Math Word Problems: Solving Number Problems Part 1

September 11, 2012 GB High School Algebra, High School Mathematics, Problem Solving and Proofs

This is the first post in the Word Problem Solving Series. This series is intended for middle school students. In this post, we are going to discuss how to solve number word problems in Algebra.

The that you should observer in this series translating words into algebraic expressions and equations. Prior to the discussion of word problems, you have already learned how to solve equations, so I will not show the solution in details.

Problem 1

The sum of two numbers is $64$ . The larger number is $18$ more than the smaller. What are the two numbers?

Solution

Suppose $15$ is a number, the larger number, which is $18$ more than it is $15 + 18$ . Therefore, if $n$ a number, then $n + 18$ is the larger number $18$ more than it. If we add the two numbers

$n + (n + 18)$

the sum is $64$ . The sentence “The sum of two numbers is $64$ ,” tells us that that the sum $n + (n + 3)$ equals $64$ . Therefore, we set up the equation,

$n + (n + 18) = 64$ .

Simplifying, we have $2n + 18 = 64$ , which gives us $n = 23$ . So, the smaller number is $23$ and the larger number is $23 + 18 = 41$ .

Checking the Answer

We can check if our solution is correct by verifying the answer. Is the larger number $18$ more than the smaller? Yes. Is the sum of the two numbers $64$ . Yes, $23 + 41 = 64$ . So, we are correct.

Problem 2

One number is twice the other number. Their sum is $84$ . What are the two numbers?

Solution

If a number is $10$ , twice that number is $2 \times 10$ . So, if the smaller number in the problem is $n$ , the larger number is $2 \times n$ or $2n$ . Again in the problem, it says that “their sum is $84$ ” which means that if we add $n$ and $2n$ , their sum is equal to $84$ .

So, we set up the equation

$n + 2n = 84$ .

This gives us $3n = 84$ which gives us $n = 28$ . The larger number is $2 \times 28 = 56$ .

Checking the Answer

Is the larger number twice the smaller? Yes. Is the sum $84$ . Yes. Therefore, we are correct.

Problem 3

The sum of three consecutive numbers is $87$ . What are the numbers?

Solution

Consecutive numbers are numbers in uninterrupted succession. For example, $6, 7, 8, 9, 10$ are five consecutive numbers.

Suppose the consecutive numbers are $5$ , $6$ , and $7$ , we can see that $5$ is the smaller. If $5$ is the smaller, then $5 + 1$ is the next number, and $5 + 2$ is the largest number. So, if $n$ is the smaller number, $n + 1$ is the next number, and $n + 2$ is the largest number. If we add the three numbers

$n + (n + 1) + (n+ 2)$

their sum is $87$ . That is,

$n + (n + 1) + (n + 2) = 87$ .

Simplifying, we have $3n + 3 = 87$ . Subtracting 3 from both sides, we have $3n = 84$ . Dividing both sides by 3, we have $n = 28$ . So the three consecutive numbers are $28$ , $29$ and $30$ .