Inverse proportion

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?

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:

Workers	$2$	$4$	$6$	$3$
Days	$12$	$6$

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?

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

a)

$x$	$2$	$3$	$4$	$6$
$y$	$18$	$12$	$9$	$6$

b)

$x$	$2$	$3$	$4$	$5$
$y$	$30$	$20$	$15$	$10$

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).

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?

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 $4$ bricklayers takes $18$ days to build the perimeter wall of a municipal sports centre. To finish earlier, the site manager decides to bring in $2$ more bricklayers with the same work rate, making a total of $6$ bricklayers.

Cian, one of the labourers, sets up the following rule of three to calculate the duration of the work:

$$4\ \text{bricklayers} \longrightarrow 18\ \text{days}$$ $$6\ \text{bricklayers} \longrightarrow x\ \text{days}$$ $$x = \frac{6 \times 18}{4} = 27\ \text{days}$$

1. a) Without doing complex numerical operations, explain why Cian's answer makes no sense from a real-world perspective.
2. b) What conceptual mistake did Cian make when setting up the proportion?
3. c) Correct the calculation and find the exact number of days that the $6$ bricklayers will take using the inverse proportionality constant (the constant product).

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 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?

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 $\text{€}\ 180$ 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 $\text{€}\ 180$.

A sports foundation distributes a financial scholarship of $\text{€}\ 1300$ among the top three athletes of a school race. To reward merit, the money is distributed in inverse proportion to the time in minutes that each runner took to cross the finish line.

The first-place finisher took $20\ \text{minutes}$ and the second-place finisher took $30\ \text{minutes}$. However, the bib with the chip of the third-place finisher fell off during the race, and their exact time was not recorded in the computer system. The official prize distribution certificate shows that the first-place finisher received exactly $\text{€}\ 600$.

1. a) From the prize obtained by the first-place finisher, calculate the distribution constant ($k$) and find how much money the second-place finisher received.
2. b) Deduce how much money was left over from the total fund of $\text{€}\ 1300$ to be awarded to the third-place finisher.
3. c) Set up the corresponding equation and reconstruct the exact time (in minutes) that the third-place finisher took to complete the race.