5.4 Ratios, Rates & Proportions
Key Takeaways
- A ratio compares two or more quantities in the same units when possible; simplify by dividing all parts by the same number.
- A rate compares different units (for example dollars per litre or kilometres per hour); a unit rate has 1 in the denominator.
- A proportion states that two ratios are equal; solve missing values by cross-multiplication when the relationship is proportional.
- Ratio tables scale all parts by the same factor so totals and shares stay consistent.
- Use money, fuel, recipes, class sizes, and travel speeds—never hose pressure or ladder-load technical formulas.
5.4 Ratios, Rates & Proportions
Quick Answer: A ratio a:b compares amounts; a rate compares different units; a unit rate is “per 1.” A proportion a/b = c/d is solved by cross-multiplication: a × d = b × c. Scale shares with a ratio table so every part multiplies by the same factor.
Ratios and proportions turn “fair shares,” map scales, recipe mixes, and “if this then how much” questions into reliable algebra-free arithmetic. This is still general Mathematics for a civil-service entrance MCQ—not fireground engineering.
Ratios: meaning and notation
A ratio compares two or more quantities. Ways to write the same ratio of 6 to 4:
- 6:4
- 6 to 4
- 6/4 (as a fraction form of the comparison)
Simplify by dividing all terms by their greatest common factor: 6:4 = 3:2.
Order matters. 3:2 is not the same as 2:3. Match the order in the stem (“teachers to students” vs “students to teachers”).
Three-part ratios
A paint mix (civilian craft example) of red:blue:white = 2:3:5 means for every 2 + 3 + 5 = 10 parts, 2 are red, 3 blue, 5 white.
If total paint is 20 litres, each “part” is 20 ÷ 10 = 2 litres → red 4 L, blue 6 L, white 10 L.
Worked example A — simplify and interpret
In a form class there are 12 boys and 18 girls. Ratio boys:girls = 12:18 = 2:3. Girls:boys = 3:2. Fraction of the class that is boys: 12/(12+18) = 12/30 = 2/5.
Ratio tables (scaling)
A ratio table multiplies or divides all parts by the same number so the ratio stays equivalent.
| Parts flour | Parts sugar | Total parts |
|---|---|---|
| 3 | 1 | 4 |
| 6 | 2 | 8 |
| 9 | 3 | 12 |
| 15 | 5 | 20 |
Example. Recipe ratio flour:sugar = 3:1. For 15 cups flour, sugar = 5 cups (scale factor 5).
Worked example B — sharing money in a ratio
Three roommates share a $360 utility bill in the ratio 2:3:1 (A:B:C).
- Total parts: 2 + 3 + 1 = 6
- Value of one part: 360 ÷ 6 = $60
- A: 2 × 60 = $120; B: $180; C: $60
- Check: 120 + 180 + 60 = 360
Trap: giving A $2, B $3, C $1 without scaling to the total.
Worked example C — adjusting one share
If B instead pays $210 under the same 2:3:1 structure, find the total bill.
B has 3 parts = 210 → one part = 70 → total 6 × 70 = $420. A pays 140; C pays 70.
Rates and unit rates
A rate compares quantities with different units: kilometres per hour, dollars per kilogram, pages per minute.
A unit rate answers “how many of the first quantity per 1 of the second.”
Worked example D — unit price
Juice costs $45 for 3 litres. Unit rate: 45 ÷ 3 = $15 per litre. Another brand is $28 for 2 litres → $14 per litre—better value if quality is equal.
Worked example E — speed as a rate
A maxi covers 150 km in 3 hours. Unit rate: 150 ÷ 3 = 50 km/h. At that constant rate, in 5 hours: 50 × 5 = 250 km. Time for 200 km: 200 ÷ 50 = 4 hours.
Worked example F — work-style rate (non-technical)
A clerk files 120 forms in 4 hours → 30 forms per hour. Forms in 7 hours at same rate: 30 × 7 = 210. Hours for 90 forms: 90 ÷ 30 = 3 hours.
Proportions
A proportion says two ratios are equal: a/b = c/d (b ≠ 0, d ≠ 0).
Cross-multiplication: a × d = b × c. This is the standard MCQ tool when one term is missing.
Worked example G — missing term
Solve: 4/10 = x/25
- Cross-multiply: 4 × 25 = 10 × x
- 100 = 10x
- x = 10
Check: 4/10 = 0.4; 10/25 = 0.4. Equal.
Worked example H — map-style scale (civilian)
On a simple sketch map, 2 cm represents 5 km. How many km does 7 cm represent?
