3.3 Ratios, Rates, Proportions, Percents & Measurement Conversions
Key Takeaways
- A proportion is an equation of two equal ratios, and cross-multiplication is valid because it is multiplication of both sides by both denominators.
- Percent change equals (new − old)/old × 100, always dividing by the original amount, which is why a 20% increase followed by a 20% decrease does not return to the starting value.
- Dimensional analysis multiplies by conversion factors written as fractions equal to 1, arranged so unwanted units cancel.
- Squared and cubed unit conversions require squaring or cubing the linear conversion factor: 1 yd² = 9 ft² and 1 yd³ = 27 ft³.
- Exponential depreciation multiplies by (1 − r) each period, so a $24,000 vehicle losing 15% annually is worth 24,000(0.85)³ after three years, not 24,000 − 3(0.15)(24,000).
3.3 Ratios, Rates, Proportions, Percents & Measurement Conversions
Competency 1 skill 6 carries an unusually long list of contexts — mixtures, comparisons, rates, measurement conversions, graphs, percent growth, taxes, depreciation. Because Florida retired the standalone measurement competency, unit conversion now lives here, which makes this one of the highest-yield skills on the test.
Ratios and proportional set-up
A ratio compares two quantities, written a : b or a/b. A rate is a ratio of unlike units (miles per hour), and a unit rate has denominator 1.
A proportion asserts a/b = c/d. Cross-multiplying to ad = bc is not a trick; it is what you get by multiplying both sides by bd.
The critical discipline is consistent placement. If the left ratio is (miles)/(hours), the right ratio must also be (miles)/(hours) — not (hours)/(miles).
A recipe uses 3 cups of flour for every 4 cups of milk. How much flour for 14 cups of milk? Set up flour/milk on both sides: 3/4 = x/14. Cross-multiply: 4x = 42, so x = 10.5 cups.
Part-to-part versus part-to-whole is the other set-up trap. If boys to girls is 3 : 5, then boys are 3/8 of the class, not 3/5. Any ratio a : b splits a whole into a + b parts.
A 48-student class has a boys-to-girls ratio of 3 : 5. Total parts = 8, so each part is 48/8 = 6 students. Boys = 3(6) = 18, girls = 5(6) = 30.
Scale factors and similar figures
A scale factor is a ratio of corresponding lengths. A map scale of 1 : 25,000 means 1 unit on the map is 25,000 of the same units in reality. If two triangles are similar with scale factor k, corresponding sides are in ratio k, areas are in ratio k², and volumes in ratio k³. That distinction is examined in the geometry competency as well and is developed further there.
The percent family
| Question type | Equation | Example |
|---|---|---|
| Percent of a number | part = percent × whole | 18% of 250 = 45 |
| What percent | percent = part/whole | 45 of 250 = 18% |
| Find the whole | whole = part/percent | 45 is 18% of 250 |
| Percent increase | (new − old)/old | 40 → 50 is 25% increase |
| Percent decrease | (old − new)/old | 50 → 40 is 20% decrease |
Note that 40 → 50 is a 25% increase while 50 → 40 is a 20% decrease, even though both involve the same 10 units. Percent change always divides by the original value, and the original changes depending on direction. This asymmetry is one of the most reliably tested ideas in the competency.
The consequence: a 20% increase followed by a 20% decrease does not restore the original. Starting at 100: 100(1.20) = 120, then 120(0.80) = 96. Successive percent changes multiply; they do not add.
Tax, tip, discount, and markup are all one-step multiplications when you use the complement:
- Add 7% sales tax: multiply by 1.07
- Take 30% off: multiply by 0.70
- Take 30% off, then 10% off the reduced price: multiply by 0.70 · 0.90 = 0.63, a 37% total discount, not 40%
Depreciation works the same way in reverse.
A $24,000 vehicle depreciates 15% per year. Value after 3 years = 24,000(0.85)³ = 24,000(0.614125) = $14,739.
The distractor is always the simple-interest style answer, 24,000 − 3(0.15)(24,000) = $13,200. Exponential decay is not repeated subtraction of the original percentage.
Dimensional analysis
Write each conversion as a fraction equal to 1 and arrange so unwanted units cancel.
Convert 45 miles per hour to feet per second. 45 mi/hr × 5,280 ft/1 mi × 1 hr/3,600 s = (45 · 5,280)/3,600 ft/s = 237,600/3,600 = 66 ft/s
Miles cancel against miles; hours cancel against hours; feet and seconds survive in the right positions.
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| Conversions worth memorizing |
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| LENGTH 1 ft = 12 in 1 yd = 3 ft 1 mi = 5,280 ft |
| 1 m = 100 cm 1 km = 1,000 m 1 in = 2.54 cm (exact) |
| MASS 1 lb = 16 oz 1 ton = 2,000 lb 1 kg = 1,000 g |
| VOLUME 1 cup = 8 fl oz 1 pt = 2 cups 1 qt = 2 pt 1 gal = 4 qt |
| 1 L = 1,000 mL 1 cm^3 = 1 mL |
| TIME 1 hr = 3,600 s 1 day = 86,400 s |
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Metric prefixes are powers of ten: kilo- 10³, hecto- 10², deka- 10¹, base, deci- 10⁻¹, centi- 10⁻², milli- 10⁻³.
Squared and cubed units — the highest-value trap
Converting area or volume requires raising the linear factor to the same power.
Since 1 yd = 3 ft, squaring gives 1 yd² = 3² ft² = 9 ft², and cubing gives 1 yd³ = 3³ ft³ = 27 ft³.
A patio measures 12 ft by 15 ft = 180 ft². In square yards that is 180 ÷ 9 = 20 yd², not 180 ÷ 3 = 60.
Similarly 1 m² = 10,000 cm² (since 100² = 10,000) and 1 m³ = 1,000,000 cm³. Candidates who divide by the linear factor are off by a factor of 3 for area and 9 for volume, and those wrong values are always among the options.
Reading proportional relationships from graphs
A relationship is proportional exactly when its graph is a straight line through the origin; then y = kx and k is the constant of proportionality, readable as the y-value at x = 1 or as the slope. A line with a nonzero y-intercept, such as a plan with a $30 fee plus $5 per hour, is linear but not proportional — doubling the hours does not double the cost. Tables reveal the same distinction: if y/x is constant across every row, the relationship is proportional.
A storage room floor measures 21 feet by 18 feet. A flooring supplier sells only by the square yard. How many square yards of flooring are needed?
A laptop priced at $800 is marked down 25%, and then the sale price is marked down an additional 20%. What is the total percent discount from the original price?
A delivery van purchased for $32,000 depreciates 12% of its current value each year. What is its value after 2 years, to the nearest dollar?