Lesson 5: Systems of Measurement

New Concept

This lesson introduces unit conversion, a key skill for working with U.S. and metric systems. You'll learn how to change units—like feet to miles or grams to kilograms—by multiplying by a clever form of 1, the multiplicative identity.

What’s next

Next, you'll practice converting units with interactive examples, starting with the U.S. system and then moving to the metric system and temperature conversions.

Property

To make unit conversions, we use the Identity Property of Multiplication. For any real number $a$, $a \cdot 1 = a$ and $1 \cdot a = a$. We write 1 as a fraction to change the units without changing the value.

To make unit conversions:

1. Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.
2. Multiply.
3. Simplify the fraction, performing the indicated operations and removing the common units.

Examples

• To convert 72 inches to feet, you multiply by a fraction that cancels inches: $72 \text{ in} \cdot \frac{1 \text{ ft}}{12 \text{ in}} = 6 \text{ ft}$.
• An African elephant weighs 4.5 tons. To find its weight in pounds, you use the conversion $1 \text{ ton} = 2000 \text{ lbs}$: $4.5 \text{ tons} \cdot \frac{2000 \text{ lbs}}{1 \text{ ton}} = 9000 \text{ lbs}$.
• To find how many minutes are in 3 weeks, you chain conversions together: $\frac{3 \text{ wk}}{1} \cdot \frac{7 \text{ days}}{1 \text{ wk}} \cdot \frac{24 \text{ hr}}{1 \text{ day}} \cdot \frac{60 \text{ min}}{1 \text{ hr}} = 30,240 \text{ min}$.

Property

In the metric system, units are related by powers of 10. The prefixes reflect this relationship (e.g., kilo- for 1000, centi- for $\frac{1}{100}$). Conversions use the same method as the U.S. system, multiplying by a conversion factor of 1.

Because the system is base-10, you can also convert by moving the decimal point:

• To multiply by 10, 100, or 1000, move the decimal to the right 1, 2, or 3 places.
• To divide by 10, 100, or 1000, move the decimal to the left 1, 2, or 3 places.

Examples

• A 5-kilometer race is $5000$ meters long. Since $1 \text{ km} = 1000 \text{ m}$, we move the decimal 3 places to the right: $5.0 \rightarrow 5000$.
• A baby weighs 3400 grams. To convert to kilograms, we know $1 \text{ kg} = 1000 \text{ g}$. We move the decimal 3 places to the left: $3400 \rightarrow 3.4 \text{ kg}$.
• To convert 2.5 liters to milliliters, we know $1 \text{ L} = 1000 \text{ mL}$. Move the decimal 3 places to the right: $2.5 \rightarrow 2500 \text{ mL}$.

Property

To convert between the U.S. and metric systems, we multiply by unit conversion factors, just as we do within each system. The conversion factors are approximations for everyday use.

Common Conversion Factors:

• Length: $1 \text{ in} = 2.54 \text{ cm}$; $1 \text{ mi} = 1.61 \text{ km}$
• Weight: $1 \text{ lb} = 0.45 \text{ kg}$; $1 \text{ kg} = 2.2 \text{ lb}$
• Volume: $1 \text{ qt} = 0.95 \text{ L}$; $1 \text{ L} = 1.06 \text{ qt}$

Examples

• A bottle holds $750 \text{ mL}$ of liquid. To convert to fluid ounces, use the factor $1 \text{ fl oz} = 30 \text{ mL}$: $750 \text{ mL} \cdot \frac{1 \text{ fl oz}}{30 \text{ mL}} = 25 \text{ fl oz}$.
• A sign says a destination is 50 kilometers away. To convert to miles, use $1 \text{ mi} = 1.61 \text{ km}$: $50 \text{ km} \cdot \frac{1 \text{ mi}}{1.61 \text{ km}} \approx 31 \text{ mi}$.
• A person weighs 150 pounds. To find their mass in kilograms, use $1 \text{ lb} = 0.45 \text{ kg}$: $150 \text{ lb} \cdot \frac{0.45 \text{ kg}}{1 \text{ lb}} = 67.5 \text{ kg}$.

Property

To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula:

To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula:

Examples

• To convert a hot day of $95°\text{F}$ to Celsius, use the formula: $C = \frac{5}{9}(95 - 32) = \frac{5}{9}(63) = 35°\text{C}$.
• If a European weather forecast predicts $15°\text{C}$, you can find the Fahrenheit temperature: $F = \frac{9}{5}(15) + 32 = 27 + 32 = 59°\text{F}$.
• To convert a freezing temperature of $14°\text{F}$ to Celsius, calculate: $C = \frac{5}{9}(14 - 32) = \frac{5}{9}(-18) = -10°\text{C}$.