Direct Proportionality
At the corner store, 3 empanadillas cost 90 pesos. If Yaritza buys 6, she pays double; if she buys 9, she pays triple. Does it always work like that? How do you find the price of any amount without guessing? That is direct proportionality, and by the end of this page you will be able to answer it.
What is direct proportionality?
At a custom print shop, 3 printed T-shirts cost $\text{RD}\$180$. Look at the table and check that the price/T-shirts quotient is always the same:
| T-shirts | 3 | 6 | 9 | 15 |
|---|---|---|---|---|
| Price (RD$) | 180 | 360 | 540 | 900 |
Here $k = 180 : 3 = 60$ pesos per T-shirt, and also $360 : 6 = 60$, $540 : 9 = 60$ and $900 : 15 = 60$.
State which of the following pairs of quantities are directly proportional. Briefly justify your answer: a) The number of people who enter a museum and the money…
State which of the following pairs of quantities are directly proportional. Briefly justify your answer:
- a) The number of people who enter a museum and the money collected from ticket sales.
- b) A person's age and their height.
- c) The weight of the apples you buy and the total price to pay.
- 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…
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).
| Number of T-shirts | $2$ | $3$ | $5$ | $10$ |
|---|---|---|---|---|
| Price ($\text{RD\$}\ $) | $24$ |
State, giving reasons, which of the following pairs of quantities are directly proportional: a) The number of concert tickets and the total price to pay (without fixed…
State, giving reasons, which of the following pairs of quantities are directly proportional:
- a) The number of concert tickets and the total price to pay (without fixed booking fees).
- b) A person's age and their height in centimeters.
- c) The weight of loose apples bought and the cost of the purchase.
- d) The speed of a car and the time taken to cover a fixed distance.
- e) The number of pages in a book and the time taken to read it (assuming a constant reading speed).
Unit rate method
If a faucet pours $15$ liters of water every $3$ minutes, how many liters will it pour if it is kept running for $10$ minutes? Solve it using the unitary method…
If a faucet pours $15$ liters of water every $3$ minutes, how many liters 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).
Marta has paid $270$ pesos 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…
Marta has paid $270$ pesos 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 $2\,112$ pesos is paid for $4$ months of service. a) Calculate the constant of direct…
A video game streaming platform offers a premium subscription. We know that $2\,112$ pesos is paid for $4$ months of service.
- a) Calculate the constant of direct proportionality $k$ (price per month).
- b) Complete the following table of values:
| Time (months) | $1$ | $3$ | $6$ | $10$ |
|---|---|---|---|---|
| Price ( pesos) |
- c) How many months could you subscribe for if you have a maximum budget of $6\,000$ pesos?
Rule of three: cross products
If you prefer to set up a proportion, you solve it with the rule of three (cross products).
Remember that solving $5x = 28{,}000$ is solving for the unknown, just like in first-degree equations.
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…
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…
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?
A forest fire monitoring drone flies at a constant speed. Control center records show that the drone covered $10.5$ kilometers in the first $15$ minutes of flight. a)…
A forest fire monitoring drone flies at a constant speed. Control center records show that the drone covered $10.5$ kilometers in the first $15$ minutes of flight.
- a) What distance will the drone cover if it maintains the same speed for a full hour?
- b) If the drone needs to cover a total distance of $63$ km, how many minutes will it take to complete the mission?
The graph: a line through the origin
If you plot the pairs of values of a direct proportionality, the points fall on a straight line and that line passes through the point $(0, 0)$: with no T-shirts, you pay nothing. The constant $k$ is the steepness of the line (the slope).
Move the slider and watch how the line changes as $k$ changes:
A smartphone fast charger is able to charge $18\%$ of the battery in just $12$ minutes. If the charging speed remains constant: a) What percentage of battery will have…
A smartphone fast charger is able to charge $18\%$ of the battery in just $12$ minutes. If the charging speed remains constant:
- a) What percentage of battery will have charged after half an hour ($30$ minutes)?
