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Rates & Unit Rates

Maths · Y8 Ratios & Rates
Year 8
Lesson Guide
Explanation
Ratios compare like with like (flour : sugar, no units). RATES join two DIFFERENT quantities with 'per': 90 km per hour, $4.50 per kg, 60 beats per minute. The slash is the word 'per'. A UNIT rate has exactly 1 on the bottom — 90 km per ONE hour — and that 1 is what makes rates comparable, calculable, and the workhorse of this lesson.
Your Turn
Rate or ratio? 3 : 2 cordial mix; $2.80/L petrol price; 5 goals per game. Check: ratio, rate, rate.
Watch Out
Keep the units in your answer — '90' is meaningless; '90 km/h' is a speed. Rates die without their units.
Explanation
Unit rates decide every best-buy: 500 g for $6 versus 800 g for $8.80. Divide price by amount to get a matching unit: A is $1.20 per 100 g, B is $1.10 per 100 g — B wins despite the bigger sticker price. Supermarket shelf labels print per-100g prices for exactly this comparison. Bigger isn't always cheaper — the unit rate is the only honest referee.
Your Turn
Better buy: 3 L of juice for $5.40, or 2 L for $3.80? Check: $1.80/L vs $1.90/L — the 3 L.
Watch Out
Compare in the SAME unit — $/kg against $/100g flatters one side tenfold. Pick one basis and convert both.
Explanation
Speed is the most famous rate: distance per time. The triangle holds all three versions — cover the letter you want and read the formula: S = D÷T, D = S×T, T = D÷S. 240 km in 3 hours → 80 km/h. At 80 km/h for 2.5 hours → 200 km. 360 km at 90 km/h → 4 hours. One triangle, three questions; the same pattern will serve density, pricing and every other rate.
Your Turn
A cyclist rides 45 km in 2.5 h — speed? A hike at 4 km/h for 3.5 h — distance? Check: 18 km/h; 14 km.
Watch Out
Time must match the rate's unit: 90 minutes at 80 km/h is 1.5 h × 80 = 120 km — never multiply by 90!
Explanation
The ambush in rate questions is mismatched time units. '45 km in 20 minutes — speed in km/h?' Don't divide by 20! Either convert (20 min = ⅓ h, so 45 ÷ ⅓ = 135) or — often slicker — scale up: three 20-minute blocks make an hour, so ×3 → 135 km/h. Know the fraction-hours cold: 15 min = ¼, 20 = ⅓, 30 = ½, 45 = ¾. The scale-up trick works for any rate: 6 L per 90 s → 4 L per minute.
Your Turn
A runner does 2.5 km in 15 minutes. Speed in km/h? Check: ×4 → 10 km/h.
Watch Out
The answer's unit tells you which time to use: km/H means hours in the division, no matter what the question measured in.
Explanation
Every rate problem is the speed triangle in a costume. Wages: $22.50/h × 6.5 h = $146.25 (rate × amount). Fuel: 7.5 L per 100 km over 640 km → 6.4 hundreds × 7.5 = 48 L (per-100 rates scale by distance ÷ 100). Typing: 1350 words at 45 words/min → 30 minutes (amount ÷ rate). Three moves cover everything: FIND the rate (divide), SCALE it (multiply), or divide BY it (how long/how many). Identify which one the question wants and the costume falls away.
Your Turn
A printer does 24 pages/min. How long for 300 pages? And a $19/h job for 7.5 h pays? Check: 12.5 min; $142.50.
Watch Out
Per-100 rates (fuel, medicine doses) trip people: divide the distance by 100 FIRST, then multiply by the rate.
Explanation
The journey killer: 120 km at 60 km/h then 120 km at 40 km/h — average speed? The seductive (60+40)/2 = 50 is WRONG. Average speed is always TOTAL distance ÷ TOTAL time: the legs take 2 h and 3 h, so 240 km ÷ 5 h = 48 km/h. The slow leg consumed more time, dragging the average below the midpoint. There is no shortcut — compute both times, then divide totals. This question appears on every Y8 exam precisely because the shortcut feels so right. Quiz and exercises!
Your Turn
Walk 6 km at 6 km/h, then 6 km at 3 km/h. Average speed? Check: 1 h + 2 h → 12 ÷ 3 = 4 km/h (not 4.5!).
Watch Out
Averaging the speeds only works when the TIMES are equal — which the question will never arrange for you.
📌 Key Points — this lesson in a nutshell
  • A rate compares DIFFERENT units; a ratio compares the same units.
  • The unit rate (cost per 1) is the fair judge in every best-buy comparison.
  • Speed triangle: S = D ÷ T, D = S × T, T = D ÷ S.
  • Convert time units to match the rate BEFORE you divide.
  • Average speed = TOTAL distance ÷ TOTAL time — never average the two speeds.
Step 1: Introduction
Step 1 of 6
Ratio vs rate RATIO same units, no name flour : sugar = 3 : 1 RATE different units, has a name 90 km/h · $4.50/kg · 60 bpm A rate joins two DIFFERENT quantities with "per" (the / sign): kilometres per hour, dollars per kilogram, beats per minute, L per 100 km. A UNIT rate has 1 on the bottom: 90 km per 1 hour. Unit rates are the universal comparison currency — next step shows why.
Which is the better buy? Box A: 500 g for $6.00 600 ÷ 500 = 1.2 c/g = $12.00 per kg Box B: 800 g for $8.80 880 ÷ 800 = 1.1 c/g = $11.00 per kg ✓ cheaper Best-buy recipe: convert BOTH to the same unit rate, compare. Divide price by amount → price per gram (or per 100 g — often neater: A: $1.20/100 g vs B: $1.10/100 g). Supermarket shelf labels print per-100g prices for exactly this reason!
The speed triangle D S T S = D ÷ T · D = S × T · T = D ÷ S 240 km in 3 h → S = 80 km/h · 2.5 h at 80 → D = 200 km 360 km at 90 km/h → T = 4 h. Cover the letter you want; the triangle shows the rest.
Minutes are the ambush "A train covers 45 km in 20 minutes. Speed in km/h?" 20 min = ⅓ h → 45 ÷ ⅓ = 135 km/h. (Or: ×3, since 3 twenties make an hour.) "A tap fills 6 L in 90 seconds. Rate in L/min?" 90 s = 1.5 min → 6 ÷ 1.5 = 4 L/min. Fractions of hours: 15 min = ¼ · 20 min = ⅓ · 30 min = ½ · 45 min = ¾ Scaling up is often easier than dividing: per-20-min × 3 = per-hour.
Same machine, many costumes Wages: $22.50/h × 6.5 h = $146.25 rate × amount, exactly like D = S × T. Fuel: a car uses 7.5 L/100 km. For 640 km → 6.4 × 7.5 = 48 L. per-100 rates: scale by (distance ÷ 100). Typing: 45 words/min. How long for a 1350-word essay? amount ÷ rate = 30 minutes — the T = D ÷ S move. Every rate question is one of three: find the rate, scale it up, or divide by it.
Average speed ≠ average of speeds! "Drive 120 km at 60 km/h, then 120 km at 40 km/h. Average speed for the whole trip?" NOT (60 + 40)/2 = 50 ✗ Leg times: 120÷60 = 2 h · 120÷40 = 3 h → total 5 h average = TOTAL distance ÷ TOTAL time = 240 ÷ 5 = 48 km/h ✓ Why? The slow leg ate MORE TIME (3 h vs 2 h), so it drags the average below the midpoint. Averages weight by TIME, not by leg. Always: total distance ÷ total time. No shortcuts exist.
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