Exercises: Direct proportion

In mathematics, we say that two quantities are directly proportional when, on multiplying one by a number, the other is multiplied by that same number. Analyse the following everyday situations and indicate which ones show direct proportion:

1. a) The number of followers of a streamer and the number of likes they receive on a video (assuming all their followers click "like").
2. b) A person's age and their height in centimetres.
3. c) The number of pizza boxes you order and the total price you pay (if all pizzas cost the same).
4. d) The speed of a car and the time it takes to reach its destination.
5. e) The kilograms of apples you buy and how much they cost.

In your favourite video game, you can buy "Gems" to get new looks for your character. We know that gems always have the same unit price. Complete the following proportionality table in your notebook to find out how much different amounts of gems cost:

A packet of football league stickers contains $6$ stickers, and $3$ of these packets cost $3.60$ €. If we want to complete the album and decide to buy $8$ identical packets, how much money will we spend in total? Use the unitary method (calculate how much a single packet costs first).

Look at the following image representing two sets of scales in balance. On the first set of scales, $3$ blue cubes weigh exactly $450$ grams. On the second set of scales, we want to know how much $5$ cubes identical to the previous ones weigh. Calculate the value of the unknown $x$.

To cook pancakes for $4$ people, exactly $200$ grams of flour are needed. If your cousins are coming over this weekend and there are $6$ people in total, what amount of flour will you need for the pancakes to turn out perfectly? Remember that the recipe quantities are directly proportional to the number of diners.

A skateboarder travels along a track at a constant speed. In the first part of the circuit, they have covered $150$ metres in $30$ seconds. If they maintain the same pace throughout the trip:

1. a) What distance will they cover in $1$ minute ($60$ seconds)?
2. b) What is their constant of proportionality (how many metres do they cover per second)?

Find the value of the unknown term $x$ in the following proportions so that the equality between the ratios holds true:

1. a) $\frac{2}{5} = \frac{10}{x}$

1. b) $\frac{x}{12} = \frac{3}{4}$

1. c) $\frac{7}{x} = \frac{14}{6}$

In direct proportion graphs, the relationship between the two quantities is always represented by a straight line passing through the point $(0,0)$. Look at the following graph relating the number of pens bought to their total cost. If $4$ pens cost $6$ €, identify the point on the graph and calculate how much $10$ pens would cost.

Cian goes with his father to buy tickets for a school theatre festival. At the ticket office, they see the sign: “$3$ tickets cost $€21$”. Cian needs to calculate how much $5$ tickets will cost for his whole family and reasons as follows: “Since $5$ tickets is $2$ tickets more than $3$, I just need to add $€2$ to the price of $€21$. Therefore, the $5$ tickets will cost $€23$”.

1. a) Explain in detail why Cian's additive reasoning is incorrect and what it really means for the number of tickets and the price to be directly proportional quantities.
2. b) Correct the calculation using the unitary method: find the price of one ticket and calculate the actual price the family must pay for the $5$ tickets.

At the local stationery shop, they have a special offer on school supplies: a pack of $3$ identical notebooks costs $\text{€}\ 5.40$. Your working group needs to buy $7$ of those same notebooks for a secondary school project.

1. a) Before doing any written calculations, mentally estimate whether the total price of the $7$ notebooks will be greater or less than $\text{€}\ 12$ and between which two whole numbers of euros it should lie. Explain how you arrived at that estimate.
2. b) Calculate the exact cost of the $7$ notebooks using the unitary method (finding the price of $1$ notebook first).
3. c) Compare the numerical value obtained with your prior estimate and explain whether your result is consistent with the starting price.

On a map of the Wicklow Mountains, the scale indicates that $2\text{ cm}$ on paper represent $5\text{ km}$ in reality. If, for a hiking route between two villages, the distance measured on the map is $8.5\text{ cm}$, how many kilometres will the hikers cover in reality?

To celebrate the end of term, a group of friends wants to buy "gems" for their favourite video game. They have seen that a pack of $1200$ gems costs $\text{€}\ 14.40$. If a friend has saved exactly $\text{€}\ 21$, how many gems will he be able to buy if the relationship is directly proportional?

Copy and complete the following table of directly proportional values relating the number of kilograms of paint to the surface area that can be covered. Also calculate the constant of proportionality $k$ expressed in $m^2/kg$.

