Math Word Problems: Solving Number Problems Part 3

This is the third part of the Solving Number Problems Series. The first part can be read here and the second part can be read here.  In this post I will continue worked examples using problems which are slightly more complicated than the problems in the previous two parts. Without further ado, lets start with the seventh problem in the series.

PROBLEM 7

Twice a number added to 18 is 5 times that number. What is the number?

Solution

In the two previous posts, we have learned that if n is a number, then twice that number is 2n. So, twice a number added to 5 is represented by 2n + 5. Now, that number, the 2n + 5 is five times that number of 5n. So, we can now set up the equation

2n + 18 = 5n.

If we solve for n, we have n = 6

Checking the Answer

Twice 6 added to 18 is 30. Five times 6 is 30. So, we are correct.

PROBLEM 8

Four times the sum of a number and 3 is 96. What is the number?

Solution

This one is quite tricky. Some interpret this as 4n + 3 = 96. But that is not correct. If you read it carefully, the equation 4n + 3 = 96 is “four times a number added to three.” However, what we want is four times the sum of a number and 3. So, if we let n be the number, then the sum of a number and 3 is n + 3. Four times the sum of a number and 3 is there 4(n+3). Now, we set up the equation

4(n + 3) = 96.

Solving, we have

4(n + 3) = 4n + 12 = 96.

This gives us 4n = 84 and n = 21.

Checking the Answer

Four times the sum of 21 and 3 is four times 24 which is equal to 96. So, we are correct.

PROBLEM 9

The sum of three numbers is 105. The first number is 3 less than the second number. The third number is four times the second number.

Solution

As you can observe, the second number has no description, so we let it be n. The first number is less than the second number by 3, so the first number is n - 3. The third number is four times the second number or 4n.

Their sum (n - 3) + n + 4n is 105, so

(n-3) + n + 4n = 105.

Simplifying, we have

6n - 3= 105

6n = 108

So n = 18 and the first number is 15. The third number is 72.

Checking the Ansewer

Left as an excersise.

***

You have probably noticed that in solving word problems, it is important to accurately convert phrases into algebraic expressions/equations and vice versa.  The word “is” for instance is the same as “equal.” Of course, these keywords are help us in understanding the problem, but we should remember to understand the problem as a whole. In the fourth part (last part) of this series, we will discuss more complicated problems.

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5 thoughts on “Math Word Problems: Solving Number Problems Part 3

  1. For question #9. It says that the first number is three less than the second number, and that the third number is four times the second number. So, aren’t the numbers then, in order from first to third (n-3, n, 4n). As such, (n-3) + (n) + (4n) = 105. Then, 6n – 3 = 105, so n=18, and the three numbers are 15, 18, and 72.

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