5 times a number increased by 4 is divided by 6 times the same number
Take this algebraic expression in parts.
Part 1: 5 times a number increased by 4
[LIST]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x: x
[*]5 times the number means multiply x by 5: 5x
[*][I]Increased by 4[/I] means we add 4 to 5x: 5x + 4
[/LIST]
Part 2: 6 times the same number
[LIST]
[*]From above, [I]a number[/I] is x: x
[*]6 times the number means we multiply x by 6: 6x
[/LIST]
The phrase [I]is divided by[/I] means we have a quotient, where 5x + 4 is the numerator, and 6x is the denominator.
[B](5x + 4)/6x[/B]
A fraction has a value of 3/4. If 7 is added to the numerator, the resulting fraction is equal to the reciprocal of the original fraction. Find the original fraction.
Let the fraction be x/y. We're given two equations:
[LIST=1]
[*]x/y = 3/4
[*](x + 7)/y = 4/3. [I](The reciprocal of 3/4 is found by 1/(3/4)[/I]
[/LIST]
Cross multiply equation 1 and equation 2:
[LIST=1]
[*]4x = 3y
[*]3(x + 7) = 4y
[/LIST]
Simplifying, we get:
[LIST=1]
[*]4x = 3y
[*]3x + 21 = 4y
[/LIST]
If we divide equation 1 by 4, we get:
[LIST=1]
[*]x = 3y/4
[*]3x + 21 = 4y
[/LIST]
Substitute equation (1) into equation (2) for x:
3(3y/4) + 21 = 4y
9y/4 + 21 = 4y
Multiply the equation by 4 on both sides to eliminate the denominator:
9y + 84 = 16y
To solve this equation for y, we type it in our math engine and we get:
y = [B]12
[/B]
We then substitute y = 12 into equation 1 above:
x = 3 * 12/4
x = 36/4
x = [B]9
[/B]
So our original fraction x/y = [B]9/12[/B]
A lottery offers 1 $1000 prize and 5 $100 prizes. 1000 tickets are sold. Find the expectation if a person buys 1 ticket for $5.
Set up the expected values E(x):
for the 1,000 price:
E(x) = (1000 - 5) * 1/1000 = 995/1000
For the 5 $100 prizes:
E(x) = (100 - 5) * 5/1000 = 475/1000
For the losing ticket. With 6 winning tickets, we have 1000 - 6 = 994 losing tickets:
E(x) = -3 * 994/1000 = -2982/1000
We get our total expected value by adding all of these expected values up. Since they all have the same denominator, we add numerators:
E(x) = (995 + 475 - 2982)/1000
E(x) = -1512/1000
E(x) = [B]-1.51[/B]
a recipe of 20 bread rolls requires 5 tablespoons of butter. How many tablespoons of butter are needed for 30 bread rolls?
Set up a proportion of bread rolls per tablespoons of butter where t is the amount of tablespoons of butter needed for 30 bread rolls:
20/5 = 30/t
Cross multiply our proportion:
Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2
20t = 30 * 5
20t = 150
Divide each side of the equation by 20:
20t/20 = 150/20
Cancel the 20's on the left side and we get:
t = [B]7.5[/B]
a stone mason builds 7 houses in 3 days. How many days does it take to build 11 houses?
The build rate of houses per days is proportional. Set up a proportion of [I]houses to days[/I] where d is the number of days it takes to build 11 houses:
7/3 = 11/d
Cross multiply:
Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2
7d = 11 * 3
7d = 33
Divide each side of the equation by 7:
7d/7 = 33/7
d = [B]4.7142857142857[/B]
at a party, there are 72 people. The ratio of men to ladies to kids is 4 to 3 to 2.
[LIST]
[*]How many men at the party?
[*]How many ladies at the party?
[*]How many kids at the party?
[/LIST]
Our total ratio denominator is 4 + 3 + 2 = 9. To find the number of each type of person, we take their ratio divided by their ratio numerator times 72 people at the party
[U]Calculate ratios:[/U]
[LIST]
[*]Men: [URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F9&frac2=72&pl=Multiply']4/9 * 72[/URL] = [B]32[/B]
[*]Ladies: [URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F9&frac2=72&pl=Multiply']3/9 * 72[/URL] = [B]24[/B]
[*]Kids: [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F9&frac2=72&pl=Multiply']2/9 * 72[/URL] = [B]16[/B]
[/LIST]
[U]Check our work:[/U]
Men + Ladies + Kids = 32 + 24 + 16
Men + Ladies + Kids = 72 <-- This checks out!
Cards in a pack are either orange or purple. 80% of the cards are orange. Write the ratio of orange cards to purple cards.
[URL='https://www.mathcelebrity.com/perc.php?num=+5&den=+8&num1=+16&pct1=+80&pct2=+90&den1=+80&pct=80&pcheck=4&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']80% as a fraction [/URL]is 4/5.
