GED Mathematical Reasoning: Word Problems

Using an equation to solve a word problem

  1. Identify the quantity you know nothing about and assign a variable to represent that value. Write the “let” statement on your paper.
  2. Create an equation that represents the situation using the information given in the problem.
  3. Solve the equation using opposite operations to find the unknown value.
  4. Be sure to answer the question. Check your answer using the parameters given in the original problem.

 

Example 1

The price of gasoline increased by $0.13 per gallon. It now costs $3.43 per gallon. Use an equation to find the price of gasoline before the price increase.

 

The first step in using an equation to solve a word problem is to identify the unknown value and assign a variable to represent that value. In this example, we don’t know the price of gasoline before the price increase so we’ll let “p” equal the initial price. It is a good idea to write this clearly on your paper:

Let p = the initial price of gasoline per gallon

Next, we need to create an equation that represents the situation using the information given in the problem. In this example, the initial price of gasoline – represented by the letter “p” – increased $0.13 per gallon and now equals $3.43 per gallon.

The word “increased” implies the operation of addition. So the equation we can use to represent this situation is:

p + \$0.13 = \$3.43

Finally, we’ll solve the equation to find the unknown value. Since the operation between the variable and number on the left side is addition, we’ll do the opposite operation and subtract $0.13 from both sides to get the variable by itself.

p + \$0.13 - \$0.13 = \$3.43 - \$0.13

p = \$3.30

Therefore, the price of gasoline before the price increase was $3.30 per gallon.

 

Example 2

Crystal donates one-tenth of her income to charity and she donated $2,200 to charity last year. Use an equation to find her total income from last year.

 

Since we’re being asked to find Crystal’s total income, total income is our unknown value. Let’s assign it the letter “m.” So we’ll write on our paper:

Let m = total income

Next, we’ll create an equation that represents the situation using the information given in the problem. In this example, Crystal donates one-tenth of her total income and the amount of the donation was equal to $2,200. The word “of” implies the operation multiplication. So the equation we can use to represent this situation is:

\frac{1}{10}m = \$2200

Finally, let’s solve the equation to find the unknown value. Since the operation between the variable and number on the left side is multiplication, we’ll do the opposite operation and divide both sides by one-tenth to get “m” by itself. Recall that to divide by one-tenth is the same as multiplying by ten over one, or just ten. This goes back to dividing with fractions by multiplying by the reciprocal.

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When we multiply $2,200 by ten, either by hand or using the calculator, the result is $22,000. Therefore, Crystal’s total income was $22,000.

One final note.

A decimal can be used to represent the value of one-tenth, making the equation:

0.10m = \$2200

To solve this equation, we divide both sides of the equation by the decimal form of one-tenth.

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Example 3

At a fundraising event, Edward raised $10 more than triple the amount raised by Robert. Together, Edward and Robert raised a total of $110. How much money did each man raise?

 

Our first step is to allow a variable to represent the quantity we know nothing about. We know something about the amount raised by Edward – he raised $10 more than triple the amount raised by Robert. That means we know nothing about the amount of money raised by Robert. So let’s let “x” equal the amount of money raised by Robert. I suggest that you write this “let” statement on your paper.

Let x = the amount of money raised by Robert

Now let’s begin step 2 – the process of creating an equation based on the information given in the problem. We are told that Edward raised $10 more than triple the amount Robert raised, which we can represent by the expression:  3 “x” plus 10. Do you see why? The word “triple” implies multiplication by 3. And the words “ten more than” imply the addition of 10.

Amount Edward raised = 3x + 10

From here, we are told that the TOTAL amount raised by both men is $110.

Since the amount of money raised by Robert is represented by “x.” And the amount of money raised by Edward is represented by the expression: 3 “x” plus 10 We can create the following equation based on these expressions and the fact that the word “total” implies that added together, the sum is 110:

Amount raised by Robert + Amount raised by Edward = $110

x + (3x + 10) = 110

Now let’s complete step 3 and solve this equation for “x.”

On the left side, although we used parenthesis to distinguish the expression representing Edward’s amount, there is no distributing to be done since there is no multiplier sitting in front of the parenthesis. So we can simply drop the parenthesis on the left side.

x + (3x + 10) = 110 \rightarrow x + 3x + 10 = 110

From here, we will combine the two “like terms” on the left side of the equal sign. “x” (or one “x”) plus 3 “x” is equal to 4x.

