Inverse proportion

State whether each pair of quantities is directly proportional, inversely proportional, or neither:

1. a) The speed of a cyclist and the time taken to arrive.
2. b) The kilograms of strawberries bought and the price paid.
3. c) The number of people harvesting a garden and the time they take.
4. d) The age of a tree and its height.

State whether the following statements are true or false, giving reasons for your answer:

1. a) If as one quantity increases the other decreases, the two quantities are inversely proportional.
2. b) If two quantities are inversely proportional, doubling one of them halves the other.
3. c) A mother's age and her daughter's age are inversely proportional quantities.
4. d) If two inversely proportional quantities take the values $(3, 20)$, they can also take the values $(5, 12)$.

Complete the table knowing that the quantities are inversely proportional:

1. a) How many days do $6$ workers take?
2. b) How many days do $3$ workers take?
3. c) How many days would $1$ worker take?

A tap delivering $120$ litres per minute fills a swimming pool in $15$ hours.

1. a) How long will another tap delivering $200$ litres per minute take to fill it?
2. b) What is the volume of the pool, in litres? Explain why it is not necessary to know this to answer part a).

A team of $6$ workers takes $10$ days to repair a park track.

1. a) How many days would $4$ workers take working at the same rate?
2. b) How many workers are needed to finish the track in $5$ days?

A car travelling at $90\ \text{km/h}$ takes $4$ hours to go from one city to another. To find out how long it would take travelling at $120\ \text{km/h}$, a pupil set up a direct proportion and got $5,\widehat{3}$ hours.

1. a) Explain what mistake they made.
2. b) Calculate the correct time at $120\ \text{km/h}$.
3. c) What speed should it travel at to take only $3$ hours?

Two cogwheels are meshed together: when one turns, it forces the other to turn. The large wheel has $36$ teeth and makes $12$ complete turns.

1. a) If the small wheel has $16$ teeth, how many turns does it make in the same time?
2. b) Why are the number of teeth and the number of turns inversely proportional quantities? What product remains constant?
3. c) If we wanted the small wheel to make $72$ turns, how many teeth should it have?

A grandmother wants to share $180\ \text{€}$ between her three grandchildren, aged $2$, $3$ and $6$ years old, in inverse proportion to their ages (so that the youngest receives more).

1. a) Sharing in inverse proportion to their ages is the same as sharing in direct proportion to $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$. Calculate the sum of these three fractions.
2. b) What fraction of the total corresponds to each grandchild?
3. c) Calculate how many euros each one receives and check that they add up to $180\ \text{€}$.