EST I HS Math · Unit 2 · Data Analysis & Probability
Lesson 2.2 · Ratios, Rates & Unit Conversions
On the paper: 1–3 questions on every real paper; the March 2026 paper starts with one
Big idea. A ratio cuts a total into equal parts. Find the size of ONE part and every ratio question is one multiplication away. A rate is a ratio with units — 'per' means divide, and the units must match before you do.
1One part 🍰
🍼 Baby steps
- 1
men : women = 7 : 4, 66 employees
2026 sample Q4.
- 2
7 + 4 = 11 parts → one part = 66 ÷ 11 = 6
Total ÷ (sum of the parts). This number is the whole question.
- 3
men 42, women 24 → difference (7 − 4) × 6 = 18
A difference is (difference of parts) × one part. You never need both shares.
- 4
$14,000 in 1 : 2 : 4 → one part 2,000 → Yasmine (2 parts) $4,000
March 2026 Q1. Three parts or ten — same move.
- 5
$12,000 in 5 : 3, winner paid $600 to enter → 7,500 − 600 = 6,900 profit
December 2024. Read the last word: PROFIT, not share.
🎛 Play with it · parts and the total
Slide the parts and the total; the bar shows one part and each share. Then use the rate converter to see km/h ↔ m/s.
ratio 7 : 4 · 11 parts · one part = 66 ÷ 11 = 6 · difference of first two = (7 − 4) × 6 = 18
Rate converter · speed = distance ÷ time
= 12 km/h = 3.33 m/s · at this speed 1 km takes 5 min
km/h → m/s: divide by 3.6. min → h: divide by 60. Convert the units BEFORE dividing, never after.
2Changing a ratio: let the unknown be the number added ➕
🍼 Baby steps
- 1
34 doctors, 68 nurses. Add d doctors so doctors : staff = 3 : 7
October 2020. Note: doctors to TOTAL staff, not to nurses.
- 2
(34 + d) / (102 + d) = 3/7
The total grows too — the new doctors are staff.
- 3
7(34 + d) = 3(102 + d) → 238 + 7d = 306 + 3d → d = 17
Cross-multiply once.
- 4
two numbers 3 : 5, subtract 9 from each → 12 : 23
March 2021. Write them as 3k and 5k: (3k − 9)/(5k − 9) = 12/23 → k = 11 → biggest 55.
- 5
35% of a = ⅔ of b → a/b = (2/3) ÷ 0.35 = 40/21
March 2021. A ratio hiding in a sentence: isolate a/b directly.
3Rates: 'per' means divide by ONE ⏱
🍼 Baby steps
- 1
$165 for 30 hours → 165 ÷ 30 = $5.50 per hour → 40 h → $220
March 2021. Find the unit rate, then scale.
- 2
4 kg for $1 = $0.25/kg; 2 kg for $0.40 = $0.20/kg → reduced by $0.05
January 2023. Compare per-kilogram, never per-bag.
- 3
4 laps = 3 miles → 0.75 mile per lap → 5 laps = 3.75
January 2023. "One additional lap" → total 5 laps.
- 4
1.2 m³ ↔ 9 gallons → 26 m³ ↔ 26 × 9 ÷ 1.2 = 195
March 2026. Proportion: multiply by the new, divide by the old.
- 5
1 machine: 600 m in 1 h. 2 machines, 300 m: 1 h × ½ (distance) × ½ (machines) = 15 min
August 2021. Halve the job, halve the time; double the machines, halve again.
Trap · units first
4Real EST questions, solved twice 📄
The ratio of men to women in an office is 7 to 4. If there are 66 employees in this office, what is the difference in numbers between the two genders?
- A.12
2 parts, or 66 − 54 — no route gives 12 except a guess.
- B.14
66 ÷ 11 = 6, then 6 + 8?
- C.16
66 − 50 — the numbers were forced.
- D.18
Safe way · ~35 s
- 11 parts. One part = 6.
