Direct proportion

State which of the following pairs of quantities are directly proportional. Briefly justify your answer:

1. a) The number of people who enter a museum and the money collected from ticket sales.
2. b) A person's age and their height.
3. c) The weight of the apples you buy and the total price to pay.
4. d) The speed of a car and the time taken to travel a fixed distance.

In a sports shop, all T-shirts of the same model have a price proportional to the quantity bought. Copy and complete the following table in your notebook and calculate the constant of proportionality $k$ (price per T-shirt).

Seán is buying oranges at his neighbourhood market. He knows that $3\text{ kg}$ of oranges cost $€4.20$ and needs to find out how much $5\text{ kg}$ will cost. To solve this, Seán reasons as follows: “Since $5$ is equal to $3 + 2$, I just need to add $€2$ to the price of $€4.20$, so I will pay $€6.20$”. Identify the mistake in Seán's reasoning, explain why his method does not respect the definition of direct proportionality, and calculate the correct price by first obtaining the constant of direct proportionality $k$ (price per kilogram).

State, giving reasons, which of the following pairs of quantities are directly proportional:

1. a) The number of concert tickets and the total price to pay (without fixed booking fees).
2. b) A person's age and their height in centimetres.
3. c) The weight of loose apples bought and the cost of the purchase.
4. d) The speed of a car and the time taken to cover a fixed distance.
5. e) The number of pages in a book and the time taken to read it (assuming a constant reading speed).

If a tap pours $15$ litres of water every $3$ minutes, how many litres will it pour if it is kept running for $10$ minutes? Solve it using the unitary method (calculating first how much it pours in one minute).

Aoife has paid $4.50$ € for $3$ boxes of felt-tip pens for her drawing project. If she needs to buy $5$ more boxes for her group members, how much money will those $5$ extra boxes cost her?

A video game streaming platform offers a premium subscription. We know that $35.20$ € is paid for $4$ months of service.

1. a) Calculate the constant of direct proportionality $k$ (price per month).
2. b) Complete the following table of values:

1. c) How many months could you subscribe for if you have a maximum budget of $100$ €?

To make a cake for $6$ people, $300$ grams of flour are needed. If we want to invite $15$ people and have everyone eat the same share of cake, how much flour will we need?

To prepare a specific shade of orange paint, an artist mixes $150$ ml of red paint with $225$ ml of yellow paint.

1. a) How many millilitres of red paint will they need if they want to use $600$ ml of yellow paint to get the same shade?
2. b) If the artist decides to make a total mixture of $1$ litre ($1000$ ml), what amount of each colour should they add while maintaining the original proportion?

To dye a batch of cotton T-shirts, a workshop uses a mixture of concentrated dye and water. The recipe states that for every $2.5\text{ litres}$ of water, $150\text{ ml}$ of dye must be added.

If the workshop has a tank with a capacity of $60\text{ litres}$ and wants to fill it while maintaining exactly the same proportion, what quantity of concentrated dye and what quantity of water should be mixed? (Be careful: the final volume of the tank is the sum of both liquids).

On a floor plan of a room with a scale of $1:50$, the length of the bed measures $4$ cm. What is the actual length of the bed expressed in metres? (Remember that a scale of $1:50$ means that $1$ cm on the drawing is equivalent to $50$ cm in reality).

On a floor plan of a house drawn to a scale of $1:250$, the living room has a length of $3.2$ cm and a width of $2.4$ cm.

1. a) Calculate the actual dimensions of the living room in metres.
2. b) Calculate the actual area of the living room in square metres.

Three siblings decide to divide the $144$ € they received as a gift from their grandparents in direct proportion to their ages, which are $10$, $12$ and $14$ years old respectively. How much money does each person get?

Three friends decide to set up a video game streaming channel. At the end of the first month, they have earned $€1\,260$ in advertising revenue. They agree to divide the money in direct proportion to the hours each of them has spent recording and editing content.

Cian spent $12$ hours, Aoife spent twice as long as Cian, and Darragh spent as many hours as half of what Cian and Aoife spent together. Calculate how much money each person gets and determine the constant of proportionality $k$ (money per hour worked).

A local council wants to encourage the use of clean energy and distributes a grant of $\text{€}\ 15\,000$ among three residents' associations that have installed solar panels. The division is made in direct proportion to the area of panels installed.

Community A has installed $120\text{ m}^2$. Community B has installed $25\%$ more than Community A. Community C has installed $60\%$ of the sum of the areas of communities A and B combined. Calculate how much money each community receives.

On a hiking map of the Wicklow Mountains, the distance between two shelters is $12.5\text{ cm}$. We know that the map scale is $1:50\,000$. Two hikers decide to do this route. If their average speed is $4\text{ km/h}$, calculate how long it will take them to get from one shelter to the other, expressing the result in hours and minutes.

A forest fire monitoring drone flies at a constant speed. Control centre records show that the drone covered $10.5$ kilometres in the first $15$ minutes of flight.

1. a) What distance will the drone cover if it maintains the same speed for a full hour?
2. b) If the drone needs to cover a total distance of $63$ km, how many minutes will it take to complete the mission?

The perimeter of a regular hexagon is directly proportional to the length of its side. If a hexagon whose side measures $4$ cm has a perimeter of $24$ cm, calculate the perimeter of another regular hexagon whose side measures $7.5$ cm.

A recycling company pays $0.65$ € for each kilogram of aluminium cans handed in. A group of students collected cans during the month and weighed them in batches:

1. a) How much money will they receive for a batch that weighs $8.4$ kg?
2. b) If by the end of the campaign they have raised a total of $27.30$ €, how many kilograms of aluminium did they manage to recycle in total?

The following graph shows the relationship between the mass of a precious metal and its market value. Note that it is a direct proportion function:

• a) Find the constant of proportionality $k$, which represents the price per gram.

• b) If a piece of jewellery contains $15\%$ of impurities (mass that has no value) and weighs $24\text{ g}$ in total, what is the value of the pure metal it contains based on the graph?