The sum of two consecutive even integers is 34. What are the integers?

### Watch This

Khan Academy Problem Solving Word Problems 2

### Guidance

To translate a problem from words into an equation, look for key words to indicate the operation used in the problem.

Once the equation is known, to solve the problem you use the same rules as when solving equations with one variable. Isolate the variable and then solve for it making sure that whatever you do to one side of the equals sign you do to the other side. Drawing a diagram is also helpful in solving some word problems.

#### Example A

Two consecutive integers have a sum of 173. What are those numbers?

**Solution:** Let \begin{align*}x =\end{align*} integer 1

Then \begin{align*}x + 1 =\end{align*} integer 2 (Because they are consecutive, they must be separated by only one number. For example: 1, 2, 3, 4,... all are consecutive.)

Translate the sentence into an equation and solve:

\begin{align*}x + (x + 1) &= 173\\ x + x + 1 &= 173 && (\text{Remove the parentheses})\\ 2x + 1 &= 173 && (\text{Combine like terms})\\ 2x + 1 {\color{red}-1} &= 173 {\color{red}-1} && (\text{Subtract} \ 1 \ \text{from both sides to isolate the variable})\\ 2x &= 172 && (\text{Simplify})\\ \frac{2x}{{\color{red}2}} &= \frac{172}{{\color{red}2}} && (\text{Divide both sides by} \ 2 \ \text{to solve for the variable})\\ x &= 86 && (\text{Simplify})\end{align*}

Therefore the first integer is 86 and the second integer is \begin{align*}(86 + 1) = 87\end{align*}. Check: \begin{align*}86 + 87=173\end{align*}.

#### Example B

When a number is subtracted from 35, the result is 11. What is the number?

**Solution:** Let \begin{align*}x =\end{align*} the number

Translate the sentence into an equation and solve:

\begin{align*}35 - x &= 11\\ 35 {\color{red}- 35} - x &= 11 {\color{red}- 35} && (\text{Subtract} \ 35 \ \text{from both sides to isolate the variable})\\ -x &= -24 && (\text{Simplify})\\ \frac{-x}{{\color{red}-1}} &= \frac{-24}{{\color{red}-1}} && (\text{Divide both sides by} \ -1 \ \text{to solve for the variable})\\ x &= 24 && (\text{Simplify})\end{align*}

Therefore the number is 24.

#### Example C

When one third of a number is subtracted from one half of a number, the result is 14. What is the number?

**Solution:** Let \begin{align*}x =\end{align*} number

Translate the sentence into an equation and solve:

\begin{align*}\frac{1}{2}x-\frac{1}{3}x=14\end{align*}

You need to get a common denominator in this problem in order to solve it. For this problem, the denominators are 2, 3, and 1. The LCD is 6. Therefore multiply the first fraction by \begin{align*}\frac{3}{3}\end{align*}, the second fraction by \begin{align*}\frac{2}{2}\end{align*}, and the third number by \begin{align*}\frac{6}{6}\end{align*}.

\begin{align*}\left({\color{red}\frac{3}{3}}\right) \frac{1}{2}x-\left({\color{red}\frac{2}{2}}\right) \frac{1}{3}x &= \left({\color{red}\frac{6}{6}}\right)14\\ \frac{3}{6}x-\frac{2}{6}x &= \frac{84}{6} && (\text{Simplify})\end{align*}

Now that the denominator is the same, the equation can be simplified to be:

\begin{align*}3x-2x &= 84\\ x &= 84 && (\text{Combine like terms})\end{align*}

Therefore the number is 84.

#### Concept Problem Revisited

The sum of two consecutive even integers is 34. What are the integers?

Let \begin{align*}x =\end{align*} integer 1

Then \begin{align*}x + 2 =\end{align*} integer 2 (Because they are even, they must be 2 numbers apart. For example: 2, 4, 6, 8,... are all consecutive even numbers.)

Translate the sentence into an equation and solve:

\begin{align*}x + (x + 2) &= 34\\ x + x + 2 &= 34 && (\text{Remove the parentheses})\\ 2x + 2 &= 34 && (\text{Combine like terms})\\ 2x + 2 {\color{red}-2} &= 34 {\color{red}-2} && (\text{Subtract} \ 2 \ \text{from both sides to isolate the variable})\\ 2x &= 32 && (\text{Simplify})\\ \frac{2x}{{\color{red}2}} &= \frac{32}{{\color{red}2}} && (\text{Divide both sides by} \ 2 \ \text{to solve for the variable})\\ x &= 16 && (\text{Simplify})\end{align*}

Therefore the first integer is 16 and the second integer is \begin{align*}(16 + 2) = 18\end{align*}. Note that \begin{align*}16 + 18\end{align*} is indeed 34.

### Vocabulary

- Algebraic Equation
- An algebraic equation contains numbers, variables, operations, and an equals sign.

- Consecutive
- The term consecutive means in a row. Therefore an example of consecutive numbers is 1, 2, and 3. An example of consecutive even numbers would be 2, 4, and 6. An example of consecutive odd numbers would be 1, 3, and 5.

### Guided Practice

1. What is a number that when doubled would equal sixty?

2. The sum of two consecutive odd numbers is 176. What are these numbers?

3. The perimeter of a square frame is 48 in. What are the lengths of each side?

**Answers:**

1. The number is 30.

\begin{align*}2x &= 60\\ \frac{2x}{{\color{red}2}} &= \frac{60}{{\color{red}2}} && (\text{Divide by} \ 2 \ \text{to solve for the variable})\\ x &= 30 && (\text{Simplify})\end{align*}

2. The first number is 87 and the second number is \begin{align*}(87 + 2) = 89\end{align*}.

\begin{align*}x + (x + 2) &= 176\\ x + x + 2 &= 176 && (\text{Remove parentheses})\\ 2x + 2 &= 176 && (\text{Combine like terms})\\ 2x+2 {\color{red}-2} &= 176 {\color{red}-2} && (\text{Subtract} \ 2 \ \text{from both sides of the equals sign to isolate the variable})\\ 2x &= 174 && (\text{Simplify})\\ \frac{2x}{{\color{red}2}} &=\frac{174}{{\color{red}2}} && (\text{Divide by} \ 2 \ \text{to solve for the variable})\\ x &= 87\end{align*}

3. The side length is 12 inches.

\begin{align*}s + s + s + s &= 48 && (\text{Write initial equation with four sides adding to the perimeter})\\ 4s &= 48 && (\text{Simplify})\\ \frac{4s}{{\color{red}4}} &= \frac{48}{{\color{red}4}} && (\text{Divide by} \ 4 \ \text{to solve for the variable})\\ s &= 12\end{align*}

### Practice

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- The sum of three consecutive numbers is 33. What are these numbers?
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