A hybrid car consumes an average of $4.5\text{ litres}$ of fuel per $100\text{ km}$ driven. If the tank has a total capacity of $54\text{ litres}$ and is currently full, what maximum distance will the vehicle be able to travel before running out of fuel?

The following graph shows the relationship between the time a water fountain is turned on and the litres it dispenses. Look at the marked point and determine how many litres the fountain will dispense if it is left running for $15\text{ minutes}$.

To cook a giant paella for $12\text{ people}$, the chef uses $1.5\text{ kg}$ of rice and $3\text{ litres}$ of stock. If more people suddenly join and there are now $18$ in total, calculate using the unitary method what amount of rice and stock the chef will need now.

A dripping tap loses approximately $1.2\text{ centilitres}$ of water every $10\text{ seconds}$. If we do not fix it, how many litres of water will it have wasted after a full hour? (Remember to convert the result to the requested unit of measurement first).

Three farmers (Aoife, Seán and Niamh) worked harvesting an orange grove in proportion to the time each dedicated. If they were paid a total of $€1500$ and their working times were:

• Aoife: $12\text{ hours}$

• Seán: $18\text{ hours}$

• Niamh: $20\text{ hours}$

How much money does each of them receive?

We have paid $\text{€}\ 25.50$ for $3$ cinema tickets. We want to buy $7$ tickets for the whole group of friends. Solve the problem in the following two ways:

1. a) By the unitary method: calculate the price of one ticket and then that of $7$.
2. b) Using direct proportion: set up the proportion $\frac{3}{25.50} = \frac{7}{x}$ and solve for $x$.
3. c) Do both results match? Which method do you find easier to use and why?

In the Natural Sciences lab, you are studying the water loss from a faulty tap that drips at a constant rate. Using a graduated cylinder, you measure that exactly $0.36\text{ litres}$ of water are collected every $12\text{ minutes}$.

1. a) You want to find out how much water will be collected after $36\text{ minutes}$. Compare two possible approaches: applying a complete rule of three or using the properties of direct proportionality (observing the direct multiplicative relationship between $12$ and $36$). Justify which of the two methods is faster and calculate that volume.
2. b) Now you need to calculate the volume collected in an interval of $25\text{ minutes}$. Is the previous mental calculation still practical, or is it better to calculate the constant of proportionality (litres per minute) beforehand? Justify your choice and calculate how many litres will have been lost in those $25\text{ minutes}$.

According to technical data for domestic solar self-consumption installations in Ireland, a single-family home equipped with $4$ identical solar panels generates, on a sunny spring day, an average of $6.8\text{ kWh}$ of electrical energy. A neighbouring family wants to install identical panels to cover a daily energy requirement of $15.3\text{ kWh}$ under the same weather conditions.

1. a) Justify why the number of solar panels and the amount of electrical energy generated are directly proportional quantities.
2. b) Determine the constant of direct proportionality $k$, stating what quantity it represents and in what units it is expressed.
3. c) Write the algebraic formula that models the energy produced ($E$, in $\text{kWh}$) as a function of the number of installed panels ($n$).
4. d) Use your model to find out how many whole solar panels the neighbouring family must install to ensure that daily production.

In a robotics workshop, pupils are programming a drone to inspect a nature park. The drone uses battery power directly proportionally to the distance travelled. If to travel $2.5\ km$ the drone uses $15\%$ of its total battery:

1. a) What is the constant of proportionality if we express the relationship as "percentage of battery per kilometre"?
2. b) If the return point is $7,200\ metres$ from the base, will it have enough battery to go and return if it takes off with a full charge? Justify your answer with calculations.

A sustainable clothing brand uses recycled fibres to manufacture its T-shirts. The relationship between the weight of plastic bottles collected and the yarn obtained is directly proportional. It is known that with $3.6\ kg$ of bottles, exactly $8$ size L T-shirts can be made.

1. a) If each size L T-shirt requires $225\ g$ of yarn, how many grams of yarn are obtained per kilogram of plastic bottles?
2. b) The local council of a city has collected $1.2\ toneladas$ of bottles. How many T-shirts could be made in total?

