Direct Proportionality
At the school co-op they sell packs of Liga MX trading cards: 1 pack costs $\$4$, 2 packs cost $\$8$, 3 packs cost $\$12$. How do you know how much you will pay for 8 packs without making a huge list? Direct proportionality solves that.
What is direct proportionality?
At a video game store, 100 gems cost $\$15$. Look at the table and check that the quotient price ÷ gems is always the same:
| Gems | 100 | 200 | 300 | 400 |
|---|---|---|---|---|
| Price (pesos) | 15 | 30 | 45 | 60 |
Here $k = 15 \div 100 = 0.15$ pesos per gem. With 200 gems: $30 \div 200 = 0.15$. It always comes out the same.
The same relationship can be drawn as an arrow diagram: each number of gems is joined to its price, and the factor $k$ is the same on every arrow.
And also as a graph: the points form a line that passes through the origin.
In math, we say that two quantities are directly proportional when, on multiplying one by a number, the other is multiplied by that same number. Analyze the following…
In math, we say that two quantities are directly proportional when, on multiplying one by a number, the other is multiplied by that same number. Analyze 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 centimeters.
- 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 favorite video game, you can buy "Gems" to get new skins for your character. We know that the gems always have the same unit price. Complete the following…
In your favorite video game, you can buy "Gems" to get new skins for your character. We know that the gems always have the same unit price. Complete the following proportion table in your notebook to find out how much different amounts of gems cost:
| Number of Gems | $100$ | $200$ | $300$ | $500$ | $1,000$ |
|---|---|---|---|---|---|
| Price ( pesos) | $30$ | $90$ |
Unit value
A pack of soccer league stickers contains $6$ stickers, and $3$ of these packs cost $72$ pesos. If we want to complete the album and decide to buy $8$ identical packs,…
A pack of soccer league stickers contains $6$ stickers, and $3$ of these packs cost $72$ pesos. If we want to complete the album and decide to buy $8$ identical packs, how much money will we spend in total? Use the unitary method (calculate how much a single pack 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.
To cook a giant paella for $12\text{ people}$, the chef uses $1.5\text{ kg}$ of rice and $3\text{ liters}$ of stock. If more people suddenly join and there are now $18$…
To cook a giant paella for $12\text{ people}$, the chef uses $1.5\text{ kg}$ of rice and $3\text{ liters}$ of stock. If more people suddenly join and there are now $18$ in total, calculate using the unitary method what amount of rice and stock the chef will need now.
Rule of three
Another way is to set up a proportion (two equal ratios) and solve it with cross products: in $\dfrac{a}{b} = \dfrac{c}{d}$ it holds that $a \cdot d = b \cdot c$.
Another case: $\dfrac{2}{5} = \dfrac{10}{x} \Rightarrow 2x = 50 \Rightarrow x = 25$.
In the figure, the terms of the same color are the ones multiplied together: 3 with $x$ (blue) and 450 with 5 (orange).
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) $\frac{7}{x} = \frac{14}{6}$
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 $x$.
We have paid $\$510$ for $3$ cinema tickets. We want to buy $7…
We have paid $\$510$ for $3$ cinema tickets. We want to buy $7$ tickets for the whole group of friends. Solve the problem in the following two ways:
- a) By the unitary method: calculate the price of one ticket and then that of $7$.
- b) Using direct proportion: set up the proportion $\frac{3}{510} = \frac{7}{x}$ and solve for $x$.
- c) Do both results match? Which method do you find easier to use and why?
The equation y = kx
If $k$ is the constant of proportionality, each value of the second quantity ($y$) is obtained by multiplying the value of the first ($x$) by $k$:
Example: a skater covers 150 m in 30 s at constant speed. Their constant is $k = 150 \div 30 = 5$ m/s, so $d = 5t$. In 60 s they cover $5 \times 60 = 300$ m.
The constant can also be negative. A 100 L tank loses 4 L per minute: the change in volume is $y = -4x$ (in $x$ minutes the volume changes by $-4x$ liters). The quantities are related in the same way, except that one increases while the other decreases.
Increasing a constant by a percent
Sometimes the constant changes: for example, the price per kilometer goes up $20\%$. Increasing a quantity by $20\%$ means adding its $20\%$ to it, that is, multiplying it by $1 + 0.20 = 1.20$.
A skateboarder travels along a track at a constant speed. In the first part of the circuit, they have covered $150$ meters in $30$ seconds. If they maintain the same…
A skateboarder travels along a track at a constant speed. In the first part of the circuit, they have covered $150$ meters in $30$ seconds. If they maintain the same pace throughout the trip:
- a) What distance will they cover in $1$ minute ($60$ seconds)?
- b) What is their constant of proportionality (how many meters do they cover per second)?
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…
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 $m^2/kg$.
| Paint ($kg$) | $1.5$ | $3$ | ? | $10$ |
|---|---|---|---|---|
| Surface area ($m^2$) | $12$ | ? | $48$ | ? |
Three 7th grade classes have decided to organize a food collection drive. The number of kilograms collected is directly proportional to the number of students in each…
Three 7th grade classes have decided to organize a food collection drive. The number of kilograms collected is directly proportional to the number of students in each class. Class A has $24$ students and has collected $36\ kg$.
- a) Complete the following table, justifying your calculations:
| Class | Students | Kilos collected |
|---|---|---|
| Class A | $24$ | $36$ |
| Class B | $30$ | $?$ |
| Class C | $?$ | $42$ |
- b) What is the constant of proportionality relating the kilos per student? Express it as an irreducible fraction.
A courier company uses electric vans. The cost of electricity consumption is directly proportional to the kilometers driven. Last month, a van covered $1,250\ km$ at a…
A courier company uses electric vans. The cost of electricity consumption is directly proportional to the kilometers driven. Last month, a van covered $1,250\ km$ at a total electricity cost of $\$750$.
- a) This month the price of electricity has increased by $20\%$. What will the new constant of proportionality be (cost per kilometer)?
- b) At the new price, how much will it cost to cover a route of $420\ km$?
The graph and non-proportional relationships
If you plot the pairs of a direct proportionality, the points lie on a straight line through the origin $(0, 0)$: with no gems, you pay nothing. Move $k$ to see how the slant changes:
To read a graph, take a point on the line, for example $(4, 120)$ in feathers and pesos: $k = 120 \div 4 = 30$ pesos per feather.
When there is a fixed charge, it is not proportional. A cell phone plan costs $\$100$ fixed plus $\$2$ for each GB:
| GB | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Cost (pesos) | 100 | 102 | 104 | 106 |
Here the quotients are not equal ($102 \div 1 = 102$, $104 \div 2 = 52$) and the line does not pass through $(0, 0)$: the equation is $y = 2x + 100$.
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 $6$ pesos, 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 liters it dispenses. Look at the marked point and determine how many…
The following graph shows the relationship between the time a water fountain is turned on and the liters it dispenses. Look at the marked point and determine how many liters 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…
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.
- a) Looking at the highlighted point, calculate how many minutes the cell phone takes to charge $1\%$ of battery.
- b) If the cell phone currently has $12\%$ of battery and you need to leave the house in $18$ minutes, what battery percentage will you leave with?
Proportionality problems
With the tools above you can solve longer problems. Steps: 1) identify the two quantities, 2) find $k$, 3) use $y = kx$, 4) check the units and whether the result makes sense.
Map scales
On a map, the distance on paper and the real distance are directly proportional. The scale $1:25{,}000$ is the constant: 1 cm on the map represents 25,000 cm in real life. Therefore:
$$\text{real distance} = \text{distance on the map} \times 25{,}000 \quad (\text{both in the same unit})$$Directly proportional sharing
Sometimes you have to split a total amount in proportion to what each person contributed (hours, kilos, money). Each person receives $k$ for every unit contributed.
On a map of Sierra Gorda, 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…
On a map of Sierra Gorda, 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 kilometers will the hikers cover in reality?
Three farmers worked harvesting an orange grove in proportion to the time each dedicated. If they were paid a total of $\$1500…
Three farmers 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:
-
Farmer A: $12\text{ hours}$
-
Farmer B: $18\text{ hours}$
-
Farmer C: $20\text{ hours}$
How much money does each of them receive?
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…
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.
- a) Calculate the actual distance between the two huts in kilometers.
- b) What time will they arrive at the second hut if they set off from the first at $08:30$ in the morning?
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,…
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.
- a) Explain why the weight of the item and the printing time are directly proportional quantities.
- b) Set up direct proportion and calculate the time it will take for the $420$-gram item.
- c) How many grams does the machine print in one hour? Check your result from part b) using this value.
Back to the start: 8 packs of cards at $\$4$ each cost $8 \times 4 = 32$ pesos, because $k = 4$ and $y = 4x$.