2.3 Ratios, Proportions & Unit Conversion
Key Takeaways
- A ratio compares two quantities (a to b, written a:b or a/b) and should be reduced to lowest terms like a fraction.
- A proportion sets two ratios equal; solve it by cross-multiplication, where a/b = c/d gives a × d = b × c.
- A rate compares quantities with different units, and a unit rate has a denominator of 1 (e.g., 60 miles per 1 hour).
- Dimensional analysis multiplies by conversion factors equal to 1 so unwanted units cancel and the target unit remains.
- Scale problems are proportions: set the map/model ratio equal to the real-world ratio and cross-multiply for the unknown.
Ratios and Rates
A ratio compares two quantities of the same kind. The ratio of 8 boys to 12 girls can be written 8:12, 8 to 12, or 8/12, and like any fraction it reduces — here to 2:3. Order matters: 2:3 is not the same as 3:2.
With three-part ratios, scale every part by the same factor. If a recipe mixes flour, sugar, and butter in 3:2:1 and you use 9 cups of flour, the factor is 9 ÷ 3 = 3, so you need 6 cups of sugar and 3 cups of butter.
A rate compares quantities with different units, such as miles per hour or dollars per pound. A unit rate rewrites the rate with a denominator of 1.
Worked Example 1. A car travels 150 miles on 5 gallons of gas. The unit rate is 150 ÷ 5 = 30 miles per gallon. Unit rates make comparison shopping easy: a better deal is simply the lower price per unit.
Solving Proportions
A proportion states that two ratios are equal: a/b = c/d. The fastest solving method is cross-multiplication, which uses the fact that a/b = c/d implies a × d = b × c.
Worked Example 2. Solve x/12 = 3/4.
- Cross-multiply: x × 4 = 12 × 3, so 4x = 36.
- Divide both sides by 4: x = 9.
- Check: 9/12 reduces to 3/4. ✓
Worked Example 3 (word problem). If 4 notebooks cost $6, how much do 10 notebooks cost? Set up matching ratios with notebooks over dollars: 4/6 = 10/x. Cross-multiply: 4x = 60, so x = $15. Keep your units consistent on top and bottom — notebooks across from notebooks, dollars across from dollars — or the answer will be wrong.
| If you know… | Set up | Solve |
|---|---|---|
| 3 apples / $2 | 3/2 = 12/x | x = $8 for 12 apples |
| 60 mi / 1.5 hr | 60/1.5 = x/4 | x = 160 mi in 4 hr |
Unit Conversion (Dimensional Analysis)
Dimensional analysis converts units by multiplying by conversion factors that equal 1, arranged so unwanted units cancel. For example, since 1 foot = 12 inches, both 12 in / 1 ft and 1 ft / 12 in equal 1; choose the one that cancels the unit you want to remove.
Useful conversions to memorize: 1 ft = 12 in, 1 yd = 3 ft, 1 mile = 5280 ft, 1 lb = 16 oz, 1 hr = 60 min, 1 km = 1000 m, 1 m = 100 cm.
Worked Example 4. Convert 5 feet to inches. 5 ft × (12 in / 1 ft) = 60 in. The 'ft' units cancel, leaving 60 inches.
Worked Example 5 (multi-step). Convert 2 miles to yards. Chain factors so each unwanted unit cancels:
2 mi × (5280 ft / 1 mi) × (1 yd / 3 ft) = 2 × 5280 ÷ 3 yd = 10560 ÷ 3 = 3520 yards.
The 'mi' cancels with the first factor and 'ft' cancels with the second, leaving yards — exactly the unit requested.
Scale Problems
A scale is a ratio between a drawing, map, or model and the real object. Scale problems are just proportions: set the scale ratio equal to the real-world ratio and cross-multiply.
Worked Example 6. A map scale reads 1 inch = 25 miles. Two cities are 3.5 inches apart on the map. How far apart are they in reality? Set up inches over miles: 1/25 = 3.5/x. Cross-multiply: 1 × x = 25 × 3.5, so x = 87.5 miles.
Worked Example 7. A model car is built at a scale of 1:18, meaning the real car is 18 times larger. If the model is 10 inches long, the real car is 10 × 18 = 180 inches (15 feet) long.
Common Mistakes
- Mismatched units in a proportion: always line up the same unit on top across both ratios (miles over miles, dollars over dollars).
- Flipping the conversion factor: pick the factor that cancels the unit you are removing; the wrong orientation leaves the units uncanceled.
- Reversing the scale: with a 1:18 model, the real object is larger — multiply, do not divide, to go from model to real.
Rates, Speed, and Converting Compound Units
Many proportional word problems are really about speed, which connects distance, rate, and time through the relationship distance = rate × time. Rearranged, rate = distance ÷ time and time = distance ÷ rate.
Worked Example 8. A train covers 240 miles in 4 hours. Its average speed is 240 ÷ 4 = 60 miles per hour. At that rate, how long to travel 390 miles? time = distance ÷ rate = 390 ÷ 60 = 6.5 hours.
Compound units such as miles per hour can themselves be converted with dimensional analysis by converting the numerator and denominator separately.
Worked Example 9. Convert 90 kilometers per hour to meters per minute. Convert km to meters and hours to minutes:
90 km/hr × (1000 m / 1 km) × (1 hr / 60 min) = 90 × 1000 ÷ 60 m/min = 90,000 ÷ 60 = 1500 meters per minute.
The 'km' cancels with the first factor and 'hr' cancels with the second, leaving meters per minute. A good sanity check on any proportion or rate problem is reasonableness: if a faster speed produced a longer travel time for the same distance, you set the ratio up upside down. Estimating the expected size of the answer before computing catches most setup errors on the placement test.
Solve the proportion: 7/x = 21/15.
Convert 3 miles to feet. (1 mile = 5280 feet)
If 5 pounds of apples cost $9, what is the cost of 8 pounds at the same rate?
On a map, 1 inch represents 40 miles. Two towns are 6 inches apart on the map. How far apart are they in reality?