To dye fabrics using natural dyes, an artisan mixes beetroot extract and water. The amount of extract is directly proportional to the volume of water. In her notebook, she has written down that for $1.5\ litros$ of water she needs $120\ ml$ of extract. However, she wants to prepare a tub with a total mixture volume (water + extract) of $10.8\ litros$. What exact amount of water and extract should she mix to maintain the proportion?

The following graph shows the relationship between the charging time of a new smartphone model and the percentage of battery charge recovered. The manufacturer states that charging is directly proportional during the first $40$ minutes.

1. a) Looking at the highlighted point, calculate how many minutes the mobile phone takes to charge $1\%$ of battery.
2. b) If the mobile phone currently has $12\%$ of battery and you need to leave the house in $18$ minutes, what battery percentage will you leave with?

On a mountain map with a scale of $1:25,000$, the distance between two mountain huts is $14.4\ cm$. A group of hikers knows that their average ascent speed is constant and that they take $45$ minutes to cover $1,500\ metros$ in reality on that terrain.

1. a) Calculate the actual distance between the two huts in kilometres.
2. b) What time will they arrive at the second hut if they set off from the first at $08:30$ in the morning?

A video game server distributes data to players directly proportionally to the connection time. A player has downloaded $1.8\ GB$ in the last $12$ minutes.

1. a) Calculate the transfer rate in $MB$ per minute (Remember that $1\ GB = 1024\ MB$).
2. b) If the total game update size is $15\ GB$, how many hours and minutes will the download take to complete if the speed remains constant?

Three Year 7 classes have decided to organise a food collection drive. The number of kilograms collected is directly proportional to the number of pupils in each class. Class A has $24$ pupils and has collected $36\ kg$.

1. a) Complete the following table, justifying your calculations:

1. b) What is the constant of proportionality relating the kilos per pupil? Express it as an irreducible fraction.

A courier company uses electric vans. The cost of electricity consumption is directly proportional to the kilometres driven. Last month, a van covered $1,250\ km$ at a total electricity cost of $€37.50$.

1. a) This month the price of electricity has increased by $20\%$. What will the new constant of proportionality be (cost per kilometre)?
2. b) At the new price, how much will it cost to cover a route of $420\ km$?

A 3D printer takes $4.5$ hours to print an object using $270$ grams of filament. We want to know how long it will take to print another object using $420$ grams, assuming the same rate.

1. a) Explain why the weight of the item and the printing time are directly proportional quantities.
2. b) Set up direct proportion and calculate the time it will take for the $420$-gram item.
3. c) How many grams does the machine print in one hour? Check your result from part b) using this value.

In an oil mill in Kilkenny, extra virgin olive oil is sold. The final price is strictly proportional to the volume of oil purchased, with no fixed packaging costs or bulk discounts. Two customers happen to be in the shop at the same time:

• The first customer buys a certain number of $5\text{-litre}$ carboys and pays an invoice of €135.

• The second customer buys exactly $7\text{ litres}$ more oil than the first and pays a total of €198.

1. a) Deduce the price of a single litre of oil (the constant of proportionality $k$) without previously knowing how many litres each one purchased. Justify with rigour the reasoning that allows you to calculate it from the differences in volume and price.
2. b) Construct the mathematical expression that relates the total cost $C$ (in euros) as a function of the total volume $V$ (in litres).
3. c) From your model, reconstruct the details of the first customer's purchase: calculate how many litres of oil they took and how many full $5\text{-litre}$ carboys they bought.

A city's municipal bicycle rental service offers two payment methods:

• Option A (Pay-as-you-go / No subscription): There is no initial fixed fee; you pay $\text{€}\ 0.60$ for each half-hour fraction of use.

• Option B (Annual subscription): An annual fixed fee of $\text{€}\ 24$ is paid, and then you pay $\text{€}\ 0.20$ for each half-hour fraction of use.

1. a) Check whether the total amount paid per year is directly proportional to the usage time in each of the two options. To justify it, calculate the ratio (total cost / half-hours) for $1$, $2$, and $10$ half-hours in both options and analyse whether it remains constant.
2. b) A cyclist subscribed to Option B claims: “If I use the bicycle twice as long this year as last year, my annual expense will double exactly”. Show with a numerical counterexample that this statement is false and explain what component of the rate prevents direct proportionality.
3. c) A person estimates that they will use the service for exactly $80$ half-hours during the year. Model algebraically the cost of both rates for that value and determine which one is more economical for them, calculating the savings in euros.