- 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 $0.65$ pesos for each kilogram of aluminum cans handed in. A group of students collected cans during the month and weighed them in batches: a)…
A recycling company pays $0.65$ pesos for each kilogram of aluminum cans handed in. A group of students collected cans during the month and weighed them in batches:
- a) How much money will they receive for a batch that weighs $8.4$ kg?
- b) If by the end of the campaign they have raised a total of $27.30$ pesos, how many kilograms of aluminum 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…
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?
Scales and proportional sharing
Scale. A $1:250$ scale says that 1 cm on the plan equals 250 cm in real life. It is a direct proportionality with $k = 250$: you multiply the plan measurement by 250 and convert from cm to m by dividing by 100.
Directly proportional sharing. To share an amount in parts proportional to some numbers, find the constant: $k = \text{total} : \text{sum of the numbers}$, and each person receives $k$ times their number.
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…
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 meters? (Remember that a scale…
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 meters? (Remember that a scale of $1:50$ means that $1$ cm on the drawing is equivalent to $50$ cm in reality).
To prepare a specific shade of orange paint, an artist mixes $150$ ml of red paint with $225$ ml of yellow paint. a) How many milliliters of red paint will they need if…
To prepare a specific shade of orange paint, an artist mixes $150$ ml of red paint with $225$ ml of yellow paint.
- a) How many milliliters of red paint will they need if they want to use $600$ ml of yellow paint to get the same shade?
- b) If the artist decides to make a total mixture of $1$ liter ($1000$ ml), what amount of each color 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. a) Calculate the actual dimensions of the…
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.
- a) Calculate the actual dimensions of the living room in meters.
- b) Calculate the actual area of the living room in square meters.
Three siblings decide to divide the $8\,640$ pesos they received as a gift from their grandparents in direct proportion to their ages, which are $10$, $12$ and $14$…
Three siblings decide to divide the $8\,640$ pesos 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 $\text{RD\$}\ 1\,260…
Three friends decide to set up a video game streaming channel. At the end of the first month, they have earned $\text{RD\$}\ 1\,260$ in advertising revenue. They agree to divide the money in direct proportion to the hours each of them has spent recording and editing content.
Marcos spent $12$ hours, Lucía spent twice as long as Marcos, and Sergio spent as many hours as half of what Marcos and Lucía spent together. Calculate how much money each person gets and determine the constant of proportionality $k$ (money per hour worked).
A municipality wants to encourage the use of clean energy and distributes a grant of $\text{RD\$}\ 900\,000…
A municipality wants to encourage the use of clean energy and distributes a grant of $\text{RD\$}\ 900\,000$ 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.
Longer problems
In the harder problems you have to do a preliminary step: convert units (hours to minutes, meters to centimeters), find an intermediate amount (a percent, a total), or use the same proportion several times.
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 $45$…
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 $45$ meters of filament of a given thickness are produced in $15$ minutes:
-
a) How many kilometers of filament will be manufactured if the machine works without interruption for a full workday of $8$ hours?
-
b) If the production cost is $\text{RD\$}\ 7.20$ for each decameter ($10\text{ m}$) of filament, what will the total production value be after those $8$ hours?
On a hiking map of the Sierra de Guadarrama, 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…
On a hiking map of the Sierra de Guadarrama, 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.
To dye a batch of cotton T-shirts, a workshop uses a mixture of concentrated dye and water. The recipe states that for every $2.5\text{ liters}$ of water,…
To dye a batch of cotton T-shirts, a workshop uses a mixture of concentrated dye and water. The recipe states that for every $2.5\text{ liters}$ of water, $150\text{ ml}$ of dye must be added.
If the workshop has a tank with a capacity of $60\text{ liters}$ 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 $18\text{ meters}$ on the ground. At that same moment, a vertical post $2.5\text{ meters}$ high located near the lighthouse casts a shadow…
A lighthouse casts a shadow of $18\text{ meters}$ on the ground. At that same moment, a vertical post $2.5\text{ meters}$ 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…
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?
Go back to the beginning: if 3 empanadillas cost 90 pesos, $k = 90 : 3 = 30$ pesos each, and any amount costs $30 \times$ the amount. It works that way only as long as the price per unit does not change: no discounts and no fixed charges.