17 results

4 consecutive integers such that the sum of the first 3 integers is equal to the 4th

4 consecutive integers such that the sum of the first 3 integers is equal to the 4th
Let n be our first consecutive integer.
[LIST=1]
[*]n
[*]n + 1
[*]n + 2
[*]n + 3
[/LIST]
The sum of the first 3 integers is equal to the 4th:
n + n + 1 + n + 2 = n + 3
Simplify by grouping like terms:
(n + n + n) + (1 + 2) = n + 3
3n + 3 = n + 3
3n = n
n = 0
n = 0
n + 1 = 1
n + 2 = 2
n + 3 = 3
Check our work:
0 + 1 +2 ? 3
3 = 3
Our final answer is [B](0, 1, 2, 3}[/B]

A boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find th

A boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find the present age of each?
Let the boy's age be b and his brother's age be c. We're given two equations:
[LIST=1]
[*]b = c + 10
[*]b + 4 = 2(c + 4)
[/LIST]
Substitute equation (1) into equation (2):
(c + 10) + 4 = 2(c + 4)
Simplify by multiplying the right side through and grouping like terms:
c + 14 = 2c + 8
[URL='https://www.mathcelebrity.com/1unk.php?num=c%2B14%3D2c%2B8&pl=Solve']Type this equation into our search engine[/URL] and we get:
c = [B]6[/B]
Now plug c = 6 into equation (1):
b = 6 + 10
b = [B]16[/B]

Alvin is 12 years younger than Elga. The sum of their ages is 60 . What is Elgas age?

Alvin is 12 years younger than Elga. The sum of their ages is 60 . What is Elgas age?
Let a be Alvin's age and e be Elga's age. We have the following equations:
[LIST=1]
[*]a = e - 12
[*]a + e = 60
[/LIST]
Plugging in (1) to (2), we get:
(e - 12) + e = 60
Grouping like terms:
2e - 12 = 60
Add 12 to each side:
2e = 72
Divide each side by 2
[B]e = 36[/B]

Factoring and Root Finding

Free Factoring and Root Finding Calculator - This calculator factors a binomial including all 26 variables (a-z) using the following factoring principles:

* Difference of Squares

* Sum of Cubes

* Difference of Cubes

* Binomial Expansions

* Quadratics

* Factor by Grouping

* Common Term

This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots

* Factors and simplifies Rational Expressions of one fraction

* Determines the number of potential*positive* and *negative* roots using Descarte’s Rule of Signs

* Difference of Squares

* Sum of Cubes

* Difference of Cubes

* Binomial Expansions

* Quadratics

* Factor by Grouping

* Common Term

This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots

* Factors and simplifies Rational Expressions of one fraction

* Determines the number of potential

If 11 times a number is added to twice the number, the result is 104

If 11 times a number is added to twice the number, the result is 104
Let [I]the number[/I] be an arbitrary variable we call x.
11 times a number:
11x
Twice the number (means we multiply x by 2):
2x
The phrase [I]is added to[/I] means we add 2x to 11x:
11x + 2x
Simplify by grouping like terms:
(11 + 2)x = 13x
The phrase [I]the result is[/I] means an equation, so we set 13x equal to 104:
13x = 104 <-- This is our algebraic expression
To solve this equation for x, [URL='https://www.mathcelebrity.com/1unk.php?num=13x%3D104&pl=Solve']we type it in our search engine[/URL] and we get:
x = [B]8[/B]

If x represents the first, or the smaller, of two consecutive odd integers, express the sum of the

If x represents the first, or the smaller, of two consecutive odd integers, express the sum of the two integers in terms of x
If x is the first of two consecutive odd integers, then we find the next consecutive odd integer by adding 2 to x:
x + 2
The sum of the two consecutive odd integers is expressed by
x + (x + 2)
Simplify by grouping like terms, we get:
[B]2x + 2[/B]

In a class there are 5 more boys than girls. There are 13 students in all. How many boys are there i

