Direct proportion
If 3 cinema tickets cost 22,50 €, 6 tickets cost 45 €. Twice as many tickets, twice the price. Three times as many tickets, three times the price. When two quantities behave like this, they are directly proportional, and calculating with them is very straightforward.
Directly proportional quantities
| Tickets ($x$) | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| Price in € ($y$) | 7,50 | 15 | 22,50 | 30 | 45 |
| $y/x$ | 7,50 | 7,50 | 7,50 | 7,50 | 7,50 |
Here $k = 7{,}50$: it is the price per unit. The constant almost always has a meaning like this: price per kilo, kilometres per hour, euros per hour…
State whether the following pairs of quantities are directly proportional. Briefly justify your answer: a) The number of followers of an influencer and the number of…
State whether the following pairs of quantities are directly proportional. Briefly justify your answer:
- a) The number of followers of an influencer and the number of likes on their latest photo.
- b) The weight of apples in kilograms that you buy and the total price to pay.
- c) A person's age and their height in centimetres.
- d) The time you leave your phone charging and the percentage of battery restored (assuming it charges at a constant rate).
A group of friends wants to go to the cinema. We know that $3$ tickets cost $22,50$ €. Complete the following table and calculate the constant of proportionality $k$…
A group of friends wants to go to the cinema. We know that $3$ tickets cost $22,50$ €. Complete the following table and calculate the constant of proportionality $k$ (price per ticket):
| Number of tickets | $1$ | $3$ | $5$ | $8$ | $10$ |
|---|---|---|---|---|---|
| Total price (€) | $22,50$ |
The graph: a straight line through the origin
If you plot the points from the table, they lie in a straight line that passes through the origin $(0, 0)$: 0 tickets, 0 €. The constant $k$ is the gradient (how much the line rises for each unit it moves along).
A radio-controlled car covers $12$ metres in just $8$ seconds, maintaining a constant speed. If it continues moving in the same way: a) What distance will it cover in…
A radio-controlled car covers $12$ metres in just $8$ seconds, maintaining a constant speed. If it continues moving in the same way:
- a) What distance will it cover in one minute ($60$ seconds)?
- b) How long will it take to cover a track of $30$ metres?
An electric scooter's battery is consumed in proportion to the distance travelled. If completing a journey of $4,5$ km used up $12 \%$ of the total battery capacity,…
An electric scooter's battery is consumed in proportion to the distance travelled. If completing a journey of $4,5$ km used up $12 \%$ of the total battery capacity, calculate using the unitary method how much battery will be used on a journey of $15$ km over the same type of terrain.
The direct rule of three
To calculate an unknown value there are two equivalent routes:
1. The unitary method: first work out the value for 1, then multiply.
2. Rule of three (proportion): set out the data and use cross-multiplication.
Scales and plans
Scales are direct proportion: real measurement $= k \cdot$ measurement on the map.
To prepare a fruit smoothie for $4$ people, $600$ ml of milk is needed. If next Saturday you are going to invite $10$ friends to your house, how much milk will you need…
To prepare a fruit smoothie for $4$ people, $600$ ml of milk is needed. If next Saturday you are going to invite $10$ friends to your house, how much milk will you need to buy so that everyone gets a smoothie? Solve using direct proportion.
A 3D printer takes $45$ minutes to manufacture a part weighing $60$ grams. If we want to print a figure of the same material weighing $100$ grams, how long will the…
A 3D printer takes $45$ minutes to manufacture a part weighing $60$ grams. If we want to print a figure of the same material weighing $100$ grams, how long will the printer take? Express the result in hours and minutes.
A hiking map of the Sierra de Guadarrama is drawn to a scale of $1:25000$. A group of students is planning a route that measures $14,4$ cm on the map. 1. How many actual…
A hiking map of the Sierra de Guadarrama is drawn to a scale of $1:25000$. A group of students is planning a route that measures $14,4$ cm on the map.
-
How many actual kilometres will they cover in total?
-
If they walk at a constant speed and take $45$ minutes to cover the first $3$ km, how long will it take them in total to complete the route?
Percentages as direct proportion
A percentage is a proportion out of 100: 15 % of 45 € is $\frac{15}{100} = \frac{x}{45}$, so $x = 6{,}75$ €. That is why a quantity and its percentage are directly proportional: 15 % of double the amount is double.
You have seen a hoodie that you love in an online shop. Its original price is $45$ €, but today it has a $15\%$ discount. How many euros do you save thanks to the…
You have seen a hoodie that you love in an online shop. Its original price is $45$ €, but today it has a $15\%$ discount. How many euros do you save thanks to the discount? (It is not necessary to calculate the final price).
At the same time of day, the shadow of a road sign $2,2$ metres high is $1,4$ metres long. If right next to it there is an electricity pylon whose shadow is $10,5$…
At the same time of day, the shadow of a road sign $2,2$ metres high is $1,4$ metres long. If right next to it there is an electricity pylon whose shadow is $10,5$ metres long, how tall is the pylon? Draw a sketch of the situation before solving the problem.
You can do it on paper and send a photo with your answer.