3F Unit Conversions

If you talked to most science teachers, one of the things they need their students to do is to be able to convert from one unit of measurement to another. For example, in a physics class, it is common to need to convert from miles per hour (mph) to feet per second (fps). How do we convert units? Fractions give us a systematic way to make the multiplicative unit conversions in this section.

To convert units, take two quantities that are equal and make a fraction out of them. This is sometimes called a unit fraction because it equals one. The key is you can use the unit fraction to convert units. Let’s look at an example.

Suppose we want to convert 34 feet into inches. Start with the two equal quantities. How many inches in a foot? 12. So we know that 1 ft = 12 in. We can make two unit fractions out of the equal quantities $\frac{1\text{ ft}}{12\text{ in}}\text{ or }\frac{12\text{ in}}{1\text{ ft}}$. Now start with the quantity we want to convert (34 ft). We want to multiply 34 ft by one of the two unit fractions. The key though is we need the units to cancel. So we need to pick the unit fraction that has feet in the denominator. Now multiply by the unit fraction and simplify. Your conversion is complete.

$$34\text{ ft}\times \frac{12\text{ in}}{1\text{ ft}}=34ft\times \frac{12\text{ in}}{1ft}=\frac{34}{1}\times \frac{12\text{ in}}{1}=\frac{34\times 12\text{ in}}{1\times 1}=\frac{408}{1}\text{ in}=408\text{ in}$$

Notice that the key was to cancel the feet and be left with inches. This process of cancelling units is often called dimensional analysis. Once we cancelled the units, we just needed to multiply the fractions and simplify. Remember if you are multiplying a whole number, such as 34, times a fraction, we can rewrite the 34 as a fraction $\frac{34}{1}$. This is also a common technique when converting units.

Let’s look at a second example: Suppose we have 10.5 cups of solution in our chemistry class. How many liters is this? To convert the units, we need to know how many cups are in 1 Liter. You can often look these quantities up in a book or online. We found that $1\text{ Liter}\approx 4.2\text{ cups}$. Let’s use this to make the conversion.

Again, use the supplied rounded conversion factor to make a fraction. Since we are trying to convert cups to something else, we need a unit fraction with cups in the denominator. $\left(\frac{1\text{ Liter}}{4.2\text{ cups}}\right)$

Now all we have to do is multiply the 10.5 cups by our unit fraction.

$$10.5\text{ cups}\times \frac{1L}{4.2\text{cups}}=10.5\text{cups}\times \frac{1L}{4.2\text{cups}}=\frac{10.5}{1}\times \frac{1}{4.2}L=\frac{10.5\times 1}{1\times 4.2}L=\frac{10.5}{4.2}L=2.5L$$

Notice again, we cancelled the cups and were left with liters. We wrote the decimal 10.5 as a fraction $\frac{10.5}{1}$ and then multiplied the fractions. Notice to simplify at the end we needed to divide decimals.

Now you try a few examples with your instructor.

