Lesson 13 ~25 min Unit 1 · Financial Maths +85 XP

Unitary Method with Rates

Find the cost (or distance, or amount) for 1, then scale up or down to any number you like.

Today's hook: 3 pens cost $\$4.50$. How much do 7 pens cost? The unitary method is your secret weapon.

0/5QUESTS

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom

Think First

warm-up

3 pens cost $\$4.50$. How much do 7 pens cost? The unitary method is your secret weapon. Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

The unitary method: find the cost/distance/value of ONE thing first, then multiply by however many you want. It works for ANY scaling problem.

3 pens cost $\$4.50$. Step 1: find the cost of 1 pen: $\$4.50 \div 3 = \$1.50$. Step 2: scale to 7 pens: $7 \times \$1.50 = \$10.50$. The method works for any rate: divide to find 1, multiply by what you need.

Cost of $n$ items $= n \times $ (cost per item)

Find 1 first

Divide by the original quantity.

Then scale

Multiply by the new quantity.

Works for any rate

Distance, time, dollars — same logic.

What You'll Master

objectives

Know

Unitary method: find unit value, then scale
Use it for pricing, distances, times, weights
Divide first, multiply second

Understand

Why dividing then multiplying works for proportional relationships
When the unitary method fails (non-proportional discounts)
How distance-speed-time problems are unitary rate problems

Can Do

Solve scaling problems by finding the unit rate first
Apply unitary method to distance-speed-time
Use unitary method for bulk pricing

Words You Need

vocabulary

Unitary methodFinding the value of ONE first, then scaling.

ProportionalWhen doubling one quantity doubles the other.

Scale upMultiply by a number bigger than 1.

Scale downMultiply by a fraction or divide.

Distance, Speed, Time$d = s \times t$; speed is a unit rate.

WorkingShowing both steps: divide then multiply.

Spot the Trap

heads-up

Wrong: "3 pens cost $\$4.50$, so 7 pens cost $7 \times 4.50 = \$31.50$" — NO. You forgot to find the cost of ONE first.

Right: Find 1 first: $\$4.50 \div 3 = \$1.50$. Then $7 \times 1.50 = \$10.50$.

Wrong: "If 5 workers take 6 hours, then 10 workers take 12 hours." — NO. More workers should take LESS time.

Right: For work problems, more workers = less time. Not directly proportional like cost.

Cost Problems

+5 XP

The most common application: find the unit cost, then scale up or down.

A box of 12 cupcakes costs $\$30$. Cost per cupcake: $30 \div 12 = \$2.50$. To buy 20 cupcakes: $20 \times 2.50 = \$50$. To buy 5 cupcakes: $5 \times 2.50 = \$12.50$. The unit rate ($\$2.50$ per cupcake) does all the heavy lifting.

Cost $= n \times $ (Cost per unit)

Divide first

Total cost $\div$ total units = unit cost.

Then multiply

Unit cost $\times$ new quantity.

Same result both ways

Could also use ratios.

Distance, Speed, Time

+5 XP

Speed is a unit rate (km/h). DST problems are unitary at their core.

You travel 240 km in 3 hours. Speed $= 240 \div 3 = 80$ km/h (the unit rate). At this speed, how far in 5 hours? $5 \times 80 = 400$ km. And how long to travel 600 km? $600 \div 80 = 7.5$ hours. The triangle $d = s \times t$ has unitary thinking baked in.

$d = s \times t$, $s = \dfrac{d}{t}$, $t = \dfrac{d}{s}$

Distance = speed × time

Multiply.

Speed = distance ÷ time

Divide.

Time = distance ÷ speed

Divide.

Watch Me Solve It · 3 examples

Watch Me Solve It · The pen problem

+15 XP per step

PROBLEM

3 pens cost $\$4.50$. How much do 7 pens cost?

1

Cost of 1 pen

$\$4.50 \div 3 = \$1.50$

Divide by 3.
2

Cost of 7 pens

$7 \times \$1.50$

Scale up.
3

Compute

$7 \times 1.50 = \$10.50$

Final cost.

Answer$\$10.50$

Watch Me Solve It · Bulk discount sanity

+15 XP per step

PROBLEM

A 5 kg bag of flour costs $\$13$. How much is 8 kg at the same rate?

1

Cost per kg

$13 \div 5 = \$2.60$

Unit rate.
2

Cost of 8 kg

$8 \times 2.60$

Multiply.
3

Compute

$8 \times 2.60 = \$20.80$

Final cost.

Answer$\$20.80$

Watch Me Solve It · DST

+15 XP per step

PROBLEM

A train covers 270 km in 3 hours. How far would it travel in 5 hours at the same speed?

1

Find the speed

$270 \div 3 = 90$ km/h

Unit rate.
2

Distance in 5 hours

$5 \times 90 = 450$ km

Scale by time.
3

Check

At $90$ km/h, $5$ h $= 450$ km ✓

Consistent.

