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Conversion Using Unit Analysis

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Conversion Using Unit Analysis
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Have you ever wondered how much of one unit is found in another unit? Sound confusing? Take a look.

Jeff made 4 liters of lemonade for a party. He is wondering how many milliliters there are in his lemonade.

Do you know how to figure this out?

Using proportions and unit analysis is the way to solve this problem. This Concept will teach you what you need to know.

Guidance

Proportions can help us convert from one unit of measure to another. For example, suppose you needed to convert from liters to milliliters.

A pitcher holds 4 liters of water. Determine how many milliliters of water the pitcher holds. Use the unit conversion: 1 liter = 1000 milliliters

First, let's set up a proportion to solve this problem.

The first ratio can use the unit conversion and compare liters to milliliters. The second ratio can compare 4 liters to the unknown number of milliliters, n .

\frac{liters}{milliliters} = \frac{1}{1000} \qquad \quad \frac{liters}{milliliters} = \frac{4}{n}

These are equivalent ratios, so we can use them to write a proportion.

\frac{1}{1000} = \frac{4}{n}

Consider which strategy to use. Should we solve for d by using proportional reasoning? Or should we cross multiply?

The relationship between the terms in the numerators is easy to see––we can multiply 1 by 4 to get 4. So, the computation will probably be simpler if we use proportional reasoning and multiply both terms of the first ratio by 4.

\frac{1}{1000} = \frac{1 \times 4}{1000 \times 4} = \frac{4}{4000} = \frac{4}{n}

From the work above, we can see that when the first term is 4, the second term is 4000. So, n = 4000 .

The pitcher holds 4000 milliliters of water.

Another strategy for solving a problem like the one with the pitcher is to use unit analysis . In unit analysis, we write ratios as fractions, just as we do when we write a proportion. However, in unit analysis, we do not want the terms in the fractions to be consistent. Instead, the units in the fractions are written so that certain units cancel one another out.

This will be easier to understand if we consider an example. Let’s go back to the last example with the liters and the pitcher. We can use unit analysis to solve it.

The problem requires us to convert 4 liters to milliliters. Our answer should be in milliliters, so the number of milliliters is unknown.

The measure we are given is 4 liters.

We know that 1 liter (L) = 1000 milliliters (mL). This can be expressed as either \frac{1L}{1000mL} or \frac{1000mL}{1L} . Each of these is a possible conversion factor by which we might multiply 4 liters.

We should start by writing 4 liters as a fraction over 1. We can do this because 4L = \frac{4L}{1} .

\frac{4L}{1}

We want our answer to be in milliliters, not in liters. So, we want the liters to cancel each other out. Since liters is in the numerator of the fraction above, we should make sure that liters is in the denominator of the conversion factor we use, like this:

\frac{4L}{1} \times \frac{1000 mL}{1L}

Since liters appears in the numerator of one factor and in the denominator of another factor, we can cancel them out, like this:

\frac{4 \bcancel{L}}{1} \times \frac{1000mL}{1 \bcancel{L}}

Now we can multiply what is left as we would multiply any fractions.

\frac{4}{1} \times \frac{1000mL}{1} = \frac{4 \times 1000 mL}{1 \times 1} = \frac{4000mL}{1} = 4000 mL

The pitcher holds 4000 milliliters of water.

Now it's your turn to try a few conversions.

Example A

How many milliliters in 2.5 liters? Write a proportion and solve.

Solution: 2500 mL

Example B

How many meters in 11 kilometers? Write a proportion and solve.

Solution: 11,000 meters

Example C

How many inches are there in 18 feet? Use a proportion and solve.

Solution: 216 inches

Here is the original problem once again.

Jeff made 4 liters of lemonade for a party. He is wondering how many milliliters there are in his lemonade.

Do you know how to figure this out?

To figure this out, we can start with the unit scale from liters to milliliters.

\frac{1}{1000}

This means that there are 1000 milliliters in 1 liter.

Now we can set up a proportion.

\frac{1}{1000}={4}{x}

Cross multiply and solve.

4000

There are 4000 mL in the lemonade container.

