A food truck sells 3 tacos for $\$9$. Your whole group wants 12 tacos. Do you have to guess the price, or is there a quick, reliable way to find it? When one quantity grows, the other grows by the same factor. That is direct proportion, and it shows up in prices, recipes, maps, speeds, and sharing money.
What is direct proportion?
A shop prints custom T-shirts. 3 shirts cost $\$18$. Check that price ÷ shirts is always the same:
Shirts
3
6
9
15
Price (dollars)
18
36
54
90
Divide each price by its number of shirts: $18 \div 3 = 6$, $36 \div 6 = 6$, $54 \div 9 = 6$, $90 \div 15 = 6$. Every ratio is 6, so $k = 6$ dollars per shirt.
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State which of the following pairs of quantities are directly proportional. Briefly justify your answer: a) The number of people entering 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 entering a museum and the money collected from tickets.
b) A person's age and their height.
c) The weight of apples you buy and the total price to pay.
d) The speed of a car and the time it takes to travel a fixed distance.
State, giving reasons, which of the following pairs of quantities are directly proportional: a) The number of tickets for a concert and the total price to pay (without…
State, giving reasons, which of the following pairs of quantities are directly proportional:
a) The number of tickets for a concert and the total price to pay (without fixed booking fees).
b) A person's age and their height in inches.
c) The weight of apples bought loose and the cost of the purchase.
d) The speed of a car and the time it takes to travel a fixed distance.
e) The number of pages in a book and the time it takes to read it (assuming a constant reading speed).
To check a table, divide $y$ by $x$ in every column. If you get the same number every time, the quantities are proportional. If even one column differs, they are not.
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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 for one T-shirt).
Sometimes the time comes out as a fraction or decimal of a minute. To write it in minutes and seconds, keep the whole minutes and multiply the leftover part by 60.
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If a faucet pours $15$ gallons of water every $3$ minutes, how many gallons will it pour if it is left open for $10$ minutes? Solve it using the unit rate method…
If a faucet pours $15$ gallons of water every $3$ minutes, how many gallons will it pour if it is left open for $10$ minutes? Solve it using the unit rate method (calculating first how much it pours in one minute).
Sarah 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 classmates, how much money will those $5…
Sarah 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 classmates, how much money will those $5$ extra boxes cost her?
A fast smartphone charger is capable of charging $18\%$ of the battery in just $12$ minutes. If the charging speed remains constant: a) What percentage of the battery…
A fast smartphone charger is capable of charging $18\%$ of the battery in just $12$ minutes. If the charging speed remains constant:
a) What percentage of the battery will have charged after half an hour ($30$ minutes)?
b) How much total time (in minutes and seconds) will the phone take to go from $0\%$ to $100\%$ charge?
A proportion says two ratios are equal. Set it up with the quantities in the same order in both ratios, put a variable where the unknown goes, and solve the equation.
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A video game takes up $12$ GB of space and has taken $8$ minutes to download with a stable connection. If the next game you want to download takes up $18$ GB and the…
A video game takes up $12$ GB of space and has taken $8$ 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 $4$ in has a perimeter of $24$ in, calculate the…
The perimeter of a regular hexagon is directly proportional to the length of its side. If a hexagon whose side measures $4$ in has a perimeter of $24$ in, calculate the perimeter of another regular hexagon whose side measures $7,5$ in.
A forest surveillance drone flies at a constant speed. The control center records that the drone has traveled $10,5$ miles in the first $15$ minutes of flight. a) What…
A forest surveillance drone flies at a constant speed. The control center records that the drone has traveled $10,5$ miles in the first $15$ minutes of flight.
a) What distance will the drone travel if it maintains the same speed for a full hour?
b) If the drone needs to cover a total distance of $63$ miles, how many minutes will it take to complete the mission?
Graph the pairs $(x, y)$ of a direct proportion and the points land on a straight line through the origin $(0, 0)$. If you buy nothing, you pay nothing. The height of the line at $x = 1$ is the constant $k$, which is the line's slope.
Drag the slider and watch the line tilt as $k$ changes.
Compare two lines: a plan that costs $\$1.50$ per mile (through the origin) and a taxi that charges a $\$3$ flat fee plus $\$1.50$ per mile (it starts at $(0, 3)$).
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A recycling company pays $\$0,65…
A recycling company pays $\$0,65$ for each pound of aluminum cans handed in. A group of students has collected cans during the month and weighed them in batches:
a) How much money will they receive for a batch weighing $8,4$ lb?
b) If at the end of the campaign they have raised a total of $\$27,30$, how many pounds of aluminum did they manage to recycle in total?
The following graph represents the relationship between the mass of a precious metal and its market value. Note that it is a direct proportion function: a) Find the…
The following graph represents 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 ounce.
b) If a piece of jewelry contains $15\%$ of impurities (mass with no value) and weighs a total of $24\text{ oz}$, what is the value of the pure metal it contains based on the graph?
Scaling a recipe is a direct proportion: a bigger batch needs ingredients multiplied by the same factor. For a mixture, find how much one batch holds, then count how many batches you need.
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To bake a cake for $6$ people, $300$ ounces of flour are needed. If we want to invite $15$ people and for everyone to eat the same proportion of cake, what amount of…
To bake a cake for $6$ people, $300$ ounces of flour are needed. If we want to invite $15$ people and for everyone to eat the same proportion of cake, what amount of flour will we need?
