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Hong KongRatio, proportion and rate
Skill 2 of Competency 2 reads "Solve problems involving ratio and proportion (e.g., scaled drawings, models, real-world scenarios)".
- It sits under geometry and measurement rather than number sense, which tells you the exam expects proportion in a spatial or unit context.
On this page9 sections
- Three words, three meanings
- The setup that prevents most errors
- Part-to-part and part-to-whole
- Rates and derived units
- Similar figures are the same idea
- Why we would prioritize this
- Interpret a ratio and keep the order fixed
- Solve proportions by preserving equivalent ratios
- Distinguish ratio, rate, and unit rate
Look where the rule filed this. Not with the arithmetic, where you would expect it. Under Knowledge of geometry and measurement, with scaled drawings and models given as the examples.
That placement is the most useful thing on this page. It tells you what a proportion question on this subtest is likely to look like, and it is a map, a model or a mixture rather than a bare pair of ratios.
Three words, three meanings
| Term | What it is | Example |
|---|---|---|
| Ratio | A comparison of two quantities | 3 pens to 5 pencils |
| Proportion | A statement that two ratios are equal | 3 to 5 equals 12 to 20 |
| Rate | A ratio of two different units | 50 miles per hour |
A proportion is an equation, which is why every proportion question can be solved the same way: write the two ratios with the same quantity on top in both, then cross multiply.
The setup that prevents most errors
Label the units. Write the ratio as a fraction with a word above and a word below, and make sure the second fraction has the same words in the same places.
Miles over hours equals miles over hours. Model centimeters over real meters equals model centimeters over real meters. If your two fractions do not have matching labels, the cross multiplication will be arithmetically perfect and answer a different question.
A scale drawing uses 1 centimeter to represent 50 kilometers. Two towns are 4 centimeters apart on the drawing. How far apart are they in reality?
- 50 kilometers
- 100 kilometers
- 200 kilometers
- 500 kilometers
Part-to-part and part-to-whole
A ratio of 3 to 5 describes two parts, so the whole is 8 parts. That single sentence resolves most ratio word problems.
If 16 students split in a ratio of 3 to 5, divide 16 by 8 to get 2 students per part, then multiply out: 6 and 10. Adding your two answers back to the original total is a free check and it costs three seconds.
Rates and derived units
Skill 4 of the same competency names derived units explicitly, giving miles per hour and dollars per gallon as examples. A rate is a ratio wearing a unit, and the word per is the division sign.
So 100 miles in 2 hours is 50 miles per hour, and the arithmetic is the label. Reading the units off the question tells you which way to divide, every time.
Similar figures are the same idea
Two similar shapes have corresponding sides in proportion, which is a proportion question with a picture attached. Scale drawings and similar figures covers the geometry side, including the trap about areas: doubling the sides does not double the area.
Why we would prioritize this
One skill of the four in Competency 2, which is about seven of the 35 questions on our split, and the same technique reappears inside unit conversion, similar figures and several data items. That estimate is ours, derived from the number of skills the rule states, and not a published weighting.
Proportion is also the single most transferable piece of arithmetic on this subtest. If you are rebuilding mathematics from a long way back, learn it before you learn anything about geometry formulas, because the formulas are on the supplied reference sheet and proportional reasoning is not.
Interpret a ratio and keep the order fixed
A ratio compares quantities by division. The ratio of 3 red counters to 5 blue counters can be written 3:5 or 3/5, but the order matters: reversing it describes blue to red. A ratio can compare parts of a whole or two separate quantities. If a recipe uses 3 cups of flour for every 5 cups of water, the total mixture has 8 parts; the ratio of flour to the total is 3:8, not 3:5.
Ratios can be simplified by dividing all terms by the same positive factor. The ratio 12:18 reduces to 2:3. For three quantities, simplify every term together: 12:18:30 becomes 2:3:5 after dividing by 6. Do not simplify only one term. Attach units where helpful, especially when comparing unlike quantities.
Solve proportions by preserving equivalent ratios
A proportion states that two ratios are equal: a/b = c/d. Cross-multiplication gives ad = bc when denominators are nonzero. For 3/5 = x/20, calculate 5x = 60, so x = 12. Substitute back to check that both ratios equal 0.6. Cross-multiplication is an algebraic shortcut; it does not replace identifying the corresponding quantities correctly.
A scale or recipe problem may be solved by multiplying both terms of a ratio by the same scale factor. If a map uses 1 inch for 12 miles, a 4-inch distance represents 48 miles. If the scale changes from 1 inch to 1.5 inches, the real distance changes by the same factor only when the same map scale applies.
Distinguish ratio, rate, and unit rate
A rate compares quantities with different units, such as 120 miles in 2 hours. A unit rate expresses the comparison per one unit: 60 miles per hour. A ratio can compare quantities with or without different units. When calculating, write units beside values so you know which quantity belongs in the numerator and which in the denominator.
A constant rate can support proportional reasoning, but not every relationship is proportional. A taxi with a base fee plus a per-mile charge does not have a constant total-cost-per-mile from zero, though the variable portion has a constant rate. Look for a fixed starting amount or other condition that changes the relationship.
- Write the order of compared quantities before forming a ratio.
- Simplify all terms by the same factor.
- Use corresponding quantities in a proportion and verify the result.
- A rate compares different units; a unit rate is per one denominator unit.
- Check whether a fixed amount prevents a relationship from being proportional.
Common questions
How do I solve a proportion question?
Write both ratios as fractions with the same units in the same positions, then cross multiply and solve. Labeling the units above and below each fraction is what prevents the most common error, which is arithmetic done correctly on the wrong pairing.
What is the difference between a ratio and a rate?
A ratio compares two quantities of the same kind, such as 3 pens to 5 pencils. A rate compares two different units, such as miles per hour or dollars per gallon. The word per tells you to divide, and the units tell you which way round.
Why is ratio filed under geometry on the FTCE?
Because the rule expects it in a spatial or measurement context. Skill 2 of Competency 2 gives scaled drawings, models and real-world scenarios as its examples, so a proportion question on this subtest is likelier to involve a map or a model than a bare pair of ratios.
How do I split a total in a given ratio?
Add the parts to get the number of shares, divide the total by that, then multiply for each side. A ratio of 3 to 5 on a total of 16 is 8 shares of 2 each, giving 6 and 10. Add your answers back to the total as a check that nothing went astray.