Direct proportion
The more cinema tickets you buy, the more you pay. The more grams of flour you put in the recipe, the more pancakes you get. In both situations, 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 video game gem shop, 100 gems cost 1,50 €. Look at the table and check that the price/gems quotient is always the same:
| Gems | 100 | 200 | 300 | 400 |
|---|---|---|---|---|
| Price (€) | 1,50 | 3 | 4,50 | 6 |
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…
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:
- a) The number of followers of a streamer and the number of likes they receive on a video (assuming all their followers click "like").
- b) A person's age and their height in centimetres.
- c) The number of pizza boxes you order and the total price you pay (if all pizzas cost the same).
- d) The speed of a car and the time it takes to reach its destination.
- 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…
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:
| Number of Gems | $100$ | $200$ | $300$ | $500$ | $1.000$ |
|---|---|---|---|---|---|
| Price (€) | $1,50$ | $4,50$ |
The unitary method
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…
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).
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…
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.
Cross-multiplication: the rule of three
Another way to solve it is to set up a proportion and apply cross-multiplication.
$$a) \ \frac{2}{5} = \frac{10}{x} \quad\Rightarrow\quad 2x = 50 \quad\Rightarrow\quad x = 25$$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…
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 $x$.
Find the value of the unknown term $x$ in the following proportions so that the equality between the ratios holds true: a) $\frac{2}{5} = \frac{10}{x}$ b)…
Find the value of the unknown term $x$ in the following proportions so that the equality between the ratios holds true:
- a) $\frac{2}{5} = \frac{10}{x}$
- b) $\frac{x}{12} = \frac{3}{4}$
- c) $\frac{7}{x} = \frac{14}{6}$
On a map of Sierra de Guadarrama, the scale indicates that $2\text{ cm}$ on paper represent $5\text{ km}$ in reality. If, for a hiking route between two villages, the…
On a map of Sierra de Guadarrama, 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?
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 gems, you pay nothing.
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…
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.
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,…
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?