthly fee. How many minutes per month must Rachel talk on the phone to make the discount a better deal?
Minutes Rachel talks = m
Current plan cost = 0.12m
New plan cost = 0.005m + 46
Set new plan equal to current plan:
0.005m + 46 = 0.12m
Solve for [I]m[/I] in the equation 0.005m + 46 = 0.12m
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 0.005m and 0.12m. To do that, we subtract 0.12m from both sides
0.005m + 46 - 0.12m = 0.12m - 0.12m
[SIZE=5][B]Step 2: Cancel 0.12m on the right side:[/B][/SIZE]
-0.115m + 46 = 0
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 46 and 0. To do that, we subtract 46 from both sides
-0.115m + 46 - 46 = 0 - 46
[SIZE=5][B]Step 4: Cancel 46 on the left side:[/B][/SIZE]
-0.115m = -46
[SIZE=5][B]Step 5: Divide each side of the equation by -0.115[/B][/SIZE]
-0.115m/-0.115 = -46/-0.115
m = [B]400
She must talk over 400 minutes for the new plan to be a better deal
[URL='https://www.mathcelebrity.com/1unk.php?num=0.005m%2B46%3D0.12m&pl=Solve']Source[/URL][/B]

The price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she pa

The price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she paid $75. What is the cost of the cheap backpack?
backpack cost = b
Cheap backpack = b - 15
The total of both items equals 75:
b + b - 15 = 75
Solve for [I]b[/I] in the equation b + b - 15 = 75
[SIZE=5][B]Step 1: Group the b terms on the left hand side:[/B][/SIZE]
(1 + 1)b = 2b
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2b - 15 = + 75
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -15 and 75. To do that, we add 15 to both sides
2b - 15 + 15 = 75 + 15
[SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE]
2b = 90
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2b/2 = 90/2
b = 45
Cheap backpack = 45 - 15 = [B]30
[URL='https://www.mathcelebrity.com/1unk.php?num=b%2Bb-15%3D75&pl=Solve']Source[/URL][/B]

The sum of -4x^2 - 5x + 7 and 2x^2 + 8x - 11 can be written in the form ax^2 + bx + c, where a, b, a

The sum of -4x^2 - 5x + 7 and 2x^2 + 8x - 11 can be written in the form ax^2 + bx + c, where a, b, and c are constants. What is the value of a + b + c?
The sum means we add the polynomials together. We do this by adding the like terms:
-4x^2 - 5x + 7 + 2x^2 + 8x - 11
(-4 +2)x^2 + (-5 + 8)x +(7 - 11)
-2x^2 + 3x - 4
We have (a, b, c) = (-2, 3, -4)
The question asks for a + b + c
a + b + c = -2 + 3 - 4
a + b + c = [B]-3[/B]

the sum of 3 consecutive natural numbers, the first of which is n

the sum of 3 consecutive natural numbers, the first of which is n
Natural numbers are counting numbers, so we the following expression:
n + (n + 1) + (n + 2)
Combine n terms and constants:
(n + n + n) + (1 + 2)
[B]3n + 3
Also expressed as 3(n + 1)[/B]

The sum of three consecutive integers is 42

Let the 3 integers be x, y, and z.
y = x + 1
z = y + 1, or x + 2.
Set up our equation:
x + (x + 1) + (x + 2) = 42
Group our variables and constants:
(x + x + x) + (1 + 2) = 42
3x + 3 = 42
Subtract 3 from each side:
3x = 39
Divide each side of the equation by 3:
[B]x = 13
So y = x + 1 = 14
z = x + 2 = 15
(x,y,z) = (13,14,15)[/B]