Vocabulary

Unit Analysis
comparing the number of units by using fractions and canceling out common values.
Scaling Strategy
looking at a common number of units to figure out a best buy. You may compare the amount in two different objects to a common unit and then figure out the best value based on the comparison.

Guided Practice

Here is one for you to try on your own.

Arnaldo needs to buy olive oil. He could buy a 15-ounce bottle of Brand A olive oil for $3, or he could buy a 20-ounce bottle of Brand B olive oil for $5. Which is the better buy?

Answer

One way to solve this problem is to find the unit price for each bottle.

Find the unit price for the 15-ounce bottle. Remember, you can find the unit price by dividing the first term by the second term.

&\$ 3 \ \text{for} \ 15 \ oz = \frac{\$3}{15oz} && \overset{ \ \ \$0.20}{15 \overline{ ) {\$3.00 \;}}}\\&&& \quad \underline{-30\;\;\;\;}\\&&& \qquad \ \ \ \ 0\\&&& \qquad  \ \ \underline{-0\;\;}\\&&& \qquad \quad \ \  0

Find the unit price for the 20-ounce bottle.

&\$ 5 \ \text{for} \ 20 \ oz = \frac{\$5}{20oz} && \overset{ \ \ \$0.25}{20 \overline{ ) {\$5.00 \;}}}\\&&& \quad \ \underline{-40\;\;\;\;}\\&&& \qquad \ 100\\&&& \quad  \ \ \underline{-100\;\;}\\&&& \qquad \quad \ 0

Since $0.20 < $0.25, the 15-ounce bottle of Brand A olive oil has the cheaper unit price and is the better buy.

Finding the unit rate is not the only strategy we could have used to solve a problem like the one in the last example. Instead of determining the unit rate, we could have imagined buying several bottles of each brand until we had the same number of ounces of Brand A oil as Brand B oil. Then we could have compared those costs. This strategy is known as the scaling strategy . Take a look at the next problem.

Find a common number of ounces for both brands.

The first few multiples of 15 are: 15, 30, 45, 60, and 75.

The first few multiples of 20 are: 20, 40, 60, 80, and 100.

The least common multiple of 15 and 20 is 60. So, we can find the cost of buying 60 ounces of each brand of oil.

A 15-ounce bottle of Brand A oil costs $3.

15 \times 4 = 60 , so \frac{ounces}{price} = \frac{15}{3} = \frac{15 \times 4}{3 \times 4} = \frac{60}{12} .

The cost of 60 ounces of Brand A oil (four 15-ounce bottles of oil) is $12.

A 20-ounce bottle of Brand B oil costs $5.

20 \times 3 = 60 , so \frac{ounces}{price} = \frac{20}{5} = \frac{20 \times 3}{5 \times 3} = \frac{60}{15} .

The cost of 60 ounces of Brand B oil (three 20-ounce bottles of oil) is $15.

Since $12 < $15, it would cost less to buy 60 ounces of Brand A olive oil than to buy 60 ounces of Brand B olive oil. So, the 15-ounce bottle of Brand A olive oil is the better buy.

Practice

Directions : Use unit analysis to solve each problem.

1. How many feet in 1 mile?

2. How many feet in 18.5 miles?

3. How many milliliters in 3.75 liters?

4. How many milliliters in 18.25 liters?

5. How many pounds in 3 tons?

6. How many pounds in 2.5 tons?

7. How many pounds in 4.75 tons?

8. How many feet in 18 yards?

9. How many inches in 4 feet?

10. How many inches in 8.75 feet?

11. How many milliliters in 29.5 liters?

Directions : Solve each problem.

12. Fred needs to buy vanilla extract to bake a cake. He could buy a 4-ounce bottle of vanilla extract for $8, or a 6-ounce bottle of vanilla extra for $15. Which bottle is the better buy?

13. A rope is 3 yards long. How many inches long is the rope? Use these unit conversions: 1 yard = 3 feet and 1 foot = 12 inches.

14. At the farmer's market, Maureen can buy 6 ears of corn for $3. At that price, how much would it cost to buy 9 ears of corn?

15. James bought a 128-ounce bottle of apple juice. How many pints of apple juice did James buy? Use these unit conversions: 1 cup = 8 fluid ounces and 1 pint = 2 cups.

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