Direct and Inverse Proportion

Compare 15 notebooks with 5 notebooks.

Find the cost of 1 metre first.

How long does the machine take to fill 1 bottle?

What would 0 kg of apples cost?

With more builders, the wall takes less time to build.

Work out the amount for 1 smoothie first.

Part (a)

1. 1 smoothie needs 300\div4=75 ml of yoghurt.
2. 10 smoothies need 75\times10=750 ml.

Part (b)

1. 1 smoothie needs 160\div4=40 g.
2. 10 smoothies need 40\times10=400 g.

Work out how long 1 cleaner would take on their own.

1. 1 cleaner would take 4\times12=48 hours.
2. 6 cleaners would take 48\div6=8 hours.

With fewer horses, the hay lasts longer. How long would it last for 1 horse?

1. The hay would feed 1 horse for 6\times10=60 days.
2. For 4 horses it would last 60\div4=15 days.

Find the cost of 1 ticket.

Part (a)

1. 1 ticket costs 30\div4=7.50 pounds.
2. 6 tickets cost 7.50\times6=45, so £45

Part (b)

1. Each ticket costs £7.50
2. 75\div7.50=10, so Ruby buys 10 tickets.

Work out the cost of 1 roll in each pack.

1. Pack A: 120\div4=30p for each roll.
2. Pack B: 150\div6=25p for each roll.
3. Pack B is cheaper for each roll, so Pack B is the better value.

In inverse proportion, when one quantity doubles, the other quantity halves.

1. Sharing £60: 2 people get £30 each, 4 people get £15 each. Doubling the people halves the amount each gets, so this is inverse proportion.
2. The apples and the wheels are direct proportion: doubling one doubles the other.
3. The plumber's charge is neither, because of the fixed £40.

What would a journey of 0 miles cost, according to the graph?

1. A direct proportion graph is a straight line through the origin (0, 0).
2. This line meets the fare axis at £3, so a 0 mile journey would cost £3, not £0.
3. Sam is not correct, because the graph does not go through (0, 0).

If the number of friends doubles, what happens to the amount each friend pays? Can it ever be £0?

1. Doubling the number of friends halves the amount each pays, so this is inverse proportion.
2. An inverse proportion graph is a curve that gets closer and closer to the axes but never touches them.
3. Graph C is a straight line that reaches the x-axis, which would mean the friends pay nothing. So the answer is Graph B.

Put the number of people in place of n and work out 60\div n.

Part (a)

1. M=60\div4=15

Part (b)

1. Friday: M=60\div3=20 minutes.
2. Saturday: M=60\div5=12 minutes.
3. 20-12=8 minutes fewer.

Find a point where the line crosses grid lines exactly, then work out the cost of 1 metre.

1. From the graph, 5 metres cost £20 (the line passes through a grid corner there).
2. So 1 metre costs 20\div5=4 pounds.
3. 15 metres cost 4\times15=60, so £60

Work out how many days 1 decorator would take. Remember you can only have a whole number of decorators.

1. 1 decorator would take 5\times12=60 days.
2. To finish in 7 days you need 60\div7=8.57... decorators.
3. 8 decorators would not finish in time, so 9 decorators are needed.

Count the work in "gardener-days": how many are needed altogether, and how many are done in the first 8 days?

1. The whole job takes 6\times20=120 gardener-days.
2. In the first 8 days, 6\times8=48 gardener-days are done.
3. Work left =120-48=72 gardener-days.
4. There are now 8 gardeners, so it takes 72\div8=9 more days.

Work out how many bottles 1 machine makes in 1 hour.

1. In 1 hour, 3 machines make 600\div4=150 bottles.
2. So 1 machine makes 150\div3=50 bottles in 1 hour.
3. 5 machines make 5\times50=250 bottles in 1 hour.
4. 1500\div250=6 hours.

For each ingredient, work out how many muffins Zara has enough for.

1. Flour: 400\div150\times12=32 muffins.
2. Sugar: 300\div100=3, so 3\times12=36 muffins.
3. Eggs: 6\div2=3, so 36 muffins.
4. Milk: 500\div200=2.5, so 2.5\times12=30 muffins.
5. The milk runs out first, so the greatest number is 30 muffins.

Find the sale price of Pack B first, then compare the cost of 1 battery in each pack.

Part (a)

1. 20\% of £6.80 is 0.2\times6.80=1.36 pounds.
2. Price after the discount =6.80-1.36=5.44 pounds.
3. Cost of 1 battery =5.44\div8=0.68 pounds.

Part (b)

1. Pack A: 9.00\div12=0.75 pounds for 1 battery.
2. Pack B: 0.68 pounds for 1 battery.
3. Pack B is cheaper for each battery, so Pack B is the better value.