Direct Proportion in Real Life

Worked example

1. 1

   $30\times 2=\pounds 60$ — The hourly part for 2 hours: £30 for each of the 2 hours.

2. 2

   $60+20=\pounds 80$ — Add the fixed £20. 2 hours cost £80.

3. 3

   $30\times 4=\pounds 120$ — The hourly part for 4 hours.

4. 4

   $120+20=\pounds 140$ — Add the fixed £20 once only. 4 hours cost £140.

5. 5

   $140\ne 2\times 80=160$ — Twice the hours did not cost twice as much, because the £20 is paid once. So this is not direct proportion.

Worked example

1. 1

   $10\text{ cakes}:250\text{ g flour}$ — Write down what the recipe says. We want 16 cakes, so first find the flour for one cake.

2. 2

   $250÷10=25\text{ g}$ — The 250 g is shared over 10 cakes, so divide by 10. That is the flour for one cake.

3. 3

   $16\times 25=400\text{ g}$ — Each of the 16 cakes needs 25 g, so multiply by 16.

Worked example

1. 1

   $600÷25=24\text{ cakes}$ — Each cake uses 25 g, so count how many 25s fit into 600.

2. 2

   $24\times 25=600\text{ g}$ — Check: 24 cakes at 25 g each use exactly 600 g.

Worked example

1. 1

   $15÷6=2.5$ — Find the multiplier: 15 people is 2.5 times as many as 6 people.

2. 2

   $240\times 2.5=600\text{ g}$ — Multiply the old amount by the same multiplier. Everything in the recipe grows 2.5 times.

Worked example

1. 1

   $360÷90=4$ — Compare butter with butter: Sam uses 4 times as much as the recipe.

2. 2

   $6\times 4=24\text{ people}$ — The number of people grows by the same multiplier, 4.

Worked example

1. 1

   $5\text{ kg apples}=\pounds 6.40$ — Start with the apples. They are the only fruit with a price of their own.

2. 2

   $6.40÷5=\pounds 1.28$ — Share £6.40 over 5 kg. That is the price of 1 kg of apples.

3. 3

   $2\times 1.28=\pounds 2.56$ — The basket has 2 kg of apples, so work out what they cost.

4. 4

   $7.21-2.56=\pounds 4.65$ — Take the apples out of the total. What is left paid for the 3 kg of pears only.

5. 5

   $4.65÷3=\pounds 1.55$ — Share that over the 3 kg of pears to get the price of 1 kg.

6. 6

   $3\times 1.55+2.56=7.21$ — Check: 3 kg of pears plus the apples gives the £7.21 total.

Worked example

1. 1

   $16\text{ biscuits}:280\text{ g sugar}$ — Write down the recipe. Find the sugar for one biscuit first.

2. 2

   $280÷16=17.5\text{ g}$ — Share 280 g over 16 biscuits. That is the sugar for one biscuit.

3. 3

   $30\times 17.5=525\text{ g}$ — Multiply by 30. This is the sugar she needs.

4. 4

   $0.5\text{ kg}=500\text{ g}$ — Change kg to grams so both amounts use the same unit. 1 kg = 1000 g.

5. 5

   $500<525$ — Compare: she has less sugar than she needs.

6. 6

   $525-500=25\text{ g}$ — Answer with numbers: “No, she needs 525 g but has only 500 g. She is 25 g short.”

Worked example

1. 1

   $150÷12=12.5\text{ g}$ — Butter for one muffin: share the recipe’s 150 g over 12 muffins.

2. 2

   $680÷12.5=54.4$ — How many muffins the butter you have could make on its own.

3. 3

   $90÷12=7.5\text{ g}$ — Sugar for one muffin.

4. 4

   $500÷7.5=66.6\dots$ — The sugar could make about 66 muffins.

5. 5

   $1.5\text{ kg}=1500\text{ g}$ — Change kg to grams to match the recipe.

6. 6

   $240÷12=20\text{ g}$ — Flour for one muffin.

7. 7

   $1500÷20=75$ — The flour could make 75 muffins.

8. 8

   $\text{smallest}:54\text{ muffins}$ — Every muffin needs all three, so the butter runs out first. 54.4 rounds down to 54.

Worked example

1. 1

   $180÷12=15\text{ g}$ — Flour for one roll.

2. 2

   $70\times 15=1050\text{ g}$ — Flour for 70 rolls.

3. 3

   $1050÷500=2.1$ — That is 2.1 bags of flour.

4. 4

   $3\text{ bags}$ — Round up. 2 bags hold only 1000 g, which is not enough, and you cannot buy part of a bag.

Worked example

1. 1

   $15÷1.75=8.57\dots$ — How many notebooks £15 covers.

2. 2

   $8\text{ notebooks}$ — Round down. She cannot pay for a 9th notebook.

3. 3

   $8\times 1.75=\pounds 14.00$ — What the 8 notebooks cost.

4. 4

   $15-14=\pounds 1.00$ — Her change. Check: £1.00 is less than £1.75, so she really cannot buy another.

Worked example

1. 1

   $400\text{ g}:\pounds 2.80$ — The small pack. 400 g is 4 lots of 100 g.

2. 2

   $2.80÷4=70\text{p}$ — Share the price over the 4 lots: 70p for 100 g in the small pack.

3. 3

   $600\text{ g}:\pounds 3.90$ — The big pack. 600 g is 6 lots of 100 g.

4. 4

   $3.90÷6=65\text{p}$ — Share the price over the 6 lots: 65p for 100 g in the big pack.

5. 5

   $65\text{p}<70\text{p}$ — Same amount of food, so now the prices can be compared directly.

6. 6

   $\text{big pack is better value}$ — It costs less for each 100 g. Write the 65p and 70p in your answer as your reason.

Worked example

1. 1

   $3\times 60=180\text{ minutes}$ — Change hours to minutes, so both times use the same unit.

2. 2

   $180÷15=12$ — Count the blocks of 15 minutes in 180 minutes.

3. 3

   $12\times 12=144\text{ bottles}$ — The machine fills 12 bottles in each block.

Worked example

1. 1

   $7÷4=1.75\text{ minutes}$ — Time for one lap.

2. 2

   $25\times 1.75=43.75\text{ minutes}$ — Time for 25 laps.

3. 3

   $43.75<45$ — Yes. Say why with numbers: 43.75 minutes is under the 45-minute limit.