- Men 42, women 24. 42 − 24 = 18.
⏱ 90-second way · ~10 s
- (7 − 4) × (66 ÷ 11) = 3 × 6 = 18.
Answer: D
An amount of $14,000 is divided among three sisters, Layla, Yasmine and Aya, in the ratio 1 : 2 : 4 respectively. How much did Yasmine receive?
- A.$2,000
One part — Layla’s share.
- B.$4,000
- C.$7,000
Half of the total; Yasmine has 2 of 7 parts, not 1 of 2.
- D.$8,000
Aya’s share (4 parts).
Safe way · ~25 s
- 1 + 2 + 4 = 7 parts. One part = 14,000 ÷ 7 = 2,000.
- Yasmine: 2 × 2,000 = 4,000.
⏱ 90-second way · ~8 s
- 14,000 × 2/7 = 4,000. Read WHICH sister: the middle one.
Answer: B
In a hospital, the medical staff is composed of 34 doctors and 68 nurses. How many additional doctors should join in order for the ratio of doctors to the total number of medical staff to become 3 to 7?
- A.21
Solves (34 + d)/68 = 3/7 — forgot that the total grows.
- B.19
Arithmetic slip.
- C.17
- D.15
Solves doctors : nurses = 3 : 4 instead of doctors : total = 3 : 7.
Safe way · ~50 s
- (34 + d)/(102 + d) = 3/7.
- 238 + 7d = 306 + 3d → 4d = 68 → d = 17.
⏱ 90-second way · ~15 s
- Test options: d = 17 → 51/119 = 3/7 ✓ (51 = 3 × 17, 119 = 7 × 17).
Answer: C
If 1.2 m³ of a pool can be filled by 9 gallons of water, how many gallons are needed to fill a 26 m³ pool?
- A.195
- B.202
Rounded 7.5 gallons/m³ to 7.8.
- C.223
26 × 9 − 11 — no.
- D.234
26 × 9, forgetting to divide by 1.2.
Safe way · ~30 s
- Gallons per m³: 9 ÷ 1.2 = 7.5.
- 26 × 7.5 = 195.
⏱ 90-second way · ~10 s
- 26 ÷ 1.2 × 9 = 195. Multiply by the new, divide by the old.
Answer: A
A machine finishes paving 600 m of road in 1 hour. At the same steady rate, how much time would two identical machines take to pave 300 m?
- A.15 minutes
- B.2 hours
Doubled the time for doubling the machines — more machines means LESS time.
- C.1 hour
Halved for distance, then forgot the second machine.
- D.30 minutes
Halved once (distance OR machines), not twice.
Safe way · ~35 s
- One machine: 300 m takes 30 min.
- Two machines share it: 15 min.
⏱ 90-second way · ~8 s
- Half the road, twice the workers → a quarter of the hour: 15 min.
Answer: A
⏱ The 90-second lane
One question = 90 seconds. These are the moves that finish this topic in one line.
- 1One part = total ÷ (sum of parts). Share = parts × one part. Difference = (difference of parts) × one part.
- 2Read the last word: share, difference, profit, largest, or the middle name.
- 3Adding people changes the total too: (a + x)/(T + x).
- 4Two numbers in ratio a : b → write them as ak and bk, then one equation.
- 5"35% of A is ⅔ of B" → A/B = (⅔) ÷ 0.35. Isolate the ratio directly.
- 6Unit rate = per ONE (per hour, per kg, per lap). Compare per-unit, then scale.
- 7Proportion: new = old × (new quantity ÷ old quantity).
- 8Machines / workers: more of them → divide the time. Less work → divide the time.
- 9Speed: convert minutes to hours (÷ 60) BEFORE dividing. km/h → m/s: ÷ 3.6.
Take with you
- ▸Ratios are parts of a total; one part unlocks every question.
- ▸Rates are ratios with units; "per" is a division, and units must agree first.
- ▸Proportional scaling and combined-worker problems are multiply-by-new, divide-by-old.