Direct Proportionality
If 3 movie tickets cost $\$225$, 6 tickets cost $\$450$. Twice the tickets, twice the price. Three times the tickets, three times the price. When two quantities behave like this, they are directly proportional, and calculating with them is very simple.
Directly proportional quantities
| Tickets ($x$) | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| Price in pesos ($y$) | 75 | 150 | 225 | 300 | 450 |
| $y/x$ | 75 | 75 | 75 | 75 | 75 |
Here $k = 75$: it is the price per ticket. The constant almost always has a meaning like this: price per kilogram, kilometers per hour, pesos 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 centimeters.
- 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 $22.50$ pesos. 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 $22.50$ pesos. Complete the following table and calculate the constant of proportionality $k$ (price per ticket):
| Number of tickets | $1$ | $3$ | $5$ | $8$ | $10$ |
|---|---|---|---|---|---|
| Total price ( pesos) | $22.50$ |
The graph: a line through the origin
If you plot the points from the table, they line up on a straight line that passes through the origin $(0, 0)$: 0 tickets, 0 pesos. The constant $k$ is the slope (how much the line rises for each unit it moves to the right).
A radio-controlled car covers $12$ meters 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 $12$ meters 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 $30$ meters?
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.
Direct rule of three
To find an unknown value there are two equivalent approaches:
1. Unit rate method: first find the value for 1, then multiply.
2. Rule of three (proportion): set up the data and use cross products.
Scales and plans
Scales are direct proportionality: actual measurement $= k \times$ 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 Nevado de Toluca area 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…
A hiking map of the Nevado de Toluca area 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 kilometers 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?
Percents as proportionality
A percent is a proportion with 100: 15 % of $\$240$ is $\frac{15}{100} = \frac{x}{240}$, so $100x = 3\,600$ and $x = 36$ pesos. That is why a quantity and its percent are directly proportional: 15 % of double is double.
You have seen a hoodie that you love in an online store. Its original price is $900$ pesos, but today it has a $15\%$ discount. How many pesos do you save thanks to the…
You have seen a hoodie that you love in an online store. Its original price is $900$ pesos, but today it has a $15\%$ discount. How many pesos 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 $2.2$ meters high is $1.4$ meters 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 $2.2$ meters high is $1.4$ meters long. If right next to it there is an electricity pylon whose shadow is $10.5$ meters long, how tall is the pylon? Draw a sketch of the situation before solving the problem.