Math Word Problems: Solving Number Problems Part 1

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.

word problems

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.

Checking the Answer 

Are the three numbers consecutive? Yes. Is their sum 87.  Yes, 28 + 29 + 30 = 87.

From the worked examples on number problems above, we have learned that the word “is” is most of the time synonymous to the phrase “is equal to.”

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