Inverse Proportionality

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 $(5, 12)$.

Check whether the quantities in these tables are inversely proportional by calculating the product of each pair of values:

a)

b)

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?

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?

Five identical taps fill a tank in $8$ hours. Let's calculate it step by step:

1. a) If there were only $1$ tap, how many hours would it take? (Think: a single tap works $5$ times slower.)
2. b) What if there were $10$ taps?

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?

A faucet delivering $120$ liters per minute fills a swimming pool in $15$ hours.

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

A bus travelling at $60\ \text{km/h}$ takes $45$ minutes to complete its route.

1. a) How long would it take travelling at $90\ \text{km/h}$?
2. b) At what speed should it travel to complete the route in $30$ minutes?

The graph shows the number of days taken by different teams of painters to paint the same house according to the number of painters in the team.

1. a) How many days does a team of $4$ painters take? And a team of $2$?
2. b) How many painters are needed to paint the house in $6$ days?
3. c) For each point on the graph, multiply the number of painters by the number of days taken. What do you notice? What type of proportion is there between the two quantities?

On a farm there is enough feed to feed $20$ horses for $30$ days.

1. a) For how many horses would the same feed last $25$ days?
2. b) If the farm buys $10$ more horses, how many days will the feed last?

At a campsite there is enough food prepared for $12$ campers to eat for $15$ days.

1. a) On the first day, $3$ more campers arrive. For how many days will the group have food?
2. b) If instead of people arriving, $2$ campers left before starting, how many days would the food last?

A team of $12$ workers can finish a building project in $20$ days, all working at the same rate.

1. a) How many days would $8$ workers take?
2. b) The work starts with the $12$ workers, but after $5$ days of work, $4$ workers leave for another project. How many more days will the remaining $8$ workers need to finish it?
3. c) How many days did the work take in total?

A tank fills in $6$ hours if $3$ identical taps are turned on at the same time.

1. a) How long would it take to fill with just one tap?
2. b) All $3$ taps are turned on, but after $2$ hours one of them breaks down and stops working. How many more hours will the remaining $2$ taps take to finish filling the tank?
3. c) How many hours in total did the tank take to fill?

A grandmother wants to share $\$3\,600$ 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 pesos each one receives and check that they add up to $\$3\,600$.