To prepare a specific orange paint, an artist mixes $150$ fl oz of red paint with $225$ fl oz of yellow paint. a) How many fluid ounces of red paint will they need if…
To prepare a specific orange paint, an artist mixes $150$ fl oz of red paint with $225$ fl oz of yellow paint.
a) How many fluid ounces of red paint will they need if they want to use $600$ fl oz of yellow paint to obtain the same shade?
b) If the artist decides to create a total mixture of $1$ gallon ($1000$ fl oz), what amount of each color must 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 $2,5\text{ liters}$ 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 keeping exactly the same proportion, what amount of concentrated dye and what amount of water must be mixed? (Be careful: the final volume of the tank is the sum of both liquids).
Sometimes money (or anything else) is shared in direct proportion to what each person contributed: hours worked, panels installed, years of age. Each person's share is $k$ times their amount.
When amounts are described by relationships ("50% more than...", "half of..."), first compute every person's amount, then share.
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Three siblings decide to share the $\$144$ they received as a gift from their grandparents in direct proportion to their ages, which are $10$, $12$ and $14…
Three siblings decide to share the $\$144$ they received as a gift from their grandparents in direct proportion to their ages, which are $10$, $12$ and $14$ years respectively. What amount of money does each one receive?
Three friends decide to set up a video game streaming channel. At the end of the first month, they have earned an income of $\$1\,260\text{ \$}$ from advertising. They…
Three friends decide to set up a video game streaming channel. At the end of the first month, they have earned an income of $\$1\,260\text{ \$}$ from advertising. They agree to split the money directly proportional to the hours each one has dedicated to recording and editing content.
Marcus dedicated $12$ hours, Lucy dedicated twice as much as Marcus, and Sam dedicated half as many hours as Marcus and Lucy dedicated together. Calculate how much money each one gets and determine the constant of proportionality $k$ (money per hour worked).
A local town council wants to promote the use of clean energy and distributes a grant of $\$15\,000\text{ \$}$ among three homeowners' associations that have installed…
A local town council wants to promote the use of clean energy and distributes a grant of $\$15\,000\text{ \$}$ among three homeowners' associations that have installed solar panels. The distribution is done directly proportional to the surface area of panels installed.
Association A has installed $120\text{ m}^2$. Association B has installed $25\%$ more than Association A. Association C has installed $60\%$ of the sum of the surface areas of associations A and B together. Calculate how much money each association receives.
A scale like $1:48$ says that 1 unit on the drawing is 48 of the same unit in real life. Real length is a direct proportion of drawing length, and the scale is the constant $k$.
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On a plan of a room with a scale of $1:50$, the length of the bed measures $4$ in. What is the real measurement of the bed expressed in feet? (Remember that the scale…
On a plan of a room with a scale of $1:50$, the length of the bed measures $4$ in. What is the real measurement of the bed expressed in feet? (Remember that the scale $1:50$ means that $1$ in on the drawing is equivalent to $50$ in in reality).
On a house plan drawn to a scale of $1:250$, the living room has a length of $3,2$ in and a width of $2,4$ in. a) Calculate the real dimensions of the living room in…
On a house plan drawn to a scale of $1:250$, the living room has a length of $3,2$ in and a width of $2,4$ in.
a) Calculate the real dimensions of the living room in feet.
b) Calculate the real area of the living room in square feet.
On a hiking map of the Great Smoky Mountains, the distance between two shelters is $12,5\text{ in}$. We know that the scale of the map is $1:50\,000$. Two hikers decide…
On a hiking map of the Great Smoky Mountains, the distance between two shelters is $12,5\text{ in}$. We know that the scale of the map is $1:50\,000$. Two hikers decide to do this route. If their average speed is $4\text{ mph}$, calculate how much time it will take them to get from one shelter to the other, expressing the result in hours and minutes.
At the same moment, the sun's rays hit every object at the same angle. So for any vertical object, $\dfrac{\text{height}}{\text{shadow}}$ is the same number. Height and shadow length are directly proportional.
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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 goes down and the shadow of the post increases by a further $30\text{ cm}$, how long will the new shadow of the lighthouse be?
Hard problems usually chain two or three steps: convert time or length to the right unit, find the unit rate, and then answer. Write each step on its own line.
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A 3D printer filament manufacturing company knows that the length of the extruded thread is directly proportional to the operating time of the extruder machine. If in…
A 3D printer filament manufacturing company knows that the length of the extruded thread is directly proportional to the operating time of the extruder machine. If in $15$ minutes $45$ meters of thread of a certain thickness are produced:
a) How many kilometers of thread will be manufactured if the machine works without interruption during a full working day of $8$ hours?
b) If the production cost is $\$0,12\text{ \$}$ for each decameter ($10\text{ m}$) of thread, what will be the total value of the production after those $8$ hours?
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 a 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 a 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 a day must the machines work to meet the deadline, assuming the production rate per hour does not change?
Back to the start: 3 tacos cost $\$9$, so $k = 9 \div 3 = 3$ dollars per taco. For 12 tacos, $12 \times 3 = \$36$. You can also see that 12 is $4 \times 3$, so the price is $4 \times 9 = \$36$.