In a class there are 5 more boys than girls. There are 13 students in all. How many boys are there in the class?
We start by declaring variables for boys and girls:
[LIST]
[*]Let b be the number of boys
[*]Let g be the number of girls
[/LIST]
We're given two equations:
[LIST=1]
[*]b = g + 5
[*]b + g = 13
[/LIST]
Substitute equation (1) for b into equation (2):
g + 5 + g = 13
Grouping like terms, we get:
2g + 5 = 13
Subtract 5 from each side:
2g + 5 - 5 = 13 - 5
Cancel the 5's on the left side and we get:
2g = 8
Divide each side of the equation by 2 to isolate g:
2g/2 = 8/2
Cancel the 2's on the left side and we get:
g = 4
Substitute g = 4 into equation (1) to solve for b:
b = 4 + 5
b = [B]9[/B]

larger of 2 numbers is 4 more than the smaller. the sum of the 2 is 40. what is the larger number?

larger of 2 numbers is 4 more than the smaller. the sum of the 2 is 40. what is the larger number?
Declare variables for the 2 numbers:
[LIST]
[*]Let l be the larger number
[*]Let s be the smaller number
[/LIST]
We're given two equations:
[LIST=1]
[*]l = s + 4
[*]l + s = 40
[/LIST]
To get this problem in terms of the larger number l, we rearrange equation (1) in terms of l.
Subtract 4 from each side in equation (1)
l - 4 = s + 4 - 4
Cancel the 4's and we get:
s = l - 4
Our given equations are now:
[LIST=1]
[*]s = l - 4
[*]l + s = 40
[/LIST]
Substitute equation (1) into equation (2) for s:
l + l - 4 = 40
Grouping like terms for l, we get:
2l - 4 = 40
Add 4 to each side:
2l - 4 + 4 = 40 + 4
Cancelling the 4's on the left side, we get
2l = 44
Divide each side of the equation by 2 to isolate l:
2l/2 = 44/2
Cancel the 2's on the left side and we get:
l = [B]22[/B]

Number Bonds

Free Number Bonds Calculator - Adds or subtracts 2 numbers and using grouping by 10 or 100. Also called number bonds or addition facts. Multiplies two numbers using tape diagrams.

Regrouping

Free Regrouping Calculator - Subtracts two numbers using regrouping

Sam has x model planes. Anton has 8 more planes than Sam does. How many model planes Does Anton have

Sam has x model planes. Anton has 8 more planes than Sam does. How many model planes Does Anton have? how many planes do they have together?
Sam has x
Anton has [B]x + 8[/B] since the word [I]more[/I] means we add
The word [I]together[/I] means we add, so we have:
Sam + Anton = x + x + 8
Grouping like terms, we have:
Sam + Anton = [B]2x + 8[/B]

Serial numbers for a product are to made using 4 letters followed by 4 numbers. If the letters are t

Serial numbers for a product are to made using 4 letters followed by 4 numbers. If the letters are to be taken from the first 5 letters of the alphabet with repeats possible and the numbers are taken from the digits 0 through 9 with no repeats, how many serial numbers can be generated?
First 5 letters of the alphabet are {A, B, C, D, E}
The 4 letters can be chosen as possible:
5 * 5 * 5 * 5
The number are not repeatable, so the 4 numbers can be chosen as:
10 * 9 * 8 * 7 since we have one less choice with each pick
Grouping letters and numbers together, we have the following serial number combinations:
5 * 5 * 5 * 5 * 10 * 9 * 8 * 7 = [B]3,150,000[/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

the difference between 7 times a number and 9 less than a number

the difference between 7 times a number and 9 less than a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
7 times a number means we multiply x by 7
7x
9 less than a number means we subtract 9 from x
x - 9
The difference between the two expressions means we subtract (x - 9) from 7x
7x - (x - 9)
Simplifying this, we have:
7x - x + 9
Grouping like terms, we get:
[B]6x + 9[/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
We have:
n + (n + 1) + (n + 2)
Grouping like terms, we have:
[B]3n + 3[/B]

The sum of 3, 7, and a number amounts to 16

The sum of 3, 7, and a number amounts to 16
Let the number be n. A sum means we add. We're given:
3 + 7 + n = 16
Grouping like terms, we get:
n + 10 = 16
[URL='https://www.mathcelebrity.com/1unk.php?num=n%2B10%3D16&pl=Solve']Typing this equation into our search engine[/URL], we get:
n = [B]6 [/B]

The sum of two consecutive integers if n is the first integer.

The sum of two consecutive integers if n is the first integer.
consecutive means immediately after, so we have:
n
n + 1
[U]The sum is written as:[/U]
n + n + 1
[U]Grouping like terms, we have:[/U]
(n + n) + 1
[B]2n + 1[/B]