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Conversion of Systems of Measure

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While vacationing in Spain, you decide to rent a car. When you fill up the gas tank, you notice that the gas is measured in liters, not gallons, like you're used to. The cost of gas in Spain is 1.44 Euros per liter. If you get 65 Euros' worth of gas, how many gallons did you buy? There are 3.8 liters in a gallon.

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Khan Academy: Unit Conversion with Fractions

Guidance

One part of word problems is the unit of measure. It can be confusing when we don’t know what the problem is asking for. For example, how many feet are in a mile? How many cups are in a gallon? Here are a few conversions between different units of measure.

Units Equivalent Unit
2 cups (c) 1 pint
2 pints (pt) 1 quart
4 quarts (qt) 1 gallon (gal)
12 inches (in) 1 foot
3 feet (ft) 1 yard
1760 yards (yd) 1 mile (mi)
16 ounces (oz) 1 pound (lb)
2000 lbs 1 ton
100 centimeters (cm) 1 meter (m)
1000 meters 1 kilometer (km)
2.2 cm 1 inch

Example A

How many cups are in a gallon?

Solution: There are 2 cups in a pint, 2 pints in a quart and 4 quarts in a gallon.

\frac{2c}{1 pt} \cdot \frac{2pt}{1 qt} \cdot \frac{4 pt}{1 gal} Cancel out the like terms and multiply across.

\frac{2c}{\cancel{1 pt}} \cdot \frac{2 \cancel{pt}}{1 \cancel{qt}} \cdot \frac{4 \cancel{pt}}{1 gal} = \frac{16 c}{1 gal} There are 16 cups in one gallon.

Make sure to always cancel any units that are in the numerator and denominator of these fractions. Fractions like these are called unit rates because the base is one unit. We write out the unit conversion problems in this way so that we ensure that all of the correct units are cancelled out.

Example B

How many feet are in 16 yards?

Solution: This problem is not a conversion problem, but asking to extend your knowledge of how many feet are in a yard. We know that there are 3 feet in a yard; therefore there will be 3 \cdot 16 = 48 \ feet in 16 yards.

Another way to solve this problem is in a ratio, below:

\frac{3 ft}{1 yd} & = \frac{x ft}{16 yd} \ \text{To solve a ratio, we cross-multiply.}\\3ft \cdot 16 yd & = 1 yd \cdot x ft\\\frac{48 ft \cdot \cancel{yd}}{1 \cancel{yd}} & = x ft \ \text{Here,} \ x = 48 \ feet \ \text{and we show that the appropriate units cancel.}

Example C

You are making soup for 6 people. The recipe you are using calls for 6 cups of broth, but it is only for 4 people. How many cups of broth will you need?

Solution: Set up a ratio, like in the previous example.

\frac{6 people}{x cups} & = \frac{4 people}{6 cups} \\6 \cdot 6 c & = 4 \cdot x c\\36&=4x\\x&=9 cups

You will need 9 cups of broth.

Intro Problem Revisit First, let's figure out how many liters of gas you put in the car.

65 = 1.44l \\l&=65 \div 1.44 \\l&=45.14

Now, we need to divide again to get the number of gallons.

g = l \div 3.8\\

Substituting for l , we get:

g&=45.14 \div 3.8\\g&=11.88

You bought 11.88 gallons of gas.

Guided Practice

1. How many centimeters are in a foot?

2. How many ounces are in 3.5 pounds?

Answers

1. This problem is just like Example A. Set up the conversion.

\frac{2.2cm}{1 in} & \cdot \frac{12 in}{1 ft} \quad \text{Cancel out the inches and multiply.}\\\frac{2.2 cm}{1 \cancel{in}} & \cdot \frac{12 \cancel{in}}{1 ft}= \frac{26.4 cm}{1 ft}

2. This problem is just like Example B. If there are 16 ounces in a pound, then there will be 16 \cdot 3.5=56 \ ounces in 3.5 pounds.

Practice

For questions 1-6, set up a unit conversion to find:

  1. Feet in a mile?
  2. Cups in a quart?
  3. Centimeters in a kilometer?
  4. Pints in a gallon?
  5. Centimeters in a mile?
  6. Gallons in a quart?
  7. How many inches are in 5.25 yards?
  8. How many pints are in 7.5 gallons?
  9. How many pounds are in 2.6 tons?
  10. How many centimeters are in 4.75 meters?
  11. Claire is making chocolate chip cookies. If the recipe calls for 3.5 cups of flour, how many cups will Claire need to use if she triples the recipe?
  12. The recipe also calls for 8 oz. of chocolate chips. Claire wants to make the cookies with three-quarters bittersweet chips and one-quarter semi-sweet chips. Again, tripling the recipe, how many ounces of each type of chocolate chip will she need?

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