Topic 1.1 · Unit 1 · CA CCSS N-Q.1–3
Quantities, units and unit conversion
The arithmetic here is easy and the setup is where the marks go. Write every conversion as a fraction and let the units cancel, rather than deciding in your head whether to multiply or divide.
- Write what you have as a fraction, even when it looks like one number. 45 miles per hour is \( \frac{45 \text{ mi}}{1 \text{ hr}} \).
- Write each conversion as a fraction equal to 1. Since \( 1 \text{ mi} = 5280 \text{ ft} \), both \( \frac{5280 \text{ ft}}{1 \text{ mi}} \) and \( \frac{1 \text{ mi}}{5280 \text{ ft}} \) equal 1, so multiplying by either changes the units and never the value.
- Orient it so the unit you want gone cancels. If "mi" is on top of what you have, put "mi" on the bottom of the conversion.
- Cancel, then multiply. Whatever units survive are your answer's units. If they are not the ones asked for, a fraction is upside down.
- Check the size. Converting to a smaller unit gives a bigger number.
Where marks are lost: squared and cubed units. To convert \( \text{ft}^2 \) to \( \text{in}^2 \) the conversion must be squared too, \( \left(\frac{12 \text{ in}}{1 \text{ ft}}\right)^2 = \frac{144 \text{ in}^2}{1 \text{ ft}^2} \), not 12. Using 12 gives an answer twelve times too small.
Example 1: a rate with both units changing. Express 60 miles per hour in feet per second.
Two units change, so two conversion fractions. Miles cancels from the top, hours from the bottom:
\( \frac{60 \text{ mi}}{1 \text{ hr}} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = \frac{60 \times 5280}{3600} = 88 \text{ ft/s} \)
Only ft over s survives, which is what was asked. Sanity check: 88 is large for a modest speed, which is right, because feet and seconds are both small units.
Example 2: a squared unit. Convert 12 square meters to square centimeters.
\( 12 \text{ m}^2 \times \left(\frac{100 \text{ cm}}{1 \text{ m}}\right)^2 = 12 \times 10000 = 120{,}000 \text{ cm}^2 \)
Multiplying by 100 instead of 10 000 is the standard error and lands a hundredfold short.
Example 3: precision. A recipe for 4 uses 350 g of flour. How much for 7?
\( 350 \times \frac{7}{4} = 612.5 \text{ g} \). But the original was given to the nearest 10 g, so 612.5 claims precision the source never had. State it as about 610 g, an answer should not be more precise than the data behind it.
Work each one on paper first. Every solution shows the full setup, so a wrong answer can be traced to the line that caused it.
- Convert 5 kilometers to meters.
Show the full solution
\( 5 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 5000 \text{ m} \). Meters are smaller, so the number grows. 5000 m
- Convert 3 hours to seconds.
Show the full solution
\( 3 \times 60 \times 60 = 10800 \). 10 800 s
- A runner covers 400 meters in 50 seconds. Find the speed in meters per second.
Show the full solution
\( \frac{400 \text{ m}}{50 \text{ s}} = 8 \text{ m/s} \). 8 m/s
- Convert 90 kilometers per hour to meters per second.
Show the full solution
\( \frac{90 \text{ km}}{1 \text{ hr}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = \frac{90000}{3600} = 25 \). 25 m/s
- Convert 4 square meters to square centimeters.
Show the full solution
Square the conversion: \( 4 \times 100^2 = 4 \times 10000 = 40000 \). 40 000 cm²
- Convert 2 cubic feet to cubic inches.
Show the full solution
Cube it: \( 2 \times 12^3 = 2 \times 1728 = 3456 \). 3456 in³
- Gasoline costs $3.60 per gallon. Express this in dollars per liter, given 1 gal = 3.785 L.
Show the full solution
Only the bottom unit changes, so gallons goes on top of the conversion to cancel: \( \frac{\$3.60}{1 \text{ gal}} \times \frac{1 \text{ gal}}{3.785 \text{ L}} \approx \$0.95 \). A liter is smaller than a gallon, so the price per liter must be smaller, a useful check. about $0.95/L
- A field is 120 m by 75 m. Find its area in hectares (1 ha = 10 000 m²).
Show the full solution
Area first: \( 120 \times 75 = 9000 \text{ m}^2 \). Then \( 9000 \times \frac{1}{10000} = 0.9 \). 0.9 ha
- In \( d = rt \), \( r \) is in miles per hour and \( t \) in hours. What units does
\( d \) have, and why?
Show the full solution
\( \frac{\text{mi}}{\text{hr}} \times \text{hr} \), the hours cancel and miles survives. Worth doing on formulas as well as conversions: if the surviving units are not a distance, the formula has been used wrongly. miles
- A car uses 8 liters per 100 km. How much fuel for a 250 km journey?
Show the full solution
\( \frac{8 \text{ L}}{100 \text{ km}} \times 250 \text{ km} = 20 \). The km cancels. 20 L
- A printer produces 18 pages per minute. How long, in hours, to print 5400 pages?
Show the full solution
Invert the rate so pages cancel: \( 5400 \times \frac{1 \text{ min}}{18} = 300 \text{ min} \), then \( 300 \div 60 = 5 \). 5 hours
- A scale drawing uses 1 cm for 2.5 m. A wall measures 6.4 cm on the drawing. How long is
the real wall?
Show the full solution
The scale is a conversion fraction like any other: \( 6.4 \times 2.5 = 16 \). 16 m
- Water flows at 2.5 cubic meters per minute. Express this in liters per second
(1 m³ = 1000 L).
Show the full solution
\( \frac{2.5 \text{ m}^3}{1 \text{ min}} \times \frac{1000 \text{ L}}{1 \text{ m}^3} \times \frac{1 \text{ min}}{60 \text{ s}} = \frac{2500}{60} \approx 41.7 \). about 41.7 L/s
- A student converts 5 m² to cm² and gets 500 cm². Find the error and give the correct
answer.
Show the full solution
They multiplied by 100 once, treating m² as m. The unit is squared so the conversion must be squared: \( 5 \times 100^2 = 50000 \). Their answer is a hundred times too small. 50 000 cm²
- A time is measured as 3.0 seconds. A student computes \( 100 \div 3.0 = 33.33333 \) m/s
and writes every digit. Why is that wrong, and what should they write?
Show the full solution
The time was measured to two significant figures, so the result cannot be known to seven, the extra digits claim precision the measurement never had. Round to match the least precise input. 33 m/s