Exercises: 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).

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).

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?

A video game takes up $12$ GB of space and took $8$ minutes to download with a stable connection. If the next game you want to download takes up $18$ GB and the download speed is the same, how many minutes will the download take?

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.

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).

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?

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).

In an artisanal bakery in Galway, organic baguettes are sold at the same unit price. Four identical baguettes cost €5.40 and you want to buy $7$ baguettes.

1. a) Before doing any written calculations, make an estimate: reason between which two consecutive integer amounts of euros the final price of the $7$ baguettes must lie, and explain your intuitive deduction.
2. b) Calculate the exact cost using the unitary method (price of a single baguette).
3. c) Compare the exact result with your initial estimate and assess whether the figure obtained is consistent with the given situation.

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 $100$ €?

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?

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).

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?

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?

A smartphone fast charger is able to charge $18\%$ of the battery in just $12$ minutes. If the charging speed remains constant:

1. a) What percentage of battery will have charged after half an hour ($30$ minutes)?
2. b) How much total time (in minutes and seconds) will it take for the phone to go from $0\%$ to $100\%$ charge?

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?

A scout group from the Guadarrama mountains is preparing a hiking route. According to their leader's logs, the group consumes $15\text{ litres}$ of drinking water for every $6\text{ km}$ walked at a constant pace. They want to predict their consumption for two different days: a flat stage of $18\text{ km}$ and a mountain stage of $7\text{ km}$.

Two procedures are proposed: Method A (unitary method, calculating how many litres they consume per kilometre) and Method B (multiplying or dividing directly using the scale factor between the known kilometres and those to be walked).

1. a) For the $18\text{ km}$ stage, justify which of the two methods allows you to calculate the required water faster and mentally. Find the result using that method.
2. b) For the $7\text{ km}$ stage, justify which method is clearer and more practical. Calculate the required litres of water with it.
3. c) What is the practical meaning of the constant of proportionality $k = \frac{15}{6}$ in this context, and in what units is it expressed?

According to data from Met Éireann, during a storm of steady intensity in County Cork, a rain gauge collected $14\text{ litres}$ of water per square metre in $40\text{ minutes}$.

1. a) Assuming that the rain falls uniformly, verify that time and the amount of collected water are directly proportional, and write the mathematical formula relating the litres per square metre collected ($y$) as a function of time in minutes ($x$).
2. b) How many litres per square metre will accumulate after $2\text{ hours and a half}$ of continuous rain?
3. c) Met Éireann issues an orange rainfall warning in that area if $40\text{ litres}$ per square metre are exceeded in one hour. With the intensity recorded in this episode, would the orange warning threshold be reached after $60\text{ minutes}$? Justify your answer with calculations.

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 3D printer filament manufacturing company knows that the length of extruded filament is directly proportional to the operating time of the extruder machine. If $45$ metres of filament of a given thickness are produced in $15$ minutes:

• a) How many kilometres of filament will be manufactured if the machine works without interruption for a full working day of $8$ hours?

• b) If the production cost is $\text{€}\ 0.12$ for each decametre ($10\text{ m}$) of filament, what will the total production value be after those $8$ hours?

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 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.

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?

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).

A lighthouse casts a shadow of $18\text{ metres}$ on the ground. At that same moment, a vertical post $2.5\text{ metres}$ high located near the lighthouse casts a shadow of $150\text{ cm}$.

• a) Calculate the height of the lighthouse.

• b) If the sun lowers and the shadow of the post increases by a further $30\text{ cm}$, how long will the new shadow of the lighthouse be?

An assembly line produces a constant number of electronic components every hour. It is known that by working $5$ days at a rate of $8$ hours per day, they have manufactured a total of $14\,000$ components.

If due to an urgent order they need to manufacture $25\,200$ components in the next $6$ days, how many hours per day must the machines work to meet the deadline, assuming that the hourly production rate does not change?

In the Science laboratory, the relationship between the mass ($M$, in grams) and the volume ($V$, in cubic centimetres) of an unknown liquid is being investigated using graduated cylinders. It is known with certainty that both magnitudes are directly proportional. After the experiment, part of the notebook was stained with ink, but three clues are preserved:

1. When the volume increases by $15\text{ cm}^3$, the mass increases by exactly $12\text{ g}$.

2. The constant of proportionality $k = \frac{M}{V}$ is a terminating decimal less than $1$.

3. The sum of the mass of $20\text{ cm}^3$ of that liquid and the volume occupied by $32\text{ g}$ of the same liquid is equal to $56$.

1. a) Deduce the exact value of the constant $k$ (density of the liquid in $\text{g/cm}^3$) from the preserved data, and prove that clues 1 and 3 are consistent with each other.
2. b) Write the analytical expression that gives the mass as a function of the volume.
3. c) If you have a container that supports a maximum load of $140\text{ g}$, what is the maximum volume of liquid that you can pour into it without exceeding that mass?

The municipal electric bicycle rental service of an Irish city applies the following fare: a fixed $\text{€}\ 1.20$ to unlock the bicycle plus $\text{€}\ 0.15$ for each minute of the ride.

Aoife argues: “If for a $10\text{-minute}$ ride I pay $1.20 + 10 \times 0.15 = \text{€}\ 2.70$, when I pedal for twice the time ($20\text{ minutes}$) I will have to pay double ($\text{€}\ 5.40$). Since doubling the time doubles the cost, both quantities are directly proportional”.

1. a) Calculate the actual cost of a $20\text{-minute}$ trip and determine whether Aoife's statement about the price is correct.
2. b) Use the ratio $\frac{\text{price}}{\text{time}}$ for $10$ and for $20\text{ minutes}$ and explain in detail why the time of use and the final price are NOT directly proportional quantities, pointing out which element of the fare breaks this proportionality and how this would be reflected in its graph compared to a line of direct proportionality.
3. c) Modify the original statement by removing or changing only one element of the fare so that the total price becomes directly proportional to the time of use. State what the new constant of direct proportionality $k$ would be and what its meaning in real life would be.