6.2 Ratios and Relative Difference

Ratio phrases are not decoration

A major Quantitative Literacy booklet skill is to interpret a ratio phrase, convert it to numbers, calculate, and interpret the result back in the same campus, clinic, or municipal story. QL remains 50 multiple-choice questions inside AQL, calculator-free, and not MAT. Independent OpenExamPrep ratio practice trains that four-beat loop. It does not reprint confidential NBTP items.

Common English that hides a ratio:

Ratio compares two quantities, often with the same unit (students to students) or two counted groups in one setting. Rate is a ratio with unlike units and a per. Proportion is a part-to-whole fraction: the part divided by the total. A percent is that proportion multiplied by 100. Mixing those four labels is how a correct calculation still picks a wrong option.

Convert the phrase, then share out the total

Worked example: dining hall 5:2. “For every 5 meal-plan students there are 2 cash-pay students.” Ratio 5:2. Total parts = 5 + 2 = 7. If 84 students eat in the hall, each part is 84 ÷ 7 = 12 students.

• Meal-plan: 5 × 12 = 60
• Cash-pay: 2 × 12 = 24

Check: 60 + 24 = 84, and 60:24 divides by 12 to 5:2. Interpret back: 60 of the 84 students are on the meal plan. An option that stops at “5 and 2” never left the phrase. An option of 42 and 42 treated the ratio as an even split.

Worked example: laboratory retests. “Three in every eight laboratory samples were retested.” That is a proportion, not a 3:8 part-to-part split of two leftover groups. Proportion retested = 3/8 = 0.375 = 37.5%. If 96 samples were processed: 96 × 3/8 = (96 ÷ 8) × 3 = 12 × 3 = 36 retested. Remaining not retested: 96 − 36 = 60. The ratio retested:not-retested is 36:60 = 3:5, which is not 3:8. The “in every eight” already used 8 as the whole.

Ratio, rate, proportion, percent, percentage points

Worked example: clinic table.

Monday 12:8 = 3:2. Tuesday 15:10 = 3:2. Wednesday 9:6 = 3:2. Combined 36:24 = 3:2. The ratio of A’s visits to B’s visits is stable.

A rate from the same table: if Monday is a 2-hour morning clinic, Nurse A’s rate is 12 visits / 2 h = 6 visits per hour. That per hour is not the same claim as the 3:2 ratio.

A proportion: Monday’s 20 visits, Nurse A has 12/20 = 0.60 = 60% of Monday’s visits. That is part-to-whole for Monday, not the 3:2 part-to-part ratio.

Percentage points versus percent. A residence occupancy board shows 72% occupied in 2025 and 80% occupied in 2026.

• Arithmetic gap: 80 − 72 = 8 percentage points
• Relative change: 8/72 = 1/9. 1 ÷ 9 = 0.111… ≈ 11.1% increase on the 2025 occupancy rate

Saying “occupancy rose 8%” is the classic QL trap. An 8% rise on 72% would be 0.08 × 72 = 5.76 percentage points, landing near 77.8%, which is not 80%. The board moved 8 percentage points, which is about an 11.1% relative rise.

Relative difference depends on the base

Worked example: municipal recycling. Suburb P collected 2.4 t. Suburb Q collected 1.6 t.

Ratio P:Q = 2.4:1.6. Multiply by 10: 24:16. Divide by 8: 3:2. P/Q = 2.4 ÷ 1.6. 24 ÷ 16 = 1.5, so P collected 1.5 times Q, which is 50% more than Q. Difference = 2.4 − 1.6 = 0.8 t. Relative to Q (the smaller base): 0.8 / 1.6 = 0.5 = 50% more. Relative to P: 0.8 / 2.4 = 1/3 ≈ 33.3% less (Q compared with P).

“50% more than Q” is not “50% of Q”. Fifty percent of Q would be 0.8 t, which is the difference, not Q’s total. Interpret back: P’s 2.4 t is one and a half times Q’s 1.6 t; Q is one-third below P.

Ratios in tables, charts, and scale diagrams

Worked example: campus map. Scale 1 cm : 40 m. A path measures 6.5 cm on the map.

Actual length: 6.5 × 40. 6 × 40 = 240; 0.5 × 40 = 20; total 260 m. If a building is 80 m long, map length = 80 ÷ 40 = 2 cm.

Trap readings: 6.5 m (forgetting the scale), 40 m (quoting the scale unit once), 260 cm (multiplying but keeping map units).

Worked example: shuttle bar chart. A printed chart uses 1 cm of bar height for 40 passengers. Morning bar 2.5 cm, midday 3.0 cm, evening 1.5 cm.

• Morning: 2.5 × 40 = 100 passengers
• Midday: 3 × 40 = 120
• Evening: 1.5 × 40 = 60

Ratio morning:midday:evening = 100:120:60. Divide by 20: 5:6:3. Proportion evening of the day’s 280 passengers: 60/280 = 6/28 = 3/14 ≈ 21.4%. The chart is a scale diagram; the ratio is in the heights only after you apply the 40-passenger scale.

A four-beat method

1. Underline the phrase (for every, in every, per, to, out of).
2. Name the type: ratio, rate, proportion, percent, or percentage points.
3. Convert and calculate with a common unit; show the share-out or the scale product.
4. Interpret back in one sentence that still names the campus, clinic, or municipality.

Traps

• Treating “3 in every 8” as a 3:8 split of two leftover groups when 8 is already the whole
• Calling an 8 percentage-point rise an 8% rise
• Changing the base of a relative difference without noticing
• Reading a map length as a real length
• Computing a tidy ratio and then choosing an option that never returns to the story

Independent OpenExamPrep QL ratio teaching is the loop, not a list of synonyms: phrase, numbers, calculation, meaning.