Direct proportion
The more phone-controller covers you buy, the more money you pay. The more white paint you mix, the more tint you need to keep the same shade. In both cases, when you multiply one quantity by a number, the other is multiplied by the same number: this is direct proportion.
What is direct proportion?
In a custom-printing shop, 3 printed T-shirts cost 18 €. Complete the table and check that the price/T-shirts quotient is always the same:
| T-shirts | 3 | 6 | 9 | 15 |
|---|---|---|---|---|
| Price (€) | 18 | 36 | 54 | 90 |
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 ($€$) | $24$ |
The unitary method
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…
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).
A video game streaming platform offers a premium subscription. We know that $35,20$ € 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 $35,20$ € 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 (€) |
- c) How many months could you subscribe for if you have a maximum budget of $100$ €?
Cross-multiplication: the rule of three
If you prefer to set up a proportion, you solve it by cross-multiplication.
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 centimetres.
- 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).
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 metres.
- b) Calculate the actual area of the living room in square metres.
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 $45$ 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 $0,12\text{ €}$ for each decametre ($10\text{ m}$) of filament, what will the total production value be after those $8$ hours?
The graph: a straight line through the origin
If you plot the pairs of values of a direct proportion, the points always lie on a straight line and that line passes through the point $(0, 0)$: with no T-shirts, you pay nothing.
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…
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).
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?