5.3 Ratio, Proportion, Rates, and Averages
Key Takeaways
- Simplify ratios like fractions: 12:18 = 2:3; a 2:3 part-to-part ratio means the parts are 2/5 and 3/5 of the whole
- Direct proportion means y = kx (more of one, more of the other); inverse proportion means xy = k (more workers, less time)
- Speed problems run on d = s × t; round-trip average speed is total distance ÷ total time, never the average of the two speeds
- Combined work rate adds the individual rates: 1/4 + 1/6 = 5/12 of the job per hour, so together they need 12/5 hours
- Never average group averages — combine groups with a weighted average using the group sizes as weights
Why Ratios, Rates, and Averages Matter on the PUPCET
Ratio and proportion items are among the most predictable questions in the Mathematics subtest: they reuse the same handful of setups — recipe scaling, map distances, workers and days, fares and fuel — with only the numbers changed. Averages appear both directly (find the mean) and inside data-interpretation items. Master the five or six standard patterns below and you can bank these points quickly, saving time for harder items.
Ratio Basics
A ratio compares two quantities and can be written three ways: 2:3, 2/3, or 2 to 3. Simplify ratios exactly like fractions — divide both terms by their GCF: 12:18 = 2:3 (dividing by 6). Ratios can have three or more terms: 2:3:5.
Distinguish part-to-part from part-to-whole. If a class has boys and girls in the ratio 2:3, that is part-to-part — but it tells you the part-to-whole fractions immediately: boys are 2/(2+3) = 2/5 of the class. In a class of 35 students: boys = 2/5 × 35 = 14, girls = 3/5 × 35 = 21. Forgetting to add the parts (using 2/3 instead of 2/5) is the standard trap.
Proportions and Cross-Multiplication
A proportion states that two ratios are equal. Solve by cross-multiplying: if 3/x = 12/20, then 12x = 3 × 20 = 60, so x = 5.
Scale example: on a map, 2 cm represents 5 km. How far apart are two barangays that are 9 cm apart on the map? Set 2/5 = 9/x; cross-multiply: 2x = 45, so x = 22.5 km.
Direct Proportion
Two quantities are directly proportional when one is a constant multiple of the other: y = kx. More of one means proportionally more of the other.
Worked example: 5 notebooks cost ₱175. How much do 8 notebooks cost? The unit price k = 175 ÷ 5 = ₱35, so 8 × 35 = ₱280. Equivalently, 5/175 = 8/x gives 5x = 1,400, x = 280.
Inverse Proportion
Two quantities are inversely proportional when their product is constant: xy = k. More of one means proportionally less of the other — the signature case is workers and time.
Worked example: 6 workers can build a barangay waiting shed in 10 days. How long would 15 workers take? The job needs 6 × 10 = 60 worker-days, so 60 ÷ 15 = 4 days. Setting this up as a direct proportion (more workers = more days) is the classic error — always ask yourself which direction the relationship goes.
Unit Rates
A unit rate gives the amount per one unit: pesos per kilo, kilometers per liter, words per minute.
- Mangoes at ₱96 for 1.5 kg: 96 ÷ 1.5 = ₱64 per kilo
- Which is the better buy — 500 g of detergent at ₱48 or 1.5 kg at ₱135? First pack: 48 ÷ 0.5 = ₱96/kg. Second: 135 ÷ 1.5 = ₱90/kg. The 1.5 kg pack is cheaper per kilo. PUPCET best-buy items are always settled with unit rates, never by comparing sticker prices.
Speed, Distance, and Time
One triangle of formulas handles everything: d = s × t, s = d ÷ t, t = d ÷ s. Keep units consistent — if speed is in km/h, time must be in hours.
- A bus travels at 60 km/h for 2.5 hours: d = 60 × 2.5 = 150 km
- Juan walks 3 km in 36 minutes: 36 min = 0.6 h, so s = 3 ÷ 0.6 = 5 km/h
The average-speed trap: average speed for a whole trip is total distance ÷ total time, never the average of the two speeds. A van goes 120 km at 60 km/h (2 hours) and returns 120 km at 40 km/h (3 hours). Average speed = 240 km ÷ 5 h = 48 km/h — not 50 km/h. The slow leg eats more time, pulling the true average below the midpoint.
Work-Rate Problems
Treat each worker's rate as the fraction of the job done per hour, then add the rates:
- Alex can paint a room alone in 4 hours (rate 1/4 per hour); Bea can do it in 6 hours (rate 1/6 per hour). Together: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 of the room per hour, so they finish in 12/5 = 2.4 hours = 2 hours 24 minutes.
Never average the times (that would give 5 hours — slower than Alex alone, which is impossible with help).
Mean, Median, Mode
| Measure | How to find it | Watch out for |
|---|---|---|
| Mean | Sum of values ÷ number of values | Pulled by extreme values |
| Median | Middle value after sorting; average the two middle values if the count is even | Must sort first |
| Mode | Most frequent value | A set may have no mode or several |
Worked example: quiz scores 7, 3, 9, 3, 10. Mean = 32 ÷ 5 = 6.4. Sorted: 3, 3, 7, 9, 10, so median = 7 (the third of five values). Mode = 3 (appears twice). If the set were 3, 3, 7, 9, the median would be (3 + 7)/2 = 5 — a value not even in the set.
Weighted Average
When values carry different weights, use weighted average = (sum of value × weight) ÷ (sum of weights).
GWA example: Paolo earns 88 in Algebra (3 units), 90 in Filipino (3 units), and 85 in Science (2 units). Weighted average = (88×3 + 90×3 + 85×2) ÷ (3+3+2) = (264 + 270 + 170) ÷ 8 = 704 ÷ 8 = 88. The plain mean (87.67) ignores the units and gives the wrong answer.
The Average-of-Averages Trap
Section A has 30 students averaging 80; Section B has 20 students averaging 90. The combined average is not (80 + 90)/2 = 85. Weight by group size: (30×80 + 20×90) ÷ 50 = (2,400 + 1,800) ÷ 50 = 4,200 ÷ 50 = 84. Whenever two groups differ in size, the combined mean leans toward the bigger group's average — 84 is closer to 80 than to 90, exactly as it should be.
Exam Tips
- Before solving any proportion item, label it direct or inverse in one second by asking: if this goes up, does that go up or down?
- In ratio items, the sum of the parts must divide the total nicely — if 35 students split 2:3 gives non-integers, you used the wrong denominator.
- On median questions, rewrite the data sorted on your scratch paper before touching the choices.
Ana can paint a room alone in 6 hours, while Ben can paint the same room alone in 3 hours. Working together at their usual rates, how long will they take to paint the room?
Section A has 30 students with an average score of 80. Section B has 20 students with an average score of 90. What is the average score of all 50 students combined?