Direct proportion

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:

1. a) The number of followers of a streamer and the number of likes they receive on a video (assuming all their followers click "like").
2. b) A person's age and their height in centimetres.
3. c) The number of pizza boxes you order and the total price you pay (if all pizzas cost the same).
4. d) The speed of a car and the time it takes to reach its destination.
5. 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 proportionality table in your notebook to find out how much different amounts of gems cost:

Cian goes with his father to buy tickets for a school theatre festival. At the ticket office, they see the sign: “$3$ tickets cost $€21$”. Cian needs to calculate how much $5$ tickets will cost for his whole family and reasons as follows: “Since $5$ tickets is $2$ tickets more than $3$, I just need to add $€2$ to the price of $€21$. Therefore, the $5$ tickets will cost $€23$”.

1. a) Explain in detail why Cian's additive reasoning is incorrect and what it really means for the number of tickets and the price to be directly proportional quantities.
2. b) Correct the calculation using the unitary method: find the price of one ticket and calculate the actual price the family must pay for the $5$ tickets.

Copy and complete the following table of directly proportional values relating the number of kilograms of paint to the surface area that can be covered. Also calculate the constant of proportionality $k$ expressed in $m^2/kg$.

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 flour will you need for the pancakes to turn out perfectly? Remember that the recipe quantities are directly proportional to the number of diners.

Three farmers (Aoife, Seán and Niamh) worked harvesting an orange grove in proportion to the time each dedicated. If they were paid a total of $€1500$ and their working times were:

• Aoife: $12\text{ hours}$

• Seán: $18\text{ hours}$

• Niamh: $20\text{ hours}$

How much money does each of them receive?

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:

1. a) $\frac{2}{5} = \frac{10}{x}$

1. b) $\frac{x}{12} = \frac{3}{4}$

1. c) $\frac{7}{x} = \frac{14}{6}$

On a map of the Wicklow Mountains, 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?

A 3D printer takes $4.5$ hours to print an object using $270$ grams of filament. We want to know how long it will take to print another object using $420$ grams, assuming the same rate.

1. a) Explain why the weight of the item and the printing time are directly proportional quantities.
2. b) Set up direct proportion and calculate the time it will take for the $420$-gram item.
3. c) How many grams does the machine print in one hour? Check your result from part b) using this value.

On a mountain map with a scale of $1:25,000$, the distance between two mountain huts is $14.4\ cm$. A group of hikers knows that their average ascent speed is constant and that they take $45$ minutes to cover $1,500\ metros$ in reality on that terrain.

1. a) Calculate the actual distance between the two huts in kilometres.
2. b) What time will they arrive at the second hut if they set off from the first at $08:30$ in the morning?

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.

The following graph shows the relationship between the time a water fountain is turned on and the litres it dispenses. Look at the marked point and determine how many litres the fountain will dispense if it is left running for $15\text{ minutes}$.

The following graph shows the relationship between the charging time of a new smartphone model and the percentage of battery charge recovered. The manufacturer states that charging is directly proportional during the first $40$ minutes.

1. a) Looking at the highlighted point, calculate how many minutes the mobile phone takes to charge $1\%$ of battery.
2. b) If the mobile phone currently has $12\%$ of battery and you need to leave the house in $18$ minutes, what battery percentage will you leave with?