Fractions to ratios can be written as numerator : denominator, so we have:
[B]4:5[/B]
difference between 2 positive numbers is 3 and the sum of their squares is 117
Declare variables for each of the two numbers:
[LIST]
[*]Let the first variable be x
[*]Let the second variable be y
[/LIST]
We're given 2 equations:
[LIST=1]
[*]x - y = 3
[*]x^2 + y^2 = 117
[/LIST]
Rewrite equation (1) in terms of x by adding y to each side:
[LIST=1]
[*]x = y + 3
[*]x^2 + y^2 = 117
[/LIST]
Substitute equation (1) into equation (2) for x:
(y + 3)^2 + y^2 = 117
Evaluate and simplify:
y^2 + 3y + 3y + 9 + y^2 = 117
Combine like terms:
2y^2 + 6y + 9 = 117
Subtract 117 from each side:
2y^2 + 6y + 9 - 117 = 117 - 117
2y^2 + 6y - 108 = 0
This is a quadratic equation:
Solve the quadratic equation 2y2+6y-108 = 0
With the standard form of ax2 + bx + c, we have our a, b, and c values:
a = 2, b = 6, c = -108
Solve the quadratic equation 2y^2 + 6y - 108 = 0
The quadratic formula is denoted below:
y = -b ± sqrt(b^2 - 4ac)/2a
[U]Step 1 - calculate negative b:[/U]
-b = -(6)
-b = -6
[U]Step 2 - calculate the discriminant ?:[/U]
? = b2 - 4ac:
? = 62 - 4 x 2 x -108
? = 36 - -864
? = 900 <--- Discriminant
Since ? is greater than zero, we can expect two real and unequal roots.
[U]Step 3 - take the square root of the discriminant ?:[/U]
?? = ?(900)
?? = 30
[U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U]
Numerator 1 = -b + ??
Numerator 1 = -6 + 30
Numerator 1 = 24
[U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U]
Numerator 2 = -b - ??
Numerator 2 = -6 - 30
Numerator 2 = -36
[U]Step 6 - calculate your denominator which is 2a:[/U]
Denominator = 2 * a
Denominator = 2 * 2
Denominator = 4
[U]Step 7 - you have everything you need to solve. Find solutions:[/U]
Solution 1 = Numerator 1/Denominator
Solution 1 = 24/4
Solution 1 = 6
Solution 2 = Numerator 2/Denominator
Solution 2 = -36/4
Solution 2 = -9
[U]As a solution set, our answers would be:[/U]
(Solution 1, Solution 2) = (6, -9)
Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution
[U]The difference of z and y means we subtract y from z[/U]
z - y
[U]Now, we form a fraction, where 10 is the numerator and z - y is the denominator[/U]
10/(z - y)
double v, add u, then divide t by what you have
Double v means we multiply the variable v by 2:
2v
Add u:
2v + u
We build a fraction, with t as the numerator, and 2v + u as the denominator
[B]t/(2v + u)[/B]
Explain the steps you would take to find an equation for the line perpendicular to 4x - 5y = 20 and sharing the same y-intercept
Get this in slope-intercept form by adding 5y to each side:
4x - 5y + 5y = 5y + 20
Cancel the 5y's on the left side and we get:
5y + 20 = 4x
Subtract 20 from each side
5y + 20 - 20 = 4x - 20
Cancel the 20's on the left side and we get:
5y = 4x - 20
Divide each side by 5:
5y/5 = 4x/5 - 4
y = 4x/5 - 4
So we have a slope of 4/5
to find our y-intercept, we set x = 0:
y = 4(0)/5 - 4
y = 0 - 4
y = -4
If we want a line perpendicular to the line above, our slope will be the negative reciprocal:
The reciprocal of 4/5 is found by flipping the fraction making the numerator the denominator and the denominator the numerator:
m = 5/4
Next, we multiply this by -1:
-5/4
So our slope-intercept of the perpendicular line with the same y-intercept is:
[B]y = -5x/4 - 4[/B]
Calculates the following factorial items:
* A factorial of one number such as n! * A factorial of a numerator divided by a factorial of a denominator such as n!m!/a!b!
* Double Factorials such as n!!
* Stirlings Approximation for n!
Fraction with variable x in numerator and 6 in the denominator.
The numerator is the top of the fraction.
The denominator is the bottom of the fraction.
[B]x/6[/B]
Harry got 42 out of 49 correct in his test. What fraction of the marks did he get correct?
The fraction correct is:
42/49
Both the numerator and denominator [URL='http://www.mathcelebrity.com/gcflcm.php?num1=42&num2=49&num3=&pl=GCF']have a common factor[/URL] of 7
Reducing top and bottom by 7, we get:
[B]6/7[/B]
I only own blue blankets and red blankets. 8 out of every 15 blankets I have are red. If have i 45 blankets, how many are blue?
