Unit 1, lesson 4

Units, Rates and Conversions

How to convert lengths, rates, areas and volumes by multiplying by conversion factors, keeping the units attached at every step so that they show whether the answer is right.

  • Theory part 7
  • Practice part 8
  • Numbers and Proportional Reasoning

Theory, part 7

Play the video

Nothing loads from YouTube until you press play.

Practice, part 8

Play the video

Nothing loads from YouTube until you press play.

A car on the highway shows 72 kilometers per hour. How many meters does it cover in one second? The number 72 alone cannot answer that, because a speed is a number together with its units. If you carry the units through the calculation like algebra, they cancel or remain just as algebraic factors do, and the units left at the end tell you whether you set the problem up correctly.

See also Geometry Formulas and Units in Foundations for Calculus.

Key terms

Unit

Definition A unit is a fixed, agreed amount of some kind of quantity, such as the meter for length, the second for time or the gram for mass. A measurement is a number times a unit, for example 5 m5 \text{ m} means 55 times one meter. Units behave like factors in algebra. They can be multiplied, divided and canceled, and only quantities of the same kind with the same unit can be added or subtracted directly.

In plain words The unit says what you are counting. A number without its unit is only half an answer.

Example Adding 2 m2 \text{ m} and 30 cm30 \text{ cm} as 2+30=322 + 30 = 32 is meaningless. Convert first, 30 cm=0.3 m30 \text{ cm} = 0.3 \text{ m}, so the total is 2.3 m2.3 \text{ m}, or equivalently 230 cm230 \text{ cm}.

Conversion factor

Definition A conversion factor is a fraction whose numerator and denominator are the same quantity written in two different units, such as

1000 m1 kmor1 km1000 m.\frac{1000 \text{ m}}{1 \text{ km}} \quad \text{or} \quad \frac{1 \text{ km}}{1000 \text{ m}}.

Its value is 11, so multiplying a measurement by it changes the number and the unit but not the amount. Choose the version with the unit you want to remove in the denominator, so that it cancels.

In plain words A conversion factor is a disguised 11. Flip it so the old unit cancels and the new unit stays.

Example Convert 3.2 km3.2 \text{ km} to meters.

3.2 km⋅1000 m1 km=3200 m.3.2 \text{ km} \cdot \frac{1000 \text{ m}}{1 \text{ km}} = 3200 \text{ m}.

Using the factor upside down would give 0.0032 km2m0.0032 \, \frac{\text{km}^2}{\text{m}}, a unit that makes no sense, which is a sign the factor is flipped.

Rate

Definition A rate is a ratio of two quantities measured in different units, such as kilometers per hour or grams per cubic centimeter. In a rate such as 72 km1 h\frac{72 \text{ km}}{1 \text{ h}}, the unit in the denominator is the reference unit, and the rate gives the amount of the numerator quantity for one reference unit. Quotient units such as density, mass per volume, are rates. Product units are formed by multiplying, for example a kilowatt-hour is the energy used by a power of 11 kilowatt over 11 hour.

In plain words “Per” means “divided by,” and the unit after “per” is what you count one of. Multiply a rate by an amount of its reference unit to get the other quantity.

Example Aluminum has a density of 2.72.7 grams per cubic centimeter. A piece with volume 40 cm340 \text{ cm}^3 has mass

2.7 gcm3⋅40 cm3=108 g.2.7 \, \frac{\text{g}}{\text{cm}^3} \cdot 40 \text{ cm}^3 = 108 \text{ g}.

Going the other way, a piece of mass 540 g540 \text{ g} has volume 540÷2.7=200 cm3540 \div 2.7 = 200 \text{ cm}^3. For a product unit, a 1.51.5 kilowatt heater running for 33 hours uses 1.5⋅3=4.51.5 \cdot 3 = 4.5 kilowatt-hours. At 0.160.16 dollars per kilowatt-hour that costs 4.5⋅0.16=0.724.5 \cdot 0.16 = 0.72 dollars.

Dimensional analysis

Definition Dimensional analysis is the method of converting a quantity by multiplying it by a chain of conversion factors, writing every unit, and canceling units that appear in both a numerator and a denominator. When every factor is a true conversion factor, the setup is right when exactly the requested units remain. Exact values should be kept until the end, and any rounding the question asks for is done only once, on the final answer.

In plain words Line up the factors so the units cancel like a domino chain. If the leftover units are right, the setup is right.

Example Convert the car’s 7272 kilometers per hour to meters per second. Kilometers go on the bottom to cancel, and hours go on top.

