6.2 Ratios & Relative Differences
Key Takeaways
- A ratio a:b compares parts; the whole has a + b parts, so a’s share of the whole is a/(a + b) when the ratio exhausts the total
- Relative difference (percent change) is absolute change ÷ original value — always divide by the starting amount named in the stem
- Phrase traps: “35% of” means × 0.35, while “35% more than” means × 1.35; “twice as many” is × 2, not + 2
- Scale diagrams and maps use a ratio 1:n — multiply map length by n, then convert units carefully (cm → m → km)
Why ratios and relative differences matter
The CEA Quantitative Literacy construct emphasises quantity together with relative difference — comparing sizes with ratios, rates of change, and proportional language drawn from tables, charts, and scale diagrams. On the AQL, stems rarely ask for a naked fraction in isolation; they ask you to turn a sentence such as “commerce to science is 2 to 5” or “fees rose by 15%” into a correct numerical representation and then compute.
This section trains three linked moves: (1) read and simplify ratio language, (2) compute absolute versus relative change from the right base, and (3) apply ratios on tables and scale diagrams with unit conversions.
Ratio language → numbers
| Phrase | Numerical meaning | Example |
|---|---|---|
| a to b / a:b | Parts a and b; whole = a + b if these are all parts | 3:5 → 3/8 and 5/8 of the whole |
| twice as many as n | 2 × n | twice as many as 40 → 80 |
| two more than n | n + 2 | two more than 40 → 42 |
| p% of n | (p/100) × n | 35% of 80 → 28 |
| p% more than n | n × (1 + p/100) | 35% more than 80 → 108 |
| p% less than n | n × (1 − p/100) | 35% less than 80 → 52 |
| relative increase from A to B | (B − A) / A | 40 → 50 → 10/40 = 25% |
Trap: “twice as many” is multiplicative; “two more” is additive. Mixing them flips easy marks.
Worked example — splitting a total in a ratio
Two campuses share 4 200 enrolled students in the ratio Campus A : Campus B = 2 : 5.
- Total parts = 2 + 5 = 7
- One part = 4 200 ÷ 7 = 600
- Campus A = 2 × 600 = 1 200
- Campus B = 5 × 600 = 3 000
Check: 1 200 + 3 000 = 4 200, and 1 200 : 3 000 simplifies by ÷600 to 2 : 5.
If the stem instead says “for every 2 students on Campus A there are 5 on Campus B” with a given A count, multiply A by 5/2 to find B — same ratio machinery, different known quantity.
Worked example — absolute vs relative difference
Residence fees rise from R8 000 to R9 200.
- Absolute difference = 9 200 − 8 000 = R1 200
- Relative difference (relative to the original) = 1 200 ÷ 8 000 = 0.15 = 15%
If someone wrongly divides by the new amount (1 200 ÷ 9 200 ≈ 13%), they report a different relative change. QL stems that say “by what percentage did fees increase?” always use the starting value as the denominator unless they explicitly name another base.
Worked example — “of” versus “more than”
A tutorial group has 80 students.
- 35% of 80 = 0.35 × 80 = 28
- 35% more than 80 = 80 + 28 = 108, or 80 × 1.35 = 108
Same digit “35,” completely different results. Underline the words of / more than / less than before calculating.
Ratios in a simple table
A water-usage table for three residences over one week:
| Residence | Litres used |
|---|---|
| Oak | 2 400 |
| Pine | 1 600 |
| Acacia | 800 |
Oak : Pine : Acacia = 2 400 : 1 600 : 800. Divide by 800 → 3 : 2 : 1.
Oak’s share of the total 4 800 L is 2 400 / 4 800 = 1/2 = 50%. Relative to Acacia, Oak uses 2 400 / 800 = 3 times as much (not “3 times more” in the loose everyday sense — stick to the unambiguous “3 times as much” = ×3).
Scale diagrams and maps
A campus map uses scale 1 : 25 000 (1 cm on the map represents 25 000 cm on the ground).
Example: A path measures 8 cm on the map. Real length?
- Ground centimetres = 8 × 25 000 = 200 000 cm
- Convert: 200 000 cm ÷ 100 = 2 000 m = 2 km
Example with 1 : 50 000: A lake spans 4 cm on the map.
- 4 × 50 000 = 200 000 cm = 2 km again
Always convert units in a chain you can check: cm → m (÷100) → km (÷1 000). Skipping a conversion by 10 or 100 is the usual scale-diagram error.
Relative difference from chart language
Suppose a bar chart shows load-shedding hours: Week 1 = 12 h, Week 2 = 9 h.
- Absolute change = 9 − 12 = −3 h (a decrease of 3 hours)
- Relative decrease = 3 ÷ 12 = 0.25 = 25%
Saying “load shedding fell by 3%” would be wrong — it fell by 3 hours, which is 25% of the Week 1 total. Confusing the unit of the absolute change with a percent is a classic relative-difference trap.
Ratio checklist under time pressure
- Identify the whole (sum of parts) or the known part.
- Simplify the ratio before splitting large totals when possible.
- For percent change, write (new − old) / old.
- For map scales, multiply then convert units step by step.
- Re-read of / more than / less than / twice as many before selecting an option.
These habits connect directly to multi-step QL items that chain a ratio split with a percent or a unit conversion in one stem.
R4 500 is shared between two societies in the ratio 2 : 7. How much does the smaller share receive?
Enrolment falls from 120 students to 90 students. What is the relative decrease?
On a map with scale 1 : 25 000, a road measures 8 cm. What is the real length of the road?
Which statement correctly matches the phrase “35% more than 80”?