5.2 Ratios, Proportions & Percentage Reasoning

Ratios as Part–Whole Reasoning

Ratio items on the JOA ask you to split quantities, scale mixtures, or compare groups — always under the same timed natural-ability frame as other numerical stems. A ratio a : b means for every a units of the first quantity there are b units of the second. The whole has a + b parts when the ratio describes a complete partition into those two categories.

Finding an actual amount from a ratio

Example: A team has apprentices and qualified members in the ratio 2 : 5. There are 28 people in total. How many are apprentices?

• Total parts = 2 + 5 = 7
• One part = 28 ÷ 7 = 4
• Apprentices = 2 × 4 = 8

Qualified members = 5 × 4 = 20. Check: 8 + 20 = 28.

Three-part ratios

Example: Fuel mix oil : petrol : additive = 1 : 20 : 2 with 46 litres total. How much petrol?

• Parts = 1 + 20 + 2 = 23
• One part = 46 ÷ 23 = 2 litres
• Petrol = 20 × 2 = 40 litres

If an option set includes 20 litres, that distractor confuses “20 parts” with “20 litres.” Always multiply parts by the size of one part.

Changing a ratio (keep the method short)

Sometimes a stem adds people or volume and asks for a new ratio. Compute new counts, then simplify.

Example: Class ratio girls : boys = 3 : 2 with 30 students (so 18 girls, 12 boys). Five more girls join. New ratio?

• Girls = 23, boys = 12 → ratio 23 : 12 (already simplified if no common factor)

Do not build lengthy algebraic systems. Count → divide by greatest common factor if needed → report.

Proportions: Scale With One Multiplier

Direct proportion means both quantities grow by the same factor. The JOA-friendly setup is:

[ \frac{\text{known A}}{\text{known B}} = \frac{\text{unknown A}}{\text{related B}} \quad \text{or simply} \quad \text{unknown} = \text{known} \times \frac{\text{new scale}}{\text{old scale}} ]

Worked example: pack cost

If 3 identical packs cost $12, what do 7 packs cost at the same unit price?

• Unit price = 12 ÷ 3 = $4
• Seven packs = 7 × 4 = $28

Or multiplier route: 7/3 × 12 = 28. Same result in one line.

Worked example: map scale / model scale

A map uses 1 cm : 2 km. A road measures 6.5 cm on the map. Real length?

• 6.5 × 2 = 13 km

If the ratio were 1 cm : 500 m, convert 500 m to 0.5 km first or work entirely in metres (6.5 × 500 = 3,250 m = 3.25 km).

Inverse-style caution (still short setups)

Some everyday rates run opposite directions (more workers, fewer hours for the same job) when the stem clearly states constant total work and equal productivity. Only invert when the story demands it.

Example: 4 people complete a job in 6 hours working at the same rate. How long for 3 people on the same job?

• Person-hours = 4 × 6 = 24
• Time for 3 people = 24 ÷ 3 = 8 hours

If you accidentally treat it as direct proportion (3/4 × 6 = 4.5), you get a wrong, faster time — a common trap. Read whether more of one quantity should make the other larger or smaller.

JOA pacing note for proportions

Proportion items are often “free marks” if numbers are friendly. Spend your seconds on which way the scale goes, not on formal cross-multiplication essays. If numbers are ugly (large primes, nested fractions), estimate whether options are in the right ballpark, compute once, and move — or flag if the arithmetic will blow the ~23.5-second budget.

Percentage Reasoning Without Long Proofs

Percent means per hundred. Core operations:

Worked examples

Percent of: 15% of 240 = 0.15 × 240 = 36.

A is what percent of B: 18 is what percent of 60? (18 ÷ 60) × 100 = 30%.

Percent change: Price rises from $80 to $100. Change = 20; percent increase = (20 ÷ 80) × 100 = 25% (not 20%, which would incorrectly divide by the new price).

Successive change trap: A value increases by 10%, then decreases by 10%. Final value is not the original.

• Start 100 → after +10% → 110 → after −10% of 110 → 99

The second percentage applies to a new base. Options that say “back to 100” punish candidates who cancel +10% with −10% mentally.

“Percent of” vs “percent more than”

• “A is 20% of B” → A = 0.20 × B
• “A is 20% more than B” → A = 1.20 × B

Misreading “more than” as “of” is a pure verbal-numeric trap — slow down for one second on the wording.

Finding the original after a percent change

If a discounted price is $84 after 30% off, original price O satisfies 0.70 × O = 84 → O = 84 ÷ 0.70 = 120.

Do not add 30% of 84 back onto 84 (that recovers a different wrong base). Divide by the remaining factor (1 − p/100).

Combining ratio and percent in one stem

Example: A budget is split 3 : 2 between training and equipment. Training is then cut by 20%. If the original whole budget was $50,000, what is training after the cut?

• Training share originally = 3/5 × 50,000 = 30,000
• After 20% cut = 0.80 × 30,000 = 24,000

Two short steps beat one tangled expression. On the JOA, chain only as many steps as the stem forces; write intermediate results mentally or on scrap paper if allowed in your sitting context, but keep each step tiny.

Estimation as a speed tool

Before fine calculation, rough-check:

• 19% of 400 is a bit under 0.20 × 400 = 80 → about 76
• 1/3 is about 33%; 2/3 about 67%
• Doubling ≈ +100%; halving ≈ −50%

If options are far apart, estimation alone can eliminate three choices in five seconds. If options are close, finish the exact arithmetic.

Process checklist for ratio / percent items

1. Identify parts vs whole and whether the whole is given.
2. For proportions, decide direct or inverse from the story.
3. For percents, lock the base (original value for percent change).
4. Apply one clear operation; avoid multi-page rearrangements.
5. Sense-check magnitude; watch successive-percent and “of”/“more than” wording.
6. If still messy after ~30–35 seconds, flag and harvest easier numerical or verbal items — the mixed JOA rewards total correct attempts, not stubbornness on one percentage chain.

Ratios, proportions, and percentages are everyday reasoning tools. Practise them as quick setups, not as textbook chapters, so they stay inside the JOA’s natural-ability and time design.