Practice Problems Section 3F

1. Convert $3⁤\frac{1}{4}$ feet into inches. (1 ft = 12 in)

   Check answer

   39 in

2. Convert 54 inches into feet. Write your answer as a mixed number. (1 ft = 12 in)

3. Convert 3.2 kilograms into grams. (1 kg = 1000 g)

   Check answer

   3,200 g

4. Convert 4600 grams into kilograms. (1 kg = 1000 g)

5. Convert 22 quarts into gallons. Write your answer as a mixed number. ( 1 gal = 4 qt )

   Check answer

   $5⁤\frac{1}{2}$ gal

6. Convert $4⁤\frac{3}{4}$ gallons into quarts. ( 1 gal = 4 qt )

7. Convert $3⁤\frac{3}{4}$ tons into pounds. (1 ton = 2000 pounds)

   Check answer

   7,500 pounds

8. Convert 3500 pounds into tons. Write your answer as a mixed number. (1 ton = 2000 pounds)

9. Convert 10.5 inches into centimeters. $(1\text{ in}=2.54\text{ cm})$ Write your answer as a decimal.

   Check answer

   26.67 cm

10. Convert 41.91 centimeters into inches. $(1\text{ in}=2.54\text{ cm})$ Write your answer as a decimal.

11. Convert 58.8 feet per second into miles per hour. (1 mph ≈ 1.47 fps)

    Check answer

    40 mph

12. Convert 65 miles per hour into feet per second. Write your answer as a decimal. (1 mph ≈ 1.47 fps)

13. Convert 2.468 grams into milligrams. (1 g = 1000 mg)

    Check answer

    2,468 mg

14. Convert 759 milligrams into grams. Write your answer as a decimal. (1 g = 1000 mg)

15. Convert 15.4 pounds into kilograms. (1 kg ≈ 2.2 pounds)

    Check answer

    7 kg

16. Convert 25 kilograms into pounds. (1 kg ≈ 2.2 pounds)

17. Convert 80 kilometers per hour into miles per hour. (1 kph ≈ 0.62 mph)

    Check answer

    49.6 mph

18. Convert 31 miles per hour into kilometers per hour. (1 kph ≈ 0.62 mph)

19. Convert $7⁤\frac{2}{3}$ feet into yards. Write your answer as a mixed number. (3 ft = 1 yd)

    Check answer

    $2⁤\frac{5}{9}$ yd

20. Convert $4⁤\frac{1}{3}$ yards into feet. (3 ft = 1 yd)

21. Convert 5,600 milliliters into liters. Write your answer as a decimal. (1 L = 1000 mL)

    Check answer

    5.6 L

22. Convert 8.25 liters into milliliters. (1 L = 1000 mL)

23. Convert $7⁤\frac{1}{4}$ quarts into cups. ( 1 quart = 4 cups )

    Check answer

    29 cups

24. Convert $13⁤\frac{1}{2}$ cups into quarts. Write your answer as a mixed number. ( 1 quart = 4 cups )

25. Convert 3,500 pounds into tons. (1 ton = 2000 pounds) Write your answer as a mixed number.

    Check answer

    $1⁤\frac{3}{4}$ ton

26. Convert 2.75 tons into pounds. (1 ton = 2000 pounds) Write your answer as a whole number.

27. Convert 21.59 centimeters into meters. Write your answer as a decimal. (1 m = 100 cm)

    Check answer

    0.2159 m

28. Convert 3.6 meters into centimeters. (1 m = 100 cm)

29. Convert 65 miles per hour into feet per second. (1 mph ≈ 1.47 fps)

    Check answer

    95.55 fps

30. Convert 3450.8 milligrams into grams. (1 g = 1000 mg)

31. Convert 10.6 kilograms into pounds. (1 kg ≈ 2.2 pounds) Write your answer as a decimal.

    Check answer

    23.32 pounds

32. Convert 52.7 miles per hour into kilometers per hour. ( 1 kph ≈ 0.62 mph )

33. Convert $5⁤\frac{3}{4}$ yards into inches. (1 yard = 36 inches)

    Check answer

    207 in

34. Convert 126 inches into yards. Write your answer as a mixed number. (1 yard = 36 inches)

35. Convert 80 kilometers per hour into feet per second. Write your answer as a decimal.$(1\text{ kph}\approx 0.911\text{ fps})$

    Check answer

    72.88 fps

36. Convert 25 meters into feet. Write your answer as a decimal. $(1m\approx 3.28\text{ ft})$

37. Convert 175 meters into feet. Write your answer as a decimal. $(1m\approx 3.28\text{ ft})$

    Check answer

    574 ft

38. Convert 150.88 feet into meters. $(1m\approx 3.28\text{ ft})$

39. In hospitals, another name for a milliliter is a “CC” (cubic centimeter). A nurse gives fluid to a patient at the rate of 1.5 Liters per hour. How many CC’s per minute is this? To set this up, complete the following fraction problem. Make sure to cancel the units.

    $$\frac{1.5\mathrm{L}}{\text{1 hour}}\times \frac{1000\text{ CC}}{1\mathrm{L}}\times \frac{\text{1 hour}}{60\text{ min}}=??$$

    Check answer

    25 CC per min

40. Let’s try another hospital problem. A nurse needs to give a patient 4.5 Liters of fluid in 6 hours. How many CC’s per minute is this? To set this up, complete the following fraction problem. Make sure to cancel the units. Write your answer as a decimal.

    $$\frac{4.5\mathrm{L}}{\text{6 hour}}\times \frac{1000\text{ CC}}{1\mathrm{L}}\times \frac{\text{1 hour}}{60\text{ min}}=??$$

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.