Mathematics • Year 8 • Unit 1 • Lesson 13
Unitary Method with Rates
Build the “find 1 first, then scale” habit. One fully-worked example, one guided example with blanks, then eight independent problems from quick price scaling up to distance-speed-time.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why the “find 1 first” move always works.
Problem. 3 pens cost $\$4.50$. How much do 7 pens cost?
Step 1 — Notice that 7 isn't a clean multiple of 3.
$7 = $ not 2 lots of 3, not 3 lots of 3. We can't scale directly.
Reason: when the new number isn't a tidy multiple, the unitary method is the safest move.
Step 2 — Find the cost of ONE pen first.
$\$4.50 \div 3 = \$1.50$ per pen
Reason: divide the total by the number you have. This is the “per 1” rate.
Step 3 — Scale up to 7 pens.
$7 \times \$1.50 = \$10.50$
Reason: once you know the per-1 rate, multiply by however many you want.
Step 4 — Sanity check.
7 pens cost a bit more than 6 pens (= $\$9$) and less than 8 pens (= $\$12$). $\$10.50$ fits. ✓
Reason: always check the answer is in the right ballpark.
Answer: 7 pens cost $\$10.50$ (per-pen rate $\$1.50$).
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. A 5 kg bag of flour costs $\$13$. At the same price-per-kg, how much do 8 kg cost?
Step 1 — Check that 8 isn't a clean multiple of 5. So we'll use the unitary method.
Step 2 — Cost of 1 kg:
$\$13 \div $ ______ $= \$$ ______ per kg
Step 3 — Cost of 8 kg:
______ $\times \$$ ______ $= \$$ ______
Step 4 — Sanity check: 5 kg costs $\$13$; 8 kg should cost (more / less) than $\$13$ but less than $\$26$ (which would be 10 kg). Tick the right word: ______ (circle).
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (cost or pay scaling). The middle two are standard (distance-speed-time). The last two are extension (scaling down, or non-proportional check).
Foundation — find 1, scale up
3.1 4 books cost $\$28$. Use the unitary method to find the cost of 9 books. 2 marks
3.2 5 hours of casual work pays $\$80$. At the same hourly rate, what does 8 hours pay? 2 marks
3.3 8 oranges cost $\$6.40$. What does 15 cost? 2 marks
3.4 300 mL of paint covers 4 m². At the same coverage, how much paint do you need for 10 m²? 2 marks
Standard — distance, speed and time
3.5 A car travels 300 km in 4 hours. (a) Find the speed in km/h. (b) At the same speed, how far in 5.5 hours? 2 marks
3.6 A car uses 12 L of fuel to travel 150 km. At the same rate, how much fuel is needed for 480 km? 2 marks
Extension — scaling down or spotting non-proportionality
3.7 A bag of 12 cupcakes costs $\$30$. (a) Find the unit cost per cupcake. (b) Find the cost of 5 cupcakes at the same rate. (c) Find the cost of 20 cupcakes. 2 marks
3.8 5 workers can paint a wall in 2 hours. Aki says “then 10 workers would take 4 hours”. Is Aki right? Explain in one sentence and find the correct time for 10 workers (assuming they each paint at the same per-person rate). 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (5 kg flour for $\$13$, find 8 kg)
Step 2: $\$13 \div \textbf{5} = \$\textbf{2.60}$ per kg.
Step 3: $\textbf{8} \times \$\textbf{2.60} = \$\textbf{20.80}$.
Step 4: 8 kg should cost more than $\$13$ but less than $\$26$ — and $\$20.80$ fits. ✓
3.1 — 9 books
Per book: $\$28 \div 4 = \$7$. For 9 books: $9 \times \$7 = \textbf{\$63}$.
3.2 — 8 hours of work
Per hour: $\$80 \div 5 = \$16/h$. For 8 hours: $8 \times \$16 = \textbf{\$128}$.
3.3 — 15 oranges
Per orange: $\$6.40 \div 8 = \$0.80$. For 15: $15 \times \$0.80 = \textbf{\$12}$.
3.4 — Paint for 10 m²
Per m²: $300 \div 4 = 75$ mL/m². For 10 m²: $10 \times 75 = \textbf{750 mL}$.
3.5 — Car speed
(a) Speed = $300 \div 4 = \textbf{75 km/h}$.
(b) In 5.5 h: $5.5 \times 75 = \textbf{412.5 km}$.
3.6 — Fuel for 480 km
Per km: $12 \div 150 = 0.08$ L/km. For 480 km: $480 \times 0.08 = \textbf{38.4 L}$.
3.7 — Cupcake bag
(a) Per cupcake: $\$30 \div 12 = \textbf{\$2.50}$.
(b) 5 cupcakes: $5 \times \$2.50 = \textbf{\$12.50}$.
(c) 20 cupcakes: $20 \times \$2.50 = \textbf{\$50}$.
3.8 — Workers and time (inverse)
Aki is wrong — more workers should take LESS time, not more. Workers and time are inversely proportional, not directly proportional. With twice as many workers (10 instead of 5) doing the same total job, the time HALVES: $2 \div 2 = \textbf{1 hour}$.