Answer$450$ km

Common Pitfalls

heads-up

Skipping the unit step

Multiplying by 7 directly without dividing by 3 first.

Fix: Always find the cost/distance/value of ONE first.

Assuming everything is proportional

“Twice as many workers $\to$ twice as long” — wrong direction.

Fix: Workers/time is INVERSE: more workers = less time.

Wrong unit on the answer

Forgetting that the result is in km, $\$$, etc.

Fix: Always include the unit in the answer.

Copy Into Your Books

Unitary Method

Find value of 1 first (divide)
Then scale to many (multiply)
Works for any rate

Cost Problems

Total $\div$ quantity = unit cost
Unit cost $\times$ new quantity
$\$30/12 = \$2.50$ each

DST

$s = d/t$
$d = s \times t$
$t = d/s$

When NOT to use

Discounts on bulk are NOT proportional
Workers/time is inverse
Check assumptions

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Unitary Method with Rates

4 problems

Four drill problems to sharpen your skills. Work each, then reveal the answer.

1 4 books cost $\$28$. Cost of 9 books?

$28 \div 4 = \$7$ each; $9 \times 7 = \$63$.$\$63$
2 A car drives 180 km in 2 h. How far in 5 h?

$180 \div 2 = 90$ km/h; $5 \times 90 = 450$ km.$450$ km
3 5 hours of work pays $\$80$. Pay for 8 hours?

$80 \div 5 = \$16$/h; $8 \times 16 = \$128$.$\$128$
4 $300$ ml of paint covers $4$ m$^2$. How much for $10$ m$^2$?

$300 \div 4 = 75$ ml/m$^2$; $10 \times 75 = 750$ ml.$750$ ml

Complete in your workbook.

Quick Check · 5 questions

5 pens cost $\$8$. The cost of 12 pens is:

+10 XP

A car travels 300 km in 4 hours. Speed is:

+10 XP

Jay's mum pays $\$24$ for 4 kg of tomatoes. The price for 7 kg is:

+10 XP

At 90 km/h, the time to travel 270 km is:

+10 XP

$\$45$ pays for 3 hours of tutoring. 5.5 hours costs:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Solve using the unitary method, showing both steps: (a) 8 oranges cost $\$6.40$ — what does 15 cost? (b) A car uses 12 L in 150 km — how much fuel for 480 km? (c) 5 workers paint a wall in 2 hours — at the same per-person rate, how long for 1 worker?

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A train covers 480 km in 6 hours. At the same speed, how far does it travel in $2\tfrac{1}{2}$ hours?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A printer can produce 270 pages in 12 minutes. (a) Find the unit rate (pages per minute). (b) How long to print 1000 pages, to the nearest minute? (c) The printer is upgraded to be 1.5× faster. How long does the same 1000 pages now take? Explain why the new rate makes the time DIVIDE rather than multiply.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $\$19.20$.

2. C — $75$ km/h.

3. C — $\$42$.

4. C — $3$ h.

5. D — $\$82.50$.

Show Your Working Model Answers

Q6 (3 marks): (a) Unit $\$0.80$; $15 \times 0.80 = \$12$ [1]. (b) Unit $12/150 = 0.08$ L/km; $480 \times 0.08 = 38.4$ L [1]. (c) 5 workers in 2 h = 10 worker-hours; 1 worker takes $10$ h [1].

Q7 (2 marks): Speed $= 480/6 = 80$ km/h [1]; Distance $= 2.5 \times 80 = 200$ km [1].

Q8 (4 marks): (a) $270 \div 12 = 22.5$ pages/min [1]. (b) $1000 \div 22.5 = 44.\overline{4}$ min $\approx 44$ min [1]. (c) New rate $= 22.5 \times 1.5 = 33.75$ pages/min; $1000 \div 33.75 \approx 30$ min [1]. Higher rate means more pages PER minute, so fewer minutes for same total — dividing by a bigger rate gives a smaller time [1].

Stretch Challenge · +25 XP, +10 coins

The Painting Project

2 painters take 5 days to paint a school. (a) How many painter-days does the job require? (b) How long would 5 painters take? (c) If the school wants the job done in 2 days, how many painters do they need? (Assume all painters work at the same rate and don't get in each other's way.)

Reveal solution

(a) $2 \times 5 = 10$ painter-days. (b) $10 \div 5 = 2$ days. (c) $10 \div 2 = 5$ painters needed. (This is INVERSE proportion: more painters, fewer days.)

Quick Review

Find 1

First divide by the original quantity

Scale

Multiply by new quantity

Cost

Per item rate × number wanted

Speed

Distance ÷ time

DST

$d = s \times t$, $s = d/t$, $t = d/s$

Watch inverse

More workers, less time

Your Badges

0 of 6

First Steps

3-Day Streak

3 in a Row

Lesson Ace

Stretch Seeker

Daily Warrior

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +85 XP and +25 coins.

Unit overview · Maths subject page