The sum of twice an integer and 3 times the next consecutive integer is 48

The sum of twice an integer and 3 times the next consecutive integer is 48
Let the first integer be n
This means the next consecutive integer is n + 1
Twice an integer means we multiply n by 2:
2n
3 times the next consecutive integer means we multiply (n + 1) by 3
3(n + 1)
The sum of these is:
2n + 3(n + 1)
The word [I]is[/I] means equal to, so we set 2n + 3(n + 1) equal to 48:
2n + 3(n + 1) = 48
Solve for [I]n[/I] in the equation 2n + 3(n + 1) = 48
We first need to simplify the expression removing parentheses
Simplify 3(n + 1): Distribute the 3 to each term in (n+1)
3 * n = (3 * 1)n = 3n
3 * 1 = (3 * 1) = 3
Our Total expanded term is 3n + 3
Our updated term to work with is 2n + 3n + 3 = 48
We first need to simplify the expression removing parentheses
Our updated term to work with is 2n + 3n + 3 = 48
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(2 + 3)n = 5n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
5n + 3 = + 48
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 3 and 48. To do that, we subtract 3 from both sides
5n + 3 - 3 = 48 - 3
[SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE]
5n = 45
[SIZE=5][B]Step 5: Divide each side of the equation by 5[/B][/SIZE]
5n/5 = 45/5
Cancel the 5's on the left side and we get:
n = [B]9[/B]

The volleyball team and the wrestling team at Clarksville High School are having a joint car wash t

The volleyball team and the wrestling team at Clarksville High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets $4 per car. In addition, they have already brought in $81 from past fundraisers. The wrestling team has raised $85 in the past, and they are making $2 per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?
Set up the earnings equation for the volleyball team where w is the number of cars washed:
E = Price per cars washed * w + past fundraisers
E = 4w + 81
Set up the earnings equation for the wrestling team where w is the number of cars washed:
E = Price per cars washed * w + past fundraisers
E = 2w + 85
If the raised the same amount in total, set both earnings equations equal to each other:
4w + 81 = 2w + 85
Solve for [I]w[/I] in the equation 4w + 81 = 2w + 85
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 4w and 2w. To do that, we subtract 2w from both sides
4w + 81 - 2w = 2w + 85 - 2w
[SIZE=5][B]Step 2: Cancel 2w on the right side:[/B][/SIZE]
2w + 81 = 85
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 81 and 85. To do that, we subtract 81 from both sides
2w + 81 - 81 = 85 - 81
[SIZE=5][B]Step 4: Cancel 81 on the left side:[/B][/SIZE]
2w = 4
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2w/2 = 4/2
w = [B]2 <-- How many cars it will take
[/B]
To get the total earnings, we take either the volleyball or wrestling team's Earnings equation and plug in w = 2:
E = 4(2) + 81
E = 8 + 81
E = [B]89 <-- Total Earnings[/B]

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.
If sum of two numbers is 20, let one number be x. Then the other number would be 20 - x.
The sum of their squares is:
x^2+(20 - x)^2
Expand this and we get:
x^2 + 400 - 40x + x^2
Combine like terms:
2x^2 - 40x + 400
Rewrite this:
2(x^2 - 20x + 100 - 100) + 400
2(x - 10)^2 - 200 + 400
2(x?10)^2 + 200
The sum of squares of two numbers is sum of two positive numbers, one of which is a constant of 200.
The other number, 2(x - 10)^2, can change according to the value of x. The least value could be 0, when x=10
Therefore, the minimum value of sum of squares of two numbers is 0 + 200 = 200 when x = 10.
If x = 10, then the other number is 20 - 10 = 10.

u varies jointly as q and the square of m

u varies jointly as q and the square of m
Varies jointly means we multiply. There exists a constant k such that:
[B]u = kqm^2[/B]

Use k as the constant of variation. L varies jointly as u and the square root of v.