4x + 10 = 110

Now that we have simplified each side of the equal sign, we will subtract 10 from both sides to get the variable term on the left side of the equation and the constant terms on the right side.

4x + 10 - 10 = 110 - 10
4x = 100

To solve for x, we will divide both sides by 4.

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So the solution is: “x” equals 25.

When we solve for “x” we aren’t finished, though. We need to be sure to follow step 4 and answer the question. In this case, we need to state the amount of money raised by both Robert AND Edward.

This is where the “let” statement comes in handy. We can refer back to it to remind ourselves that “x” represents the amount of money raised by Robert.

To determine how much money Edward raised, we need to triple this amount and add $10. 3 times 25 equals $75. Adding $10 results in the amount $85. So Robert raised $25 and Edward raised $85.

x = \$25 \leftarrow the amount of money raised by Robert

3(\$25) + \$10 = \$85 \leftarrow the amount of money raised by Edward

I do not recommend checking word problems like this using the original equation because it is possible that we made a mistake and the equation we used for solving was incorrect. Instead, let’s check our result based on the parameters given in the original question.

Do these two amounts total to be $110? $25 plus $85 DOES equal $110, so we can be confident our solution is correct.

 

Example 4

The sum of three consecutive odd numbers is 75. What are the three numbers? 

 

Consecutive numbers are numbers that follow one right after the other in counting order. For example, the numbers 1, 2, 3 are three consecutive numbers. The numbers 4, 6, 8 are three consecutive EVEN numbers. And the numbers 9, 11, 13 are three consecutive ODD numbers.

Note that if you know the first number in the sequence, you can determine the others. If I ask you to state three consecutive odd numbers beginning with the number 5, you would simply add two to arrive at the second consecutive odd number, which would be 7. And you’d add four to arrive at the third consecutive odd number, which would be 9.

Going back to example 2, then, it’s the first number of the sequence that we know nothing about. So let’s let x equal the first odd number and make a note of it on our paper.

Let x = the first odd number

Now we’ll begin step 2 – creating our equation based on the information given in the original problem. As we just discussed, if “x” equals the first odd number, then the expression “x” plus 2 represents the second consecutive odd number and the expression “x” plus 4 represents the third consecutive odd number.

x + 2 = the second consecutive odd number
x + 4 = the third consecutive odd number

From here, we’re told that the total of the three numbers is 75. This means that if we add the first, second, and third number, the total should be 75. Using this information and the expressions that represent each number, we can create the following equation:

the first number + the second number + the third number = 75

x + (x + 2) + (x + 4) = 75

Now let’s complete step 3 and solve this equation for “x.”

On the left side, although we used parenthesis to distinguish the expressions representing the second and third numbers, there is no distributing to be done since there are no multipliers sitting in front of the parenthesis. So we can drop the parenthesis on the left side.

x + (x + 2) + (x + 4) = 75 \rightarrow x + x + 2 + x + 4 = 75

From here, let’s combine the “like terms” on the left side of the equal sign.

x + x + 2 + x + 4 = 75
1x + 1x + 2 + 1x + 4 = 75
3x + 6 = 75

So we have 3x plus 6 equals 75.

Now that we have simplified each side of the equal sign, we will subtract 6 from both sides to get the variable term on the left side of the equation and the constant terms on the right side.

3x + 6 = 75
3x + 6 - 6= 75 - 6
3x = 69

x = 23

So the solution is: x=23.

But remember – when we solve for “x” we aren’t finished. We need to be sure to follow step 4 and answer the question. In this case, we need to state all three consecutive odd integers. Let’s look back at our “let” statement to remind ourselves that “x” represents the first odd number in the sequence.

To determine the second consecutive odd number, we’ll add 2 to 23, which is equal to 25. To determine the third consecutive odd number, we’ll add 4 to 23, which is equal to 27. So the three consecutive odd numbers that add to be 75 are:  23, 25, and 27. Let’s double check to be sure:  23 plus 25 plus 27 DOES equal 75.

23 + 25 + 27 = 75

Before we end this session, you may be wondering: What do I do if my answer doesn’t check correctly?

In that case, go back through all your work and look for a mistake. Especially examine the equation you’re using to solve the problem. You might try a different combination of expressions and then re-solve. Sometimes the best way to learn is by learning from our mistakes.

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