## How to Solve Number Word Problems Part 1

In the previous post, we have learned How to Solve Number Problems Mentally. In this post, we are going to solve the same word problems algebraically. The objective of this post is for you to be able to learn how to set up equations based on given problems. Once you know how to set up equations for easy problems, it will be easier for you to do so using harder problems which we will discuss in the latter parts of this series. Note that before solving these problems, it is already assumed that you know how to solve equations.

Problem 1

One number is 3 more than the other. Their sum is 45. What are the numbers?

Scratch Work

The strategy in solving algebraic problems is to take a specific case. For instance, in the problem above, if one number is say, $5$, then the larger number is $5 + 3$ because it is $3$ greater than the first number. Since we do not know the numbers yet, we can represent the smaller number by $x$ and the larger number by $x + 3$.

The next clue is the word “sum,” their sum is $45$. So, sum means you have to add the two numbers which are $x$ and $x + 3$. In sentence form, the sum of $x$ and $x + 3$ is $45$ or

$x + (x + 3) = 45$

in equation form. Now, we write the solution.

Solution

Let $x$ be the smaller number and $x + 3$ be the larger number.

$x + (x + 3) = 45$

$2x + 3 = 45$

$2x = 42$

$x = 21$

So, the smaller number is $21$ and the larger number is

$x + 3 = 21 + 3 = 24$.

Of course, after this, you can always do the checking by looking at the conditions. Is the larger number greater than $3$? Is the sum of two numbers $45$? Once the answer satisfies all the conditions in the given problem, then it is correct.

Problem 2

The sum of two numbers is 53. One number is 7 less than the other. What are the numbers?

Scratch Work

In this example, if the larger number is $100$, then the other number is $7$ less than the $100$ or $100-7$. So, if the number is $x$, the smaller number is $x - 7$. The next sentence is their sum is $53$, so we have to add $x$ and $x -7$ forming the equation

$x + (x - 7) = 53$.

We now write the solution.

Solution

Let $x$ be the larger number and $x - 7$ be the smaller number.

$x + (x-7) = 53$

$2x - 7 = 53$

$2x = 60$

$x = 30$

Therefore, the larger number is $30$ and the smaller number is $x - 7 = 30 - 7 = 23$. Again, you can check the answer by if it satisfies the conditions above.

Problem 3

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

Scratch Work

If one number is $5$, then the number twice it is $10$, or $2(5)$. Therefore, if one number is $x$, the number twice it is $2(x)$ or $2x$. Next, their sum is $45$. It means that if you add $x$ and $2x$, the sum is $45$. Or, $x + 2x = 45$.

Solution

Let $x$ be the smaller number and $2x$ be the larger number.

$x + 2x = 45$

$3x = 45$

Dividing both sides by $3$, we have

$x = 15$.

Therefore, the smaller number is $15$ and the larger number is $2(15) = 30$. Checking it, $30$ is indeed twice $15$, and yes, their sum is $45$.

Note: This post has a video version. In the video, these problems were solved in Taglish.

In the second part of this series, we will learn to solve more complicated number word problems.