Use k as the constant of variation. L varies jointly as u and the square root of v.
Since u and v vary jointly, we multiply by the constant of variation k:
[B]l = ku * sqrt(v)[/B]

Wendy is paid $7.50 per hour plus a bonus of $80 each week. Last week Wendy earned $312.50. How many

Wendy is paid $7.50 per hour plus a bonus of $80 each week. Last week Wendy earned $312.50. How many hours did Wendy work last week?
Setup the earnings equation with h hours:
7.5h + 80 = 312.50
Solve for [I]h[/I] in the equation 7.5h + 80 = 312.50
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 80 and 312.50. To do that, we subtract 80 from both sides
7.5h + 80 - 80 = 312.50 - 80
[SIZE=5][B]Step 2: Cancel 80 on the left side:[/B][/SIZE]
7.5h = 232.5
[SIZE=5][B]Step 3: Divide each side of the equation by 7.5[/B][/SIZE]
7.5h/7.5 = 232.5/7.5
h = [B]31
[URL='https://www.mathcelebrity.com/1unk.php?num=7.5h%2B80%3D312.50&pl=Solve']Source[/URL][/B]

What is the annual nominal rate compounded daily for a bond that has an annual yield of 5.4%? Round

What is the annual nominal rate compounded daily for a bond that has an annual yield of 5.4%? Round to three decimal places. Use a 365 day year.
[U]Set up the accumulation equation:[/U]
(1+i)^365 = 1.054
[U]Take the natural log of each side[/U]
365 * Ln(1 + i) = 1.054
Ln(1 + i) = 0.000144089
[U]Use each side as a exponent to eulers constant e[/U]
(1 + i) = e^0.000144089
1 + i = 1.000144099
i = 0.000144099 or [B].0144099%[/B]

When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the num

When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
4 times a number means we multiply x by 4:
4x
Increased by 40 means we add 40 to 4x:
4x + 40
100 decreased by the number means we subtract x from 100:
100 - x
The phrase [I]is the same as[/I] means equal to, so we set 4x + 40 equal to 100 - x
4x + 40 = 100 - x
Solve for [I]x[/I] in the equation 4x + 40 = 100 - x
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 4x and -x. To do that, we add x to both sides
4x + 40 + x = -x + 100 + x
[SIZE=5][B]Step 2: Cancel -x on the right side:[/B][/SIZE]
5x + 40 = 100
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 40 and 100. To do that, we subtract 40 from both sides
5x + 40 - 40 = 100 - 40
[SIZE=5][B]Step 4: Cancel 40 on the left side:[/B][/SIZE]
5x = 60
[SIZE=5][B]Step 5: Divide each side of the equation by 5[/B][/SIZE]
5x/5 = 60/5
x = [B]12[/B]
Check our work for x = 12:
4(12) + 40 ? 100 - 12
48 + 40 ? 100 - 12
88 = 88

Which of the following can increase power?

Which of the following can increase power?
a. Increasing standard deviation
b. Decreasing standard deviation
c. Increasing both means but keeping the difference between the means constant
d. Increasing both means but making the difference between the means smaller
[B]b. Decreasing standard deviation[/B]
[LIST=1]
[*]Power increases if the standard deviation is smaller.
[*]If the difference between the means is bigger, the power is bigger.
[*]Sample size increase also increases power
[/LIST]

while scuba diving jerey rose directly toward the surface of the water at a constant velocity for 2.

while scuba diving jerey rose directly toward the surface of the water at a constant velocity for 2.0 minutes. he rose 9.0 meters in that time. what was his velocity?
9 meters / 2 minutes = [B]4.5 meters / minute[/B]

Write an equation that relates the quantities. G varies jointly with t and q and inversely with the

Write an equation that relates the quantities. G varies jointly with t and q and inversely with the cube of w . The constant of proportionality is 8.25 .
[LIST]
[*]Varies jointly or directly means we multiply
[*]Varies inversely means divide
[*]The cube of w means we raise w to the 3rd power: w^3
[/LIST]
Using k = 8.25 as our constant of proportionality, we have:
[B]g = 8.25qt/w^3[/B]

