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Mental Math to Solve Proportions

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Chase, Marc and Kris are all working at the supermarket in the stock room. After school, they work to stock the shelves at the local supermarket with all kinds of cans. In fact, they have gotten quite fast at it and they love to have contests to see who is the fastest. Often the one who loses has to treat the others to ice cream after work. It takes Chase 15 minutes to stock three shelves of canned goods, and it takes Marc 45 minutes to stock nine shelves.

“I am definitely faster,” Chase tells Marc one afternoon.

“I don’t think so, I think we are both the same.” Marc disagrees.

Chase and Marc continue to argue. Who is correct? Their friend Kris stocks shelves at the same rate as Chase does. If he stocks 12 shelves at this rate, how many minutes does it take him?

Previously we worked on how to identify a proportion. In this Concept, you will learn how to use mental math to solve proportions. Solving proportions using mental math is exactly what Marc and Chase need to help them with their argument.

Guidance

In an earlier Concept on unit rates and rates, you practiced figuring out a rate or a unit rate. Essentially, there was a missing part of the rate that you had to figure out.

\frac{25\ campers}{1\ tent} = \frac{75\ campers}{?\ tents}

Here we have a unit rate of twenty-five campers for one tent. Then we have another rate that says that we have 75 campers and we are trying to figure out how many tents are needed for the 75 campers.

These two ratios are equal and they form a proportion.

How can we figure out the missing number of tents?

We can use equal ratios to do this, or we can simply examine the problem and use mental math. Sometimes it makes more sense to simply figure out an answer in your head!

Our answer is three tents.

Often we can use mental math to quickly figure out the missing part of a proportion. We call this “solving a proportion.”

\frac{4}{8} = \frac{x}{16} We can think, “Four is half of eight, what is half of sixteen?” Our answer is eight.

Practice using mental math to solve the following proportions.

Example A

\frac{1}{4} = \frac{x}{16}

Solution: 4

Example B

\frac{3}{9} = \frac{x}{18}

Solution: 6

Example C

\frac{5}{15} = \frac{1}{x}

Solution: 3

Now let's go back and help Marc and Chase to figure out their argument.

Chase, Marc and Kris are all working at the supermarket as stock guys. After school, they work to stock the shelves at the local supermarket with all kinds of cans. In fact, they have gotten quite fast at it and they love to have contests to see who is the fastest. Often the one who loses has to treat the ice cream after work. It takes Chase 15 minutes to stock three shelves of canned goods, and it takes Marc 45 minutes to stock nine shelves.

“I am definitely faster,” Chase tells Marc one afternoon.

“I don’t think so, I think we are both the same.” Marc disagrees.

Chase and Marc continue to argue. Who is correct? Their friend Kris stocks shelves at the same rate as Chase does. If he stocks 12 shelves at this rate, how many minutes does it take him?

The first problem is to figure out which boy is faster at stocking shelves. To do this, we need to write two ratios and see if they form a proportion. If they form a proportion, then Marc is correct.

\frac{15}{3} = \frac{45}{9}

We can look at the relationship between the numerators and the denominators to see that the ratios are the same. Three is one - fifth of fifteen and nine is one - fifth of forty - five. Using mental math has helped us to solve this dilemma.

The ratios form a proportion, so Marc is correct. The boys both work at the same pace.

Now Kris works at the same rate as Chase (Marc too). He stocked 12 shelves. Given the rate, how long did it take him? Once again, we need to write two ratios to form a proportion. The unknown will be the time that it takes Kris.

\frac{15}{3} = \frac{x}{12}

If we look at this proportion and ask ourselves, “What times 3 equals twelve?” The answer is four. We can use that to form an equal ratio.

15 \times 4 = 60.

It takes Kris 60 minutes to stock his twelve shelves. All three boys are equal in their rate of speed. They decide to take turns buying the ice cream.

Vocabulary

Proportion
two equal ratios.
Ratio
a comparison of two quantities can be written in fraction form, with a colon or with the word “to”.
Cross Products
to multiply the diagonals of each ratio of a proportion.

Guided Practice

Here is one for you to try on your own.

\frac{2}{3} = \frac{x}{33}

Answer

To figure this out, you can look at the relationship between the denominators and the numerators. Then using mental math, you can solve for the missing value.

 3 \times 11 = 33

Therefore, we multiply the given numerator by 11 to find the unknown numerator.

Our answer is 22.

Video Review

Khan Academy, Solving Proportions

James Sousa, Introduction to Proportions

Practice

Directions: Use mental math to solve the unknown part of each proportion.

1. \frac{1}{2} = \frac{x}{8}

2. \frac{1}{2} = \frac{5}{x}

3. \frac{1}{3} = \frac{4}{x}

4. \frac{2}{3} = \frac{x}{6}

5. \frac{1}{2} = \frac{x}{16}

6. \frac{5}{6} = \frac{x}{12}

7. \frac{14}{16} = \frac{x}{8}

8. \frac{1}{2} = \frac{x}{18}

9. \frac{1}{4} = \frac{x}{20}

10. \frac{1}{4} = \frac{x}{24}

11. \frac{1}{4} = \frac{x}{40}

12. \frac{2}{4} = \frac{x}{40}

13. \frac{25}{50} = \frac{2}{x}

14. \frac{4}{12} = \frac{x}{48}

15. \frac{6}{7} = \frac{36}{x}

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