# Math Word Problems: Solving Number Problems Part 2

This is the continuation of the previous post on solving number problems. In this post, I will give three more examples on how to solve word problems about numbers.

PROBLEM 4

One number is smaller than the other by $12$. Their sum is $102$. What is the smaller number?

Solution

In the previous post, we talked about two numbers, one is being larger than the other. In this problem, the other number is smaller. If a number is $15$, and the other number is $6$ smaller than it, then that number is $15 - 6$. So, in the problem above, if we let $n$ be the number, then $n - 12$ is the smaller number.  Again, their sum is $102$, so we can now set up the equation

${\color{Green} n}+{\color{Red} (n-12)}=102.$

Simplifying, we have $2n - 12 = 102$. This gives us $2n = 114$ which means that $n = 57$. The smaller number is $57-12 = 45$.

Is $45$ twelve less than $57$? Is their sum equal to $102$? If the answer to both questions is yes, then we are correct.

PROBLEM 5

Divide 71 into two parts such that one part exceeds the other by 8. What are the two numbers?

Solution 1

Let \$let n\$ be the smaller and $71 - n$ be the larger number.  Now, since the larger number exceeds the smaller number by 7, we can form the equation

larger numbersmaller number = $7$

which is equivalent to

${\color{Blue} (71-n)}-{\color{Red} n}=7.$

Simplifying, we have $71 - 2n = 7$. This gives us $2n = 78$ which implies that the larger number is ${\color{Blue} 39}$. The smaller  is $71-39={\color{Red} 32}$.

Solution 2

Same as the solution of Problem 1. I leave this to you as an excercise.

Do the two numbers add up to $71$. Does the number exceeds the smaller number by $7$?

PROBLEM 6

The sum of three consecutive even numbers is 90. What are the three numbers?

Solution

In the previous post, we talked about consecutive numbers. We know that consecutive numbers are numbers in uninterrupted succession. For instance, $10, 11$ and $12$ are three consecutive numbers.  In effect, if $n$ is the smaller number, the next two numbers are $n + 1$ and $n + 2$. However, consecutive even numbers increases by $2$ such as $16, 18$, and $20$. In effect, if $n$ is the smallest number, then $n + 2$ is the middle number and $n + 4$  is the largest number. Knowing their sum, we can now set up the equation

${\color{Green} n}+{\color{Blue} (n+2)}+{\color{Red} (n+4)}=90.$

Simplifying, we have $3n + 6 = 90$. This gives us $3n = 84$ and $n = 28$. So the consecutive even numbers are $28$, $30$ and $32$.

Are the three numbers even and consecutive? Is their sum equal to $90$? If the answers to both question is yes, then we are correct.