8.3 Ratio, Proportion, and Rates

8.3 Ratio, Proportion, and Rates

Quick Answer: A ratio compares quantities; a proportion says two ratios are equal. Convert word problems into a:b or a/b form, simplify, then cross-multiply—or use unit rates when units differ.

Ratio and proportion sit at the center of Grade 11 General Mathematics and appear constantly on the USTET: sharing costs, scaling recipes, map distances, speed–time–distance, and mixture problems all reduce to the same tools.

Ratios

The ratio of a to b is written a:b or a/b, provided b ≠ 0. Ratios should use the same units before simplifying.

Worked example: Express 40 minutes to 2 hours as a simplified ratio.

Convert 2 hours = 120 minutes. Ratio = 40:120 = 1:3.

Worked example: A class has 18 girls and 12 boys. Find the ratio of girls to boys, and girls to the whole class.

Girls:boys = 18:12 = 3:2. Girls:total = 18:(18+12) = 18:30 = 3:5.

Part-sharing with a ratio

If an amount T is shared in the ratio m:n, the first share is (m/(m+n))×T and the second is (n/(m+n))×T.

Worked example: PHP 4,800 is shared between Ana and Ben in the ratio 5:3. How much does each receive?

Total parts = 5 + 3 = 8. Ana: (5/8)×4800 = PHP 3,000. Ben: (3/8)×4800 = PHP 1,800.

Check: 3000 + 1800 = 4800.

Proportions

A proportion is an equality of two ratios: a/b = c/d. Cross-multiplying gives ad = bc.

Worked example: Solve for x: 3/5 = x/20.

Cross-multiply: 3 × 20 = 5x → 60 = 5x → x = 12.

Worked example: If 4 notebooks cost PHP 120, how much do 7 identical notebooks cost?

Set 4/120 = 7/x, or better: cost scales with count → 4/7 = 120/x is wrong form. Use notebooks/cost consistently:

4 notebooks / 120 pesos = 7 notebooks / x pesos → 4x = 840 → x = PHP 210.

Unit-rate approach: 120/4 = PHP 30 per notebook; 7 × 30 = PHP 210.

Direct proportion

Two quantities are in direct proportion if their ratio is constant: y = kx. Doubling x doubles y.

Worked example: A car travels 150 km in 3 hours at constant speed. How far in 5 hours?

Distance ∝ time → 150/3 = d/5 → d = 250 km.

Inverse proportion

Two quantities are in inverse proportion if their product is constant: xy = k. Doubling x halves y.

Worked example: 6 workers finish a job in 10 days (same work rate). How many days for 4 workers?

Workers × days = constant → 6 × 10 = 4 × d → d = 15 days.

Worked example: Speed and time for a fixed distance: traveling 120 km at 40 km/h takes 3 hours; at 60 km/h, time = 120/60 = 2 hours. Check product: 40×3 = 60×2 = 120.

Rates and Unit Rates

A rate compares different kinds of units (kilometers per hour, pesos per kilogram, words per minute). A unit rate has a denominator of 1.

Worked example — better buy: Brand A is PHP 85 for 500 g; Brand B is PHP 100 for 600 g. Which is cheaper per gram?

A: 85/500 = PHP 0.17 per gram. B: 100/600 ≈ PHP 0.1667 per gram. Brand B is slightly cheaper per gram.

Worked example — speed: Distance = rate × time. A bus averages 45 km/h for 2.5 hours. Distance = 45 × 2.5 = 112.5 km.

Scale and Similarity

Map scales are ratios. If a map scale is 1:50,000, then 1 cm on the map represents 50,000 cm = 500 m = 0.5 km in reality.

Worked example: On a 1:50,000 map, two towns are 8 cm apart. Actual distance?

8 × 50,000 = 400,000 cm = 4,000 m = 4 km.

Mixture Snapshot

Mixtures combine ratios of parts.

Worked example: A blend mixes coffee A and B in the ratio 2:3. How many kilograms of A are in 20 kg of blend?

A fraction = 2/(2+3) = 2/5; amount of A = (2/5)×20 = 8 kg.

Putting It Together: Multi-Step Problem

A juice concentrate is mixed with water in the ratio 1:4. You need 15 liters of drink.

1. Total parts = 5; concentrate = (1/5)×15 = 3 L; water = 12 L.
2. If concentrate costs PHP 80 per liter, concentrate cost = 3 × 80 = PHP 240.
3. If you later dilute further so the concentrate:water ratio becomes 1:5 for a new batch of 18 L, concentrate needed = (1/6)×18 = 3 L still—but water becomes 15 L. Always recompute parts when the ratio changes.

USTET Timing Tips

1. Convert units before writing the ratio.
2. Label every fraction (what is on top?) to avoid inverted proportions.
3. Prefer unit rates when comparing packages or speeds.
4. Ask: if one quantity goes up, should the other go up (direct) or down (inverse)?
5. Cross-multiply carefully; a single swapped term flips the answer.

Ratio, proportion, and rates unlock later algebra word problems and many Science calculation items that quote densities, concentrations, or scaled measurements.