A ratio compares two quantities, a rate compares quantities with different units, and a proportion says two ratios are equal. Together they cover recipes, maps, prices, speeds and a large share of SAT and ACT word problems. The skill that matters most is setting the problem up correctly. The arithmetic after that is short.
Ratios
A ratio of 3 to 5 can be written 3:5, 3 to 5, or 3/5. Order matters: if a class has 12 girls and 18 boys, the ratio of girls to boys is 12:18, and boys to girls is 18:12.
Simplify a ratio like a fraction, by dividing both parts by their greatest common factor. 12:18 divides by 6 to give 2:3.
Example A class has 12 girls and 18 boys. What fraction of the class is girls?
The ratio of girls to boys is 2:3, so out of every 2 + 3 = 5 students, 2 are girls. The fraction is 2/5, or 40%.
Part-to-part is not part-to-whole
A girls-to-boys ratio of 2:3 does not mean 2/3 of the class is girls. Add the parts to get the whole first.
Sharing in a ratio
To split an amount in a given ratio, add the parts, find the value of one part, then multiply.
Example Split $240 between Ana and Ben in the ratio 5:3
- Total parts: 5 + 3 = 8.
- One part: 240 ÷ 8 = $30.
- Ana gets 5 × 30 = $150 and Ben gets 3 × 30 = $90.
- Check: 150 + 90 = 240. ✓
Rates and unit rates
A rate has units: 150 miles in 3 hours, $4.50 for 3 pounds. A unit rate is the rate for one unit of the second quantity, found by dividing.
- 150 miles ÷ 3 hours = 50 miles per hour.
- $4.50 ÷ 3 pounds = $1.50 per pound.
Unit rates make comparisons easy.
Example Which is the better buy: 12 oz for $3.48 or 20 oz for $5.40?
- 3.48 ÷ 12 = $0.29 per ounce.
- 5.40 ÷ 20 = $0.27 per ounce.
- The 20 oz size is cheaper per ounce.
Proportions and cross-multiplying
A proportion sets two ratios equal: a/b = c/d. To solve for a missing value, cross-multiply: a × d = b × c.
Example Solve x/12 = 5/8
- Cross-multiply: 8x = 60.
- Divide by 8: x = 7.5.
The real skill is writing the proportion so the units line up. Put the same kind of quantity in the same position on both sides.
Example A recipe uses 3 cups of flour for 24 cookies. How much flour is needed for 40 cookies?
- Flour over cookies on both sides: 3/24 = f/40.
- Cross-multiply: 24f = 120.
- Divide: f = 5 cups.
Example On a map, 2 cm represents 15 km. Two towns are 9 cm apart on the map. How far apart are they?
- Map over real on both sides: 2/15 = 9/d.
- Cross-multiply: 2d = 135.
- d = 67.5 km.
This is the same idea behind similar triangles, where matching sides are proportional. The Pythagorean theorem guide and the SOH CAH TOA guide both rely on it.
Unit conversion
Converting units is multiplying by fractions equal to 1. Write each conversion factor so the unit you want to get rid of cancels.
Example Convert 60 miles per hour to feet per second
- 60 miles/hour × 5,280 feet/1 mile × 1 hour/3,600 seconds.
- Miles and hours cancel, leaving feet per second.
- 60 × 5,280 ÷ 3,600 = 88 feet per second.
If a unit does not cancel, the factor is upside down. Flip it.
Direct and inverse variation
Two quantities vary directly when their ratio stays the same: y = kx. Double x and y doubles. Pay and hours worked at a fixed wage vary directly.
Two quantities vary inversely when their product stays the same: y = k/x. Double x and y halves. Speed and travel time for a fixed distance vary inversely.
Example y varies directly with x, and y = 18 when x = 4. Find y when x = 10.
- k = y/x = 18/4 = 4.5.
- y = 4.5 × 10 = 45.
Example It takes 6 workers 10 days to build a wall. How long would 4 workers take, at the same pace?
- Workers × days stays the same: 6 × 10 = 60 worker-days.
- 4 × d = 60, so d = 15 days.
- Setting up 6/10 = 4/d would give fewer days for fewer workers, which makes no sense. Always ask whether the answer should go up or down.
Percentages are ratios out of 100, so the same setup solves percent problems too. The percentages guide covers percent change, discounts and interest.
Check yourself
Simplify the ratio 45:60.
Show answer
3:4. Both parts divide by 15.
A bag has red and blue marbles in the ratio 4:7. There are 66 marbles. How many are blue?
Show answer
42. There are 11 parts, so one part is 6 marbles, and blue is 7 × 6 = 42.
Solve (x + 2)/9 = 4/6.
Show answer
x = 4. Cross-multiply: 6(x + 2) = 36, so x + 2 = 6.
A car uses 8 gallons to travel 236 miles. How many gallons does it need for 413 miles?
Show answer
14 gallons. 8/236 = g/413, so 236g = 3,304 and g = 14.
Convert 3 square feet to square inches.
Show answer
432 square inches. One square foot is 12 × 12 = 144 square inches, and 3 × 144 = 432.
y varies inversely with x, and y = 12 when x = 5. Find y when x = 15.
Show answer
y = 4. k = xy = 60, so y = 60/15.