Proportional Relationships and Unit Rates
At a food truck outside the stadium, Aaliyah sees that 3 tacos cost $\$7.50$. Her whole team wants 10 tacos. How much will she pay? You don't need to guess, because tacos and price are linked by one number that never changes. That link is a proportional relationship.
What is a proportional relationship?
Movie tickets cost the same each, so the total price is proportional to the number of tickets:
| Tickets | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Price | $\$9$ | $\$18$ | $\$27$ | $\$36$ |
Divide price by tickets in any column: $9 \div 1 = 9$, $18 \div 2 = 9$, $27 \div 3 = 9$. The constant is $k = 9$ dollars per ticket.
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…
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:
- a) The number of followers a streamer has and the number of likes they receive on a video (assuming all their followers "like" it).
- b) A person's age and their height in inches.
- 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 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…
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:
| Number of Gems | $100$ | $200$ | $300$ | $500$ | $1.000$ |
|---|---|---|---|---|---|
| Price ($\$$) | $1,50$ | $4,50$ |
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…
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?
The unit rate method
Why does it work? In a proportional relationship every single unit is worth the same amount. Once you know what one unit is worth, you only have to count how many units you have.
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…
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).
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…
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.
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 $18$…
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.
Tables and the constant of proportionality
A table is the best way to organize a proportional relationship. Find $k = y \div x$ from any complete column. Then:
-
to find a missing $y$, multiply: $y = k \cdot x$;
-
to find a missing $x$, divide: $x = y \div k$.
The constant can be read in two ways. Here $k = 3.5$ dollars per gallon, or $\dfrac{1}{3.5} = \dfrac{2}{7}$ gallon per dollar. Always write the units of $k$ so you know which one you found. Fractions are fine too: if 24 students collect 36 lb, then $k = \dfrac{36}{24} = \dfrac{3}{2}$ pounds per student, in simplest form.
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…
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:
- a) What distance will it travel in $1$ minute ($60$ seconds)?
- b) What is its constant of proportionality (how many feet does it travel per second)?
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…
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 $m^2/kg$.
| Paint ($kg$) | $1,5$ | $3$ | ? | $10$ |
|---|---|---|---|---|
| Area ($m^2$) | $12$ | ? | $48$ | ? |
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…
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 $36\ kg$.
- a) Complete the following table by reasoning the calculations:
| Class | Students | Pounds collected |
|---|---|---|
| Class A | $24$ | $36$ |
| Class B | $30$ | $?$ |
| Class C | $?$ | $42$ |
- b) What is the constant of proportionality that relates the pounds per student? Express it as a fraction in simplest form.
Solving with proportions
A proportion is an equation that says two ratios are equal, such as $\dfrac{4}{36} = \dfrac{7}{x}$. To solve it, use cross products: multiply diagonally.
Why? Multiply both sides of the proportion by $b \cdot d$. Then the denominators cancel and you are left with $a \cdot d = b \cdot c$.
Sometimes a shortcut is faster. In $\dfrac{2}{5} = \dfrac{10}{x}$, the top went from 2 to 10, which is $\times 5$. So the bottom is also $\times 5$: $x = 5 \cdot 5 = 25$. Cross products give the same answer: $2x = 50$.
Find the value of the unknown term $x$ in the following proportions so that the equality between the ratios holds: 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:
- 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 balanced scales. On the first scale, $3$ blue cubes weigh exactly $450$ ounces. On the second scale, we want to know how…
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 $x$.
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…
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?
Graphs of proportional relationships
Plot the pairs $(x, y)$ of a proportional relationship. The points always fall on a straight line through the origin $(0, 0)$, because zero tacos cost zero dollars. The point with $x = 1$ is $(1, k)$, so the height of the line at $x = 1$ is the unit rate.
Try the slider below. Changing $k$ tilts the line, and it always passes through the origin.
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 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.
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…
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 $15\text{ minutes}$.
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…
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.
- 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 percentage of battery will you leave with?
Converting units and mixed rates
A unit conversion is also a proportional relationship: $1$ pound is always $16$ ounces, so ounces are proportional to pounds with $k = 16$. Rates often mix units (quarts per second, miles per hour), so convert to the unit the question asks for.
| Conversion | Value |
|---|---|
| 1 mi | about 1.609 km (so 1 km ≈ 0.621 mi) |
| 1 lb | 16 oz |
| 1 oz | about 28.35 g |
| 1 lb | about 454 g |
| 1 kg | about 2.2 lb |
| 1 gal | 4 qt (and about 3.785 L) |
| 1 L | 1,000 mL (and 1 cL = 10 mL) |
| 1 kg | 1,000 g (1 metric ton = 1,000 kg) |
| 1 h | 60 min = 3,600 s |
| 1 GB | 1,024 MB |
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…
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?
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…
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).
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. a) Calculate…
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.
- a) Calculate the transfer rate in $MB$ per minute (Remember that $1\ GB = 1024\ MB$).
- 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?
Multi-step problems: scales, percents and changing rates
Hard problems chain several steps. Break them into small proportional pieces, and keep track of the units.
Map scales. A scale of $1 : 100{,}000$ means 1 unit on the map is 100,000 of the same unit in real life. So real distance = map distance $\times 100{,}000$.
Percents as rates. "15% of the battery per 2 miles" is still a rate, with the percent just being a number. You can find the percent used per mile (or miles per 1%) and scale it. Remember a full battery is 100%.
Changing rates. If a rate goes up by 20%, the new rate is $1.20 \times$ the old rate. If it goes down by 20%, it is $0.80 \times$ the old rate.
Back to the taco truck: 3 tacos cost $\$7.50$, so one taco costs $7.50 \div 3 = 2.50$ dollars, and 10 tacos cost $10 \times 2.50 = 25$ dollars. Aaliyah pays $\$25$.
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…
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:
- a) What is the constant of proportionality if we express the relationship as "percentage of battery per mile"?
- 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.
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…
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.
- a) Calculate the real distance between the two shelters in miles.
- b) What time will they arrive at the second shelter if they leave the first one at $08:30$ in the morning?
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…
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.
- a) This month the price of electricity has gone up by $20\%$. What will be the new constant of proportionality (cost per mile)?
- b) With the new price, how much will it cost to travel a route of $420\ km$?