Exercises: Direct proportion

In mathematics, we say that two quantities are directly proportional when, upon multiplying one by a number, the other is multiplied by that same number. Analyze the following everyday situations and indicate which ones are of direct proportion:

1. a) The number of followers a streamer has and the number of likes they receive on a video (assuming all their followers "like" it).
2. b) A person's age and their height in inches.
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 pounds of apples you buy and the money they cost.

In your favorite video game, you can buy "Gems" to get new skins 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:

A pack of soccer league stickers contains $6$ stickers, and $3$ of these packs cost $\$3,60$. If we want to complete the album and decide to buy $8$ identical packs, how much money will we spend in total? Use the unit rate method (first calculate how much a single pack costs).

Look at the following image representing two balanced scales. On the first scale, $3$ blue cubes weigh exactly $450$ ounces. On the second scale, we want to know how much $5$ cubes identical to the previous ones weigh. Calculate the value of the unknown $x$.

To cook pancakes for $4$ people, exactly $200$ ounces of flour are needed. If your cousins are coming over this weekend and there are a total of $6$ people, what amount of flour will you need for the pancakes to turn out perfect? Remember that the recipe amounts are directly proportional to the number of diners.

A skater travels along a track at a constant speed. In the first part of the circuit, they have traveled $150$ feet in $30$ seconds. If they maintain the same pace throughout the journey:

1. a) What distance will it travel in $1$ minute ($60$ seconds)?
2. b) What is its constant of proportionality (how many feet does it travel per second)?

Find the value of the unknown term $x$ in the following proportions so that the equality between the ratios holds:

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

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

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

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 that relates the number of pens bought and their total cost. If $4$ pens cost $\$6$, identify the point on the graph and calculate how much $10$ pens would cost.

On a map of the Rocky Mountains, the scale indicates that $2\text{ in}$ on paper represents $5\text{ mi}$ in reality. If to do a hiking route between two villages the distance measured on the map is $8,5\text{ in}$, how many miles will the hikers travel in reality?

To celebrate the end of the semester, a group of friends wants to buy "gems" for their favorite video game. They have seen that a pack of $1200$ gems costs $14,40\text{ \$}$. If a friend has saved exactly $21\text{ \$}$, how many gems will they be able to buy if the relationship is directly proportional?

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

A hybrid car consumes an average of $4,5\text{ liters}$ of fuel for every $100\text{ km}$ traveled. If the tank has a total capacity of $54\text{ liters}$ and is currently full, what maximum distance will the vehicle be able to travel before running out of fuel?

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

To cook a giant paella for $12\text{ people}$, the chef uses $1,5\text{ kg}$ of rice and $3\text{ liters}$ of broth. If suddenly more people join and there are now $18$ in total, calculate using the unit rate method what amount of rice and broth the chef will need now.

A dripping faucet loses approximately $1,2\text{ centiliters}$ of water every $10\text{ seconds}$. If we do not fix it, how many gallons of water will it have wasted after a full hour? (Remember to first convert the result to the requested unit of measurement).

Three farmers have worked on harvesting an orange orchard in proportion to the time each one has dedicated. If they have earned a total of $1500\text{ \$}$ and the working times have been:

• Farmer A: $12\text{ hours}$

• Farmer B: $18\text{ hours}$

• Farmer C: $20\text{ hours}$

How much money corresponds to each of them?

In a robotics workshop, the students are programming a drone to inspect a nature reserve. The drone consumes battery in direct proportion to the distance traveled. If to travel $2,5\ km$ the drone consumes $15\%$ of its total battery:

1. a) What is the constant of proportionality if we express the relationship as "percentage of battery per mile"?
2. b) If the return point is $7.200\ meters$ from the base, will it have enough battery to go and return if it takes off fully charged? Justify your answer with calculations.

A sustainable clothing brand uses recycled fibers to make its t-shirts. The relationship between the weight of the plastic bottles collected and the yarn obtained is directly proportional. It is known that with $3,6\ kg$ of bottles, exactly $8$ size L t-shirts can be made.

1. a) If each size L t-shirt needs $225\ g$ of yarn, how many ounces of yarn are obtained for each pound of plastic bottles?
2. b) A city council has collected $1,2\ toneladas$ of bottles. How many t-shirts could be made in total?

To dye fabrics with natural dyes, an artisan mixes beet extract and water. The amount of extract is directly proportional to the volume of water. In her notebook, she has noted that for $1,5\ litros$ of water she needs $120\ ml$ of extract. However, she wants to prepare a vat with a total mixture volume (water + extract) of $10,8\ litros$. What exact amount of water and extract must she mix to maintain the proportion?

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

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

On a high mountain map with a scale of $1:25.000$, the distance between two shelters 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 real distance between the two shelters in miles.
2. b) What time will they arrive at the second shelter if they leave the first one at $08:30$ in the morning?

A video game server distributes data to players in direct proportion to their connection time. A player has downloaded $1,8\ GB$ in the last $12$ minutes.

1. a) Calculate the transfer rate in $MB$ per minute (Remember that $1\ GB = 1024\ MB$).
2. b) If the total game update weighs $15\ GB$, how many hours and minutes will the download take to complete if the speed remains constant?

Three 6th grade classes have decided to organize a food collection campaign. The number of pounds collected is directly proportional to the number of students in each class. Class A has $24$ students and has collected $36\ kg$.

1. a) Complete the following table by reasoning the calculations:

1. b) What is the constant of proportionality that relates the pounds per student? Express it as a fraction in simplest form.

A courier company uses electric vans. The cost of electricity consumption is directly proportional to the miles traveled. Last month, a van traveled $1.250\ km$ with a total cost of $37,50\ \$$ in electricity.

1. a) This month the price of electricity has gone up by $20\%$. What will be the new constant of proportionality (cost per mile)?
2. b) With the new price, how much will it cost to travel a route of $420\ km$?