X varies directly with the cube root of y when x=1 y=27. Calculate y when x=4

X varies directly with the cube root of y when x=1 y=27. Calculate y when x=4
Varies directly means there is a constant k such that:
x = ky^(1/3)
When x = 1 and y = 27, we have:
27^1/3(k) = 1
3k = 1
To solve for k, we[URL='https://www.mathcelebrity.com/1unk.php?num=3k%3D1&pl=Solve'] type in our equation into our search engine[/URL] and we get:
k = 1/3
Now, the problem asks for y when x = 4. We use our variation equation above with k = 1/3 and x = 4:
4 = y^(1/3)/3
Cross multiply:
y^(1/3) = 4 * 3
y^(1/3) =12
Cube each side:
y^(1/3)^3 = 12^3
y = [B]1728[/B]

y varies directly as the reciprocal of x

y varies directly as the reciprocal of x
The reciprocal of x is written as:
1/x
The phrase [I]varies directly[/I] means there exists a constant k such that
[B]y = k/x[/B]

y varies directly as x and inversely as i

y varies directly as x and inversely as I
Note:
Direct variation means we multiply. Inverse variation means we divide.
There exists a constant k such that:
[B]y = kx/i[/B]

You prepare 18 scoops of dog food for 6 dogs, and prepare 24 scoops of dog food for 8 dogs. What is

You prepare 18 scoops of dog food for 6 dogs, and prepare 24 scoops of dog food for 8 dogs. What is the constant of proportionality for the amount of dog food to the number of dogs? How many scoops of dog food should you prepare for 9 dogs?
18/6 = 24/8 = 3 as the constant of proportionality for the amount of dog food to the number of dogs.
What this means is for every dog, we give them 3 scoops of food.
So for 9 dogs, we give 9 dogs * 3 scoops of food per dog = 27 scoops

z is directly proportional to the square of x and y

z is directly proportional to the square of x and y
Directly proportional means there exists a constant k such that:
z = [B]kx^2y
[MEDIA=youtube]J3ByZkcX38E[/MEDIA][/B]

z is jointly proportional to the square of x and the cube of y

z is jointly proportional to the square of x and the cube of y
The square of x means we raise x to the power of 2:
x^2
The cube of y means we raise y to the power of 3:
y^3
The phrase [I]jointly proportional[/I] means we have a constant k such that:
[B]z = kx^2y^3[/B]

z varies directly with x and inversely with y

z varies directly with x and inversely with y
[LIST]
[*]The phrase directly means we multiply.
[*]The phrase inversely means we divide
[*]Variation means there exists a constant k such that:
[/LIST]
[B]z = kx/y[/B]

z varies inversely as the square of t. if z=4 when t=2, find z when t is 10

z varies inversely as the square of t. if z=4 when t=2, find z when t is 10
Varies inversely means there exists a constant k such that:
z = k/t^2
If z = 4 when t = 2, we have:
4 = k/2^2
4 = k/4
Cross multiply and we get:
k = 4 * 4
k = 16
Now the problem asks to find z when t is 10:
z = k/t^2
z = 16/10^2
z = 16/100
z = [B]0.16[/B]

z varies inversely with w, x, and y

z varies inversely with w, x, and y
Inversely means their exists a constant k such that:
[B]z = k/wxy[/B]

Z varies jointly as the 4th power of x and the 5th power of y

Z varies jointly as the 4th power of x and the 5th power of y
The 4th power of x means we raise x to the power of 4:
x^4
The 5th power of y means we raise y to the power of 5:
y^5
The phrase [I]varies jointly[/I] means we have a constant k such that:
z = [B]kx^4y^5[/B]

z varies jointly as x and y. If z=3 when x=3 and y=15, find z when x=6 and y=9

z varies jointly as x and y. If z=3 when x=3 and y=15, find z when x=6 and y=9
Varies jointly means there exists a constant k such that:
z = kxy
We're given z = 3 when x = 3 and y = 15, so we have:
3 = 15 * 3 * k
3 = 45k
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=3%3D45k&pl=Solve']equation solver,[/URL] we see that:
k = 1/15
So our joint variation equation is:
z = xy/15
Then we're asked to find z when x = 6 and y = 9
z = 6 * 9 / 15
z = 54/15
[URL='https://www.mathcelebrity.com/search.php?q=54%2F15&x=0&y=0']z =[/URL] [B]18/5[/B]