72 km1 h⋅1000 m1 km⋅1 h3600 s=72,000 m3600 s=20 ms.\begin{aligned} &\frac{72 \text{ km}}{1 \text{ h}} \cdot \frac{1000 \text{ m}}{1 \text{ km}} \cdot \frac{1 \text{ h}}{3600 \text{ s}} \\[0.4em] &\quad = \frac{72{,}000 \text{ m}}{3600 \text{ s}} = 20 \, \frac{\text{m}}{\text{s}}. \end{aligned}

A conversion with two unit changes works the same way. With 11 mile equal to 52805280 feet, a speed of 3030 feet per second is

30 ft1 s⋅3600 s1 h⋅1 mi5280 ft=22511 mih≈20.45 mih.\begin{aligned} &\frac{30 \text{ ft}}{1 \text{ s}} \cdot \frac{3600 \text{ s}}{1 \text{ h}} \cdot \frac{1 \text{ mi}}{5280 \text{ ft}} \\[0.4em] &\quad = \frac{225}{11} \, \frac{\text{mi}}{\text{h}} \approx 20.45 \, \frac{\text{mi}}{\text{h}}. \end{aligned}

Keep the exact fraction 22511\frac{225}{11} if the answer is used again later, and round only when the question asks.

Squared unit

Definition A squared unit such as m2\text{m}^2 is the area of a square whose side is 11 of the length unit. If 1 A=c B1 \text{ A} = c \text{ B} for two length units A and B, then

1 A2=c2 B2,1 \text{ A}^2 = c^2 \text{ B}^2,

so area conversions use the square of the length conversion factor. Equivalently, apply the length factor twice.

In plain words A square meter is 100100 centimeters on each side, so it holds 100⋅100100 \cdot 100 square centimeters, not 100100.

Example A floor is 1212 feet by 1515 feet, so its area is 180 ft2180 \text{ ft}^2. With 1 yd=3 ft1 \text{ yd} = 3 \text{ ft}, one square yard is 9 ft29 \text{ ft}^2, and

180 ft2⋅1 yd29 ft2=20 yd2.180 \text{ ft}^2 \cdot \frac{1 \text{ yd}^2}{9 \text{ ft}^2} = 20 \text{ yd}^2.

Dividing by 33 instead would give 6060, the linear scaling error. A check with the sides confirms the right answer, since the floor is 44 yards by 55 yards.

Cubed unit

Definition A cubed unit such as cm3\text{cm}^3 is the volume of a cube whose edge is 11 of the length unit. If 1 A=c B1 \text{ A} = c \text{ B} for two length units, then

1 A3=c3 B3,1 \text{ A}^3 = c^3 \text{ B}^3,

so volume conversions use the cube of the length conversion factor. For example 1 m3=1003 cm3=1,000,000 cm31 \text{ m}^3 = 100^3 \text{ cm}^3 = 1{,}000{,}000 \text{ cm}^3. The liter is defined by 1 L=1000 cm31 \text{ L} = 1000 \text{ cm}^3, so 1 m3=1000 L1 \text{ m}^3 = 1000 \text{ L}.

In plain words A cube that is 1010 small cubes along each edge holds 10⋅10⋅10=100010 \cdot 10 \cdot 10 = 1000 small cubes. Volume grows with the cube of the scale.

Example A tank measures 2 m2 \text{ m} by 1.5 m1.5 \text{ m} by 0.8 m0.8 \text{ m}. Its volume is 2.4 m32.4 \text{ m}^3, and in liters

2.4 m3⋅1000 L1 m3=2400 L.2.4 \text{ m}^3 \cdot \frac{1000 \text{ L}}{1 \text{ m}^3} = 2400 \text{ L}.

In cubic centimeters it is 2.4⋅1,000,000=2,400,000 cm32.4 \cdot 1{,}000{,}000 = 2{,}400{,}000 \text{ cm}^3. Converting each edge to centimeters first, 200⋅150⋅80200 \cdot 150 \cdot 80, gives the same number.

Check yourself

  1. A cyclist rides at 5454 kilometers per hour. What is this speed in meters per second? (A) 1515 (B) 54,00054{,}000 (C) 194.4194.4 (D) 0.90.9
Show the answer

The answer is (A), since

54⋅10003600=15.54 \cdot \frac{1000}{3600} = 15.

Choice (B) converts kilometers to meters but forgets the hours. Choice (C) multiplies by 3.63.6 instead of dividing, and choice (D) divides by 6060 as if only minutes were involved.

  1. A rug has an area of 2.52.5 square yards. What is its area in square feet? Use 11 yard =3= 3 feet.
Show the answer

One square yard is 32=93^2 = 9 square feet, so the area is

2.5⋅9=22.5 square feet.2.5 \cdot 9 = 22.5 \text{ square feet.}

The value 7.57.5 comes from using the linear factor 33, which is wrong for area.

  1. A cube has edges of length 3030 centimeters. What is its volume in cubic meters? (A) 0.0270.027 (B) 0.30.3 (C) 2.72.7 (D) 27,00027{,}000
Show the answer

The answer is (A). The edge is 0.3 m0.3 \text{ m}, so the volume is 0.33=0.027 m30.3^3 = 0.027 \text{ m}^3. Choice (B) is the edge length, not the volume. Choice (C) divides 27,000 cm327{,}000 \text{ cm}^3 by 10,00010{,}000, the area factor, and choice (D) is the volume in cubic centimeters.

How this lesson connects