Multistep Proportion Problems
 

Worksheet Objective:
To learn how to solve multi-step word problems using proportions.

There are multiple worksheets that cover decimals. When you complete this one, be sure to move on to the next.




Multistep Proportion Problems

For some proportion problems on the high school equivalency test, one of the numbers to include in the ratios may not be given.† However, you will be able to find this number by adding or subtracting the numbers that are given.

Example:† Carl got 2 problems wrong for every 5 problems right on a math test.† How many problems did Carl get wrong if there were 35 problems altogether on the test?

Letís try to set up the proportion the normal way.
         2 (wrong) = x (wrong)
         5 (right)†††††† 35 (total)

Notice that our labels do not match like they did on all of the other proportion problems.† Wrong matches wrong, but right †does not match total.† The labels must match before we can solve a proportion.†

Since the questions asks us to the wrong items compared to the total, we must leave that ratio the way it is.† Working backwards from total to right, is there a way that we can change the 5 (number of items right) to a total?† If Carl gets 2 items wrong and 5 right, how many total questions were there?† Good!.....2 + 5 = 7.† There were 7 total.† So if we change the 5 to a 7, our labels will match.
         2 (wrong) = x (wrong)
         7 (total)†††††† 35 (total)†† †††††††††††††††††††††††††††††††††††††††††††

Now everything matches and we can solve the proportion.
         2 * 35 = 70 †††††††††† 70 ÷ 7 = 10†
If Carl misses 2 out of every 7 items, on a 35 question test, he would miss 10.

 


Letís try another example.
The ratio of men to women working at Central Hospital is 2: 9. Thirty men work at the hospital. In all how many people work there?

Step 1: Set up the proportion the regular way.
         2 (men)†††††† = 30 (men)
         9(women)†† =†† x (total)

Step2: Find out which part doesn't match and then figure out how to get it to match the problem.
Men
matches men on the top, so we are okay there.† However, women and total †do not match so we must change one of them.† Since the problem asks us to find the total number of people working at the hospital, we have to leave that label.† How can we change women to total?† You are right!† If we add the men (2) and women (9), we will get a total (11).

Now letís solve the proportion with the new numbers.
2 = 30††††††††††††30 * 11 = 330††††††††††††† 330 ÷ 2 = 165
11††† x
So there are 165 people who work at the hospital.

 


Letís try one more.
For every $10 Helen takes home, her employer withholds $3 for taxes and social security. Helenís gross pay each month is $1950. How much does she take home each month?

Step 1: Set up the proportion as you normally would.
         10 (take home) = x †(take home)
†         3 (withheld)††† = 1950 (gross pay)

Notice that the top labels are the sameótake home
The problem is that the bottom labels do not† match, so we canít work the proportion without changing the form.

Step 2: Letís rewrite the proportion.

How can we change the withholding number so that it reflects gross pay?
Thatís right!† By adding the take home pay (10) and the withheld amount (3), we get the gross pay.
         10 (take home) = x (take home)
         13 (gross pay)† †† 1950 (gross pay)

Letís solve the proportion now:
1.† Cross multiply††††††††† 1950 x 10 = 19,500
2.† Divide by the remaining number††††††19,500 ÷ 13 = $1,500
Helen takes home $1,500 each month.

 

Now you practice

Be careful when answering these questions.  If the question includes the label in the problem, you do not need to include the label in the answer.

1.† The ratio of the number of men to the number of women working at Forms, Inc. is 7:2.† Altogether there are 360 workers at the company.† How many of the workers are women.
Answer

2.† Recently 300 people in Central County took a civil service exam.† For every 5 people who passed the exam, one person failed.† How many people passed?
Answer

3.†† For every $9 that Fumiko makes she spends $7.† The rest goes into her savings account.† If her weekly take-home pay is $540, how much does she save each week?
Answer

4.† The ratio of domestic cars to imported cars in the lot at Alís Auto Shop is 3 : 2.† There are 42 domestic cars in the shop.† How many cars are there altogether?
Answer

5.  In a recent interview in Greensport, 7 out of 10 people said they prefer coffee in the morning.† The rest said that they prefer tea.† 27 people said they prefer tea.† How many people were interviewed?
Answer

 

Answers:

1.† 80 † 7 + 2 = 9 †††††††††2 = w ††††††††††† 360 * 2 = 720 ÷ 9 = 80     (back to questions)
††††††††††††††† ††††††††††††††††††††9††† 360

 

 

 

2.† 250 5 + 1 = 6†††††††††5 = p†††††††††††† 300 * 5 = 1500 ÷ 6 = 250     (back to questions)
††††††††††††††††††††††††††††††††††††6††† 300

 

 

 

3.† $120††† 9 Ė 7 = 2†††††††††††††††††2 = s††††††††††††† 540 * 2 = 1080 ÷ 9 = 120     (back to questions)
††††††††††††††††††††††††††††††††††††††††††††††††9††† 540

 

 

 

4.† 70 † 3 + 2 = 5 ††††††††††††††3 = 42†††††††††† 42 * 5 = 210 ÷ 3 = 70     (back to questions)
† ††††††††††††††††††††††††††††††††††††††††5†††† x

 

 

 

5.† 90 † 10 Ė 7= 3†††††††††3 = 27†††††††††† 27 * 10 = 270 ÷ 3 = 90     (back to questions)
††††† †††††††††††††††††††† ††††††††† 10†† †x

 

 

 

 

 

 

 

 

 

 



Resources:
GED Test 5: Mathematics. (2001). Contemporary Books, NTC/Contemporary Publishing Group: Lincolnwood, IL.

Howett, Jerry. (1998). GED Program: Mathematics. Cambridge Adult Education: Upper Saddle River, NJ.

Pre-GED: Mathematics. (2000). Steck-Vaughn: Austin, TX.