Proportion: 2/5 = 7/x or 2 cm / 5 km = 7 cm / x km.
Using 2/5 = 7/x → 2x = 35 → x = 17.5 km.
Using unit scale: 1 cm = 2.5 km; 7 × 2.5 = 17.5 km. Same answer.
Worked example I — recipe proportion
If 6 roti use 750 g of flour, how much flour for 10 roti (same recipe)?
6/750 = 10/x → 6x = 7,500 → x = 1,250 g.
Or unit rate: flour per roti = 750 ÷ 6 = 125 g; 10 × 125 = 1,250 g.
Worked example J — currency-style conversion rate
Suppose a shop posts a simple practice rate of $1 USD exchanges for $6.80 TTD for a classroom word problem (illustrative only—not live market data). How many TTD for USD 25?
1/6.80 = 25/x → x = 25 × 6.80 = $170 TTD.
Direct proportion vs inverse proportion (awareness)
- Direct proportion: as one quantity doubles, the other doubles (constant rate). More hours at fixed speed → more distance.
- Inverse proportion: as one doubles, the other halves (product constant). More workers sharing a fixed total of person-hours may finish sooner—only when the stem clearly states that inverse relationship.
Most entrance-style items at foundation level use direct proportion and unit rates. Do not force inverse logic unless the wording requires it.
Direct check: if x₁/y₁ = x₂/y₂, ratios of corresponding pairs match.
Inverse check: if x₁ × y₁ = x₂ × y₂, products match.
Worked example K — inverse only when stated
“A job takes 6 people 8 days (same work rate each). How many days for 4 people?” If the stem assumes inverse proportion: 6 × 8 = 4 × d → 48 = 4d → d = 12 days. Only use this when equal-work inverse is implied; otherwise stay with direct rates given explicitly.
Connecting ratios to fractions and percents
Ratio men:women = 3:5 → men are 3/8 of the group, women 5/8. Percent men: (3/8) × 100% = 37.5%.
This link lets you move among chapters of the Mathematics subject without new formulas.
Multi-step civilian example — fuel and trip budget
A driver budgets for a trip: the car uses fuel at a steady 8 litres per 100 km (rate given in the problem). Trip length 250 km. Fuel price $7.50 per litre.
- Fuel needed: (8/100) × 250 = 20 litres
- Cost: 20 × 7.50 = $150
If two drivers split fuel 2:3, total parts 5; one part $30; shares $60 and $90.
No pump pressure or firefighting hydraulics—only rates and ratios.
Common trap table (ratios, rates, proportions)
| Trap | Wrong move | Fix |
|---|---|---|
| Reversed ratio | 2:5 read as 5:2 | Match named order |
| Adding ratio numbers as amounts | Ratio 2:3 of $100 → $2 and $3 | Parts of total 5 → $40 and $60 |
| Cross-multiply upside down | From a/b = c/d write a×b = c×d | a×d = b×c |
| Mixed units | km/h with minutes unconverted | Convert to consistent units |
| Unit rate forgotten | Compare $18/2 L to $25/3 L by looking at $ only | Compute price per litre |
| Scaling only one part | Flour ×2 but sugar unchanged | Multiply every ratio term |
| Treating all word problems as inverse | Halving time when speed doubles without checking | Use direct rate unless inverse is stated |
Building fluency for MCQs
- Rewrite the comparison as a fraction or ratio with labels (boys/girls).
- Simplify early to make cross-multiplication lighter.
- For “best buy,” compute unit rates the same way for each option.
- After solving a proportion, substitute back to verify equality of decimals or simplified fractions.
- Estimate: if 3 items cost about $30, 12 items should be about $120—reject $40 or $1,200.
Section checkpoint
You can simplify ratios, share totals with part values, compute unit rates, and solve proportions by cross-multiplication. Together with whole numbers, fractions, decimals, and percents, these foundations support the applied arithmetic chapters that follow—still without any firefighting technical math.
Chapter wrap-up (foundations)
Chapter 5 built the number core of the SCD Mathematics subject: order of operations, fraction and decimal fluency, percent change, and proportional reasoning. Practise until each worked pattern feels automatic under multiple-choice pressure. Keep logistics honest: public materials do not fix a universal item count, time limit, pass mark, or domain weight for this entrance exam—your edge is accurate arithmetic, not invented exam statistics.
Simplify the ratio 24:36.
A $480 grocery bill is split among three flatmates in the ratio 3:2:1. How much does the person with 2 parts pay?
Solve for x: 5/8 = x/24.
Rice is sold at $54 for 3 kg. What is the unit price per kilogram, and what would 7 kg cost at that rate?