6.2 Proportion

• Direct Proportion: When one quantity increases, the other increases at the same rate
• Inverse Proportion: When one quantity increases, the other decreases at a related rate

Both types maintain a constant relationship between the quantities!

Direct Proportion:
• More hours worked → more pay earned
• More fuel → longer distance traveled
• More ingredients → more servings

Inverse Proportion:
• More workers → less time to complete a job
• Faster speed → less time to travel
• More people sharing → smaller share each

If $y \propto x$, then:

$$y = kx$$
where $k$ is the constant of proportionality

Also means: $\frac{y_1}{x_1} = \frac{y_2}{x_2}$

5 apples cost £2.50. How much do 8 apples cost?

Method 1: Unitary Method
Step 1: Find cost of 1 apple
Cost of 1 apple = $£2.50 \div 5 = £0.50$

Step 2: Find cost of 8 apples
Cost of 8 apples = $8 \times £0.50 = £4.00$

Method 2: Ratio Method
$$\frac{£2.50}{5} = \frac{x}{8}$$
$$x = \frac{£2.50 \times 8}{5} = £4.00$$

$y$ is directly proportional to $x$. When $x = 4$, $y = 20$. Find $y$ when $x = 7$.

Step 1: Find constant $k$
$y = kx$
$20 = k \times 4$
$k = 20 \div 4 = 5$

Step 2: Write the formula
$y = 5x$

Step 3: Find $y$ when $x = 7$
$y = 5 \times 7 = 35$

Answer: When $x = 7$, $y = 35$

A recipe for 6 people uses 450g of flour. How much flour is needed for 10 people?

Step 1: Find flour per person
Flour per person = $450 \div 6 = 75$ g

Step 2: Calculate for 10 people
Flour needed = $75 \times 10 = 750$ g

Alternative using ratios:
$$\frac{450}{6} = \frac{x}{10}$$
$$x = \frac{450 \times 10}{6} = 750 \text{ g}$$

✓ As one increases, the other increases
✓ The ratio $\frac{y}{x}$ stays constant
✓ The graph is a straight line through the origin
✓ Doubling one quantity doubles the other

If $y \propto \frac{1}{x}$, then:

$$y = \frac{k}{x}$$ or $$xy = k$$
where $k$ is the constant

Also means: $x_1 \times y_1 = x_2 \times y_2$

It takes 6 workers 8 hours to paint a house. How long would it take 4 workers?

Step 1: Find the constant
$k = 6 \times 8 = 48$ worker-hours

Step 2: Calculate time for 4 workers
$4 \times t = 48$
$t = 48 \div 4 = 12$ hours


Logic: Fewer workers → more time ✓

A car traveling at 60 km/h takes 3 hours to complete a journey. How long would it take at 90 km/h?

Step 1: Find the constant (total distance)
$k = 60 \times 3 = 180$ km

Step 2: Calculate time at 90 km/h
$90 \times t = 180$
$t = 180 \div 90 = 2$ hours

Alternative method:
$$\frac{60}{90} = \frac{t}{3}$$
$$t = \frac{60 \times 3}{90} = 2 \text{ hours}$$

Logic: Faster speed → less time ✓

$y$ is inversely proportional to $x$. When $x = 5$, $y = 12$. Find $y$ when $x = 3$.

Step 1: Find constant $k$
$xy = k$
$5 \times 12 = k$
$k = 60$

Step 2: Write the formula
$y = \frac{60}{x}$

Step 3: Find $y$ when $x = 3$
$y = \frac{60}{3} = 20$

Answer: When $x = 3$, $y = 20$

A cake recipe for 8 people uses:
• 300g flour
• 200g sugar
• 4 eggs

Adapt the recipe for 12 people.

Step 1: Find the scale factor
Scale factor = $\frac{12}{8} = 1.5$

Step 2: Multiply all ingredients
• Flour: $300 \times 1.5 = 450$ g
• Sugar: $200 \times 1.5 = 300$ g
• Eggs: $4 \times 1.5 = 6$ eggs

Which is better value?
• Small box: 400g for £2.80
• Large box: 650g for £4.55

Step 1: Calculate price per 100g
Small box: $\frac{£2.80}{400} \times 100 = £0.70$ per 100g
Large box: $\frac{£4.55}{650} \times 100 = £0.70$ per 100g

Product	Size	Price	Price per 100g
Small Box	400g	£2.80	£0.70
Large Box	650g	£4.55	£0.70

Conclusion: Both are the same value per 100g!

Which washing powder is the best value?

Brand	Weight	Price	Price per kg	Best Buy?
Brand A	2 kg	£8.50	£4.25
Brand B	3 kg	£11.70	£3.90	✓ Best Buy
Brand C	5 kg	£20.00	£4.00

Working:
• Brand A: $£8.50 \div 2 = £4.25$ per kg
• Brand B: $£11.70 \div 3 = £3.90$ per kg ✓ Cheapest
• Brand C: $£20.00 \div 5 = £4.00$ per kg

Step 1: Calculate unit price (price per gram, per kg, per litre, etc.)
Step 2: Compare unit prices
Step 3: Lowest unit price = best value

Remember:
• Convert to the same units first!
• Bigger size doesn't always mean better value
• Consider if you'll use the larger amount

A printer prints 45 pages in 3 minutes. How many pages can it print in 7 minutes?

Identify: Direct proportion (more time → more pages)

Step 1: Find pages per minute
Pages per minute = $45 \div 3 = 15$ pages/minute

Step 2: Calculate for 7 minutes
Pages in 7 minutes = $15 \times 7 = 105$ pages

8 taps can fill a swimming pool in 6 hours. How long would it take 12 taps?

Identify: Inverse proportion (more taps → less time)

Step 1: Find constant
$k = 8 \times 6 = 48$ tap-hours

Step 2: Calculate for 12 taps
$12 \times t = 48$
$t = 48 \div 12 = 4$ hours

Alternate method:
More taps means proportionally less time:
$\frac{8}{12} = \frac{t}{6}$
$t = \frac{8 \times 6}{12} = 4$ hours

£1 = $1.30. How many dollars do you get for £50?

Direct proportion:
$\frac{£1}{\$1.30} = \frac{£50}{x}$

$x = £50 \times \$1.30 = \$65$

Answer: £50 = $65