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Rates & proportional relationships FAQ

Frequently asked questions about rates and proportional relationships

Why do we need to learn about rates and proportions?

Rates and proportions are important concepts in math that can help us understand relationships between different quantities. They come up frequently in everyday life - for example, when we're comparing prices, figuring out how much of an ingredient to use in a recipe, or determining how fast we need to travel to reach a destination on time.

What is a constant of proportionality?

A constant of proportionality is a number that relates two quantities in a proportional relationship. For example, if we say that y is proportional to x, we might write the equation y, equals, k, x, where k is the constant of proportionality.
The constant of proportionality is another name for the unit rate. Suppose Zion skips rope at a constant rate, and they skip over a jump rope 135 times in 3 minutes. There are two unit rates:
  • start fraction, 135, divided by, 3, end fraction, which simplifies to 45, skips per minute
  • start fraction, 3, divided by, 135, end fraction, which simplifies to start fraction, 1, divided by, 45, end fraction, minutes per skip
To tell which unit rate to use, we need to know the meaning of x in the equation. Suppose x represents the number of minutes and y represents the number of skips. Then the constant of proportionality would be 45, the unit rate that had the number of minutes in the denominator.
y, equals, 45, x

How do we identify proportional relationships?

There are a few ways we can tell if two quantities are proportional to each other. We might see that the ratio between the two quantities is always the same, or we might see that a graph of the two quantities forms a straight line through the origin (the point where both x and y are equal to 0).

How do we write and solve proportions?

To write a proportion, we set two rates equal to each other. For example, we might say start fraction, 2, divided by, 4, end fraction, equals, start fraction, 6, divided by, 12, end fraction.
Solving a proportion just means finding the missing value in the proportion. For example, if we know that 8 out of 12 oranges are ripe, and we want to figure out how many out of 24 oranges would be ripe at the same rate, we can set up a proportion like this: start fraction, 8, divided by, 12, end fraction, equals, start fraction, x, divided by, 24, end fraction. Notice that we use the same rate start fraction, start text, r, i, p, e, space, o, r, a, n, g, e, s, end text, divