Direct Proportionality

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:

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 centimeters.
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 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:

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 pack of soccer league stickers contains $6$ stickers, and $3$ of these packs cost $216$ 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 flour will you need for the pancakes to turn out perfectly? Remember that the recipe quantities are directly proportional to the number of diners.

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:

1. a) What distance will they cover in $1$ minute ($60$ seconds)?
2. b) What is their constant of proportionality (how many meters do they cover per second)?

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 $\text{RD\$}\ 864$. If a friend has saved exactly $\text{RD\$}\ 1\,260$, how many gems will he be able to buy if the relationship is directly proportional?

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.

A dripping faucet loses approximately $1.2\text{ centiliters}$ of water every $10\text{ seconds}$. If we do not fix it, how many liters of water will it have wasted after a full hour? (Remember to convert the result to the requested unit of measurement first).

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 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 kilometers will the hikers cover in reality?

We have paid $\text{RD\$}\ 1\,530$ for $3$ cinema tickets. We want to buy $7$ tickets for the whole group of friends. Solve the problem in the following two ways:

1. a) By the unitary method: calculate the price of one ticket and then that of $7$.
2. b) Using direct proportion: set up the proportion $\frac{3}{1\,530} = \frac{7}{x}$ and solve for $x$.
3. c) Do both results match? Which method do you find easier to use and why?

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.

Three farmers worked harvesting an orange grove in proportion to the time each dedicated. If they were paid a total of $\text{RD\$}\ 1500$ 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?

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 $36\ kg$.

1. a) Complete the following table, justifying your calculations:

1. 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 total electricity cost of $\text{RD\$}\ 2\,250$.

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

A sustainable clothing brand uses recycled fibers to manufacture its T-shirts. The relationship between the weight of 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 requires $225\ g$ of yarn, how many grams of yarn are obtained per kilogram of plastic bottles?
2. b) The municipality of a city has collected $1.2\ toneladas$ of bottles. How many T-shirts could be made in total?

To dye fabrics using natural dyes, an artisan mixes beetroot extract and water. The amount of extract is directly proportional to the volume of water. In her notebook, she has written down that for $1.5\ litros$ of water she needs $120\ ml$ of extract. However, she wants to prepare a tub with a total mixture volume (water + extract) of $10.8\ litros$. What exact amount of water and extract should she mix to maintain the proportion?

A video game server distributes data to players directly proportionally to the 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 size is $15\ GB$, how many hours and minutes will the download take to complete if the speed remains constant?

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.

A hybrid car consumes an average of $4.5\text{ liters}$ of fuel per $100\text{ km}$ driven. 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 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 that charging 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 battery percentage will you leave with?

In a robotics workshop, pupils are programming a drone to inspect a nature park. The drone uses battery power directly proportionally to the distance travelled. If to travel $2.5\ km$ the drone uses $15\%$ of its total battery:

1. a) What is the constant of proportionality if we express the relationship as "percentage of battery per kilometer"?
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 with a full charge? Justify your answer with calculations.

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 kilometers.
2. b) What time will they arrive at the second hut if they set off from the first at $08:30$ in the morning?