If 8 out of 15 blankets are red, then 15 - 8 = 7 are blue
So 7 out of every 15 blankets are blue.
Set up a proportion of blue blankets to total blankets where b is the number of blue blankets in 45 blankets
7/15 = b/45
Cross multiply:
If 2 proportions are equal, then we can do the following:
Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2
15b = 45 * 7
15b = 315
To solve for b, divide each side of the equation by 15:
15b/15 = 315/15
Cancel the 15's on the left side and we get:
b = [B]21[/B]
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]
In Trina's desk drawer, there are 15 paper clips and 18 rubber bands. In Kirk's office supply tray, there are 13 paper clips and 16 rubber bands. Who has a higher ratio of paper clips to rubber bands?
Trina: 15/18
Kirk: 13/16
We want common denominators to compare, so we get a greatest common factor (GCF) for 16 and 18.
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=16&num2=18&num3=&pl=GCF+and+LCM']Running this through our search engine[/URL], we get GCF(16, 18) = 144
For Trina, 144/18 = 8
For Kirk, 144/16 = 9
We multiply Trina's fraction, top and bottom by 8:
15 * 8 / 18 * 8
120/144
We multiply Trina's fraction, top and bottom by 8:
13 * 8 / 16 * 8
104/144
[B]Trina[/B] has more in her numerator, so her ratio of paper clips to rubber bands is greater.
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]
numerator of a fraction is 5 less than its denominator. if 1 is added to the numerator and to the denominator the new fraction is 2/3. find the fraction.
Let n be the numerator.
Let d be the denominator.
We're given 2 equations:
[LIST=1]
[*]n = d - 5
[*](n + 1)/(d + 1) = 2/3
[/LIST]
Substitute equation (1) into equation (2) for n:
(d - 5 + 1) / (d + 1) = 2/3
(d - 4) / (d + 1) = 2/3
Cross multiply:
3(d - 4) = 2(d + 1)
To solve this equation for d, we type it in our search engine and we get:
d = 14
Substitute d = 14 into equation (1) to solve for n:
n = 14 - 5
n = 9
Therefore, our fraction n/d is:
[B]9/14[/B]
One positive number is one-fifth of another number. The difference between the two numbers is 192, find the numbers.
Let the first number be x and the second number be y. We're given two equations:
[LIST=1]
[*]x = y/5
[*]x + y = 192
[/LIST]
Substitute equation 1 into equation 2:
y/5 + y = 192
Since 1 equals 5/5, we rewrite our equation like this:
y/5 = 5y/5 = 192
We have fractions with like denominators, so we add the numerators:
(1 + 5)y/5 = 192
6y/5 = 192
[URL='https://www.mathcelebrity.com/prop.php?num1=6y&num2=192&den1=5&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']Type this equation into our search engine[/URL], and we get:
[B]y = 160[/B]
Substitute this value into equation 1:
x = 160/5
x = [B]32[/B]
Calculates the relational items between a fraction, a decimal (including repeating decimal and terminating decimal), a percentage, and the numerator and denominator piece of that fraction. Also calculates the percentage change going from one number to another or the amount increase or decrease of a percentage above/below a number. Round decimals. decimals into fractions
raise t to the 10th power, then find the quotient of the result and s
Raise t to the 10th power means we use t as our variable and 10 as our exponent:
t^10
The quotient means a fraction, where the numerator is t^10 and the denominator is s:
[B]t^10/s[/B]
Suppose that Sn = 3 + 1/3 + 1/9 + ... + 1/3(n-2)
a) Find S10 and S?
b) If the common difference in an arithmetic sequence is twice the first term, show that Sn/Sm = n^2/m^2
a) Sum of the geometric sequence is
a = 3 and r = 1/3
(a(1 - r)^n)/(1 - r)
(3(1 - 1/3)^9)/(1 - 1/3)
[B]S10 = 4.499771376[/B]
For infinity, as n goes to infinity, the numerator goes to 1
so we have [B]S? = 3(1)/2/3 = 4.5[/B]
b) Sum of an arithmetic sequence formula is below:
n(a1 + an)/2
an = a1 + (n - 1)2a1 since d = 2a1
n(a1 + a1 + (n - 1)2a1)/2
(2a1n + n^2 - 2a1n)/2
n^2/2
For Sm
m(a1 + am)/2
am = a1 + (m - 1)2a1 since d = 2a1
m(a1 + 1 + (m - 1)2a1)/2
(2a1m + m^2 - 2a1m)/2
m^2/2
Sn/Sm = n^2/m^2 (cancel the 2's)
S10/S1 = 10^2/1^2
We know S1 = 3
So we have 100(3)/1
[B]S10 = 300[/B]
Let x equal "a number".
Double the number is 2x.
The average is (x + 2x)/2
Combine the terms in the numerator:
3x/2
The word [I]is[/I] means equal to, so we set 3x/2 equal to 25.5
3x/2 = 25.5
Cross multiply the 2:
3x = 51
Divide each side by 3
[B]x = 17[/B]
The denominator of a fraction is 4 more than the numerator. If 4 is added to the numerator and 7 is added to the denominator, the value of the fraction is 1/2. Find the original fraction.
Let the original fraction be n/d.
We're given:
[LIST=1]
[*]d = n + 4
[*](n + 4) / (d + 7) = 1/2
[/LIST]
Cross multiply Equation 2:
2(n + 4) = d + 7
2n + 8 = d + 7
Now substitute equation (1) into tihs:
2n + 8 = (n + 4) + 7
2n + 8 = n + 11
[URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B8%3Dn%2B11&pl=Solve']Type this equation into our search engine[/URL], and we get:
n = 3
This means from equation (1), that:
d = 3 + 4
d = 7
So our original fraction n/d = [B]3/7[/B]
The fraction has a value of 3/5. The sum of the numerator and the denominator was 40. What was the fraction?
We're given two equations with a fraction with numerator (n) and denominator (d):
[LIST=1]
[*]n + d = 40
[*]n/d = 3/5
[/LIST]
Cross multiply equation 2, we get:
5n = 3d
Divide each side by 5:
5n/5 = 3d/5
n = 3d/5
Substitute this into equation 1:
3d/5 + d = 40
Multiply through both sides of the equation by 5:
5(3d/5) = 5d = 40 * 5
3d + 5d =200
To solve this equation for d, we [URL='https://www.mathcelebrity.com/1unk.php?num=3d%2B5d%3D200&pl=Solve']type it in our search engine and we get[/URL]:
d = [B]25
[/B]
Now substitute that back into equation 1:
n + 25 = 40
Using [URL='https://www.mathcelebrity.com/1unk.php?num=n%2B25%3D40&pl=Solve']our equation solver again[/URL], we get:
n = [B]15[/B]
the quotient of a variable and 7.
A variable means an arbitrary number, let's call it x.
A quotient means a fraction, where x is the numerator and 7 is the denominator:
[B] x
---
7[/B]
the quotient of the cube of a number x and 5
[LIST]
[*]A number means an arbitrary variable, let's call it x
[*]The cube of a number means raise it to the 3rd power, so we have x^3
[*]Quotient means we have a fraction, so our numerator is x^3, and our denominator is 5
[/LIST]
[B]x^3
----
5[/B]
Which of the followings can increase the value of t? (select all the apply)
a. Increase the standard deviation of difference scores
b. Decrease the standard deviation of difference scores
c. Increase the difference between means
d. Decrease the difference between means
[B]b. Decrease the standard deviation of difference scores
c. Increase the difference between means[/B]
[I]Increase numerator or decrease denominator of the t-value formula[/I]
Write a fraction with a denominator of 9. The fraction should be less than 1/2
Let n be the numerator. We have:
n/9 < 1/2
multiply each side by 9:
9n/9 < 9/2
n < 9/2
Examples are 8/2, 7/2, 6/2, 5/2, 4/2, 3/2,
You have 30 DONUTS. 1/6 of them are Boston Cream. 2/5 of them are Maple. 3/10 of them are Chocolate Dip and 1/3 are Sprinkled. IS THIS EVEN POSSIBLE?? How many donuts OVER or UNDER am I? (Show your work and use EQUIVALENTS.)
We use 30 as our common denominator. Let's get [I]equivalent fraction[/I]s for each donut type with a denominator of 30:
[LIST]
[*][URL='https://www.mathcelebrity.com/equivalent-fractions.php?num=1%2F6&pl=Equivalent+Fractions']1/6[/URL] = 5/30
[*][URL='https://www.mathcelebrity.com/equivalent-fractions.php?num=2%2F5&pl=Equivalent+Fractions']2/5 [/URL]= 12/30
[*][URL='https://www.mathcelebrity.com/equivalent-fractions.php?num=3%2F10&pl=Equivalent+Fractions']3/10[/URL] = 9/30
[*][URL='https://www.mathcelebrity.com/equivalent-fractions.php?num=1%2F3&pl=Equivalent+Fractions']1/3[/URL] = 10/30
[/LIST]
Add up our numerators of the common denominator of 30:
5 + 12 + 9 + 10 = 36
So our fraction is 36/30. This makes our scenario [B]impossible[/B]. Fractions of the donut should add up to 1. Which would mean our numerators need to sum to 1 or less. Since 36 > 30, this scenario is [B]impossible.[/B]
