Number Word Problems 1 – How to Solve Number Problems Mentally

This is the first part of the Solving Number Problems Series for Grade 6-8 students, a-sub series of the Math Word Problem Solving Series.

Some of you probably need paper and pencil to solve number problems. You will be surprised that if you think harder and work backward, you can actually solve these problems in your head. Consider the following examples.

Example 1

One number is 1 more than the other. Their sum is 47. What are the numbers?

Explanation and Solution

First, one number is 1 more than the other. That means that if we subtract 1 from the larger number the two numbers will be equal. That’s our first clue.

Second, if we subtract 1 from the larger number, then we should also subtract 1 from the sum (Can you see why?). That makes the sum 46. Now, since the two numbers are equal, we can divide 46 by 2. That gives us 23 which is the smaller number. Now, since the other number is 1 larger than 23, then it is 24.

Answers: 23 and 24

Check:

Is one number is 1 more than the other? Yes, 24 is 1 more than 23.

Is their sum 47? Yes, 23 + 24 = 47.

Example 2

One number is twice the other number. Their sum is 45. What are the numbers?

Explanation and Solution

If one number is twice the other number, then their sum is thrice the smaller number. For example, if the number is equal to 3, then the other number is 6. And therefore, if 3 is one group, and 6 is two groups, then adding them will result to thrice 3 which is equal to 9. That means that 45 is thrice the smaller number.

If 45 is thrice the smaller number, then we can divide 45 by 3. This gives us 15. If 15 is the smaller number, then twice 15 or 30 is the larger number.

Answers: 15 and 30

Check:

Is the larger number twice the smaller? Yes, 30 is twice 15.

Is the sum of the two numbers 45? Yes, 30 + 15 = 45.

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Example 3

The sum of two odd consecutive numbers is 64. What is the larger number?

Explanation and Solution

Examples of consecutive odd numbers are 3, 5, 7 and 9. As we can observe, we add 2 each time. This means that the larger number is 2 more than the smaller number. Just like in example 1, if we subtract 2 from the larger number, they will be equal. But we also have to subtract 2 from the total. So, if the total is 62 (two subtracted from 64), then the two are equal. Therefore, we divide 64 by 2 which gives us 31. We are looking for the larger numbers, so we add 2 to 31 which gives us 33.

Answer: 33

Check:

Are the two numbers consecutive odd numbers? Yes, 31 and 33 are consecutive odd numbers.

Is their sum 64? Yes, 31 + 33 = 64.

The three examples above will give you ideas on how to use mental mathematics and backward method to solve similar problems without pen and paper. You will be surprised that there are a lot of other problems that you can solve using this method. For instance, extending the procedure, you can solve problems about three consecutive numbers easily (can you see why?).

To be able to practice solving mentally, do the problem in your head first and think as hard as you can. Solve using pencil and paper or other tools only if you see that the problem is impossible to solve mentally. In the next part of this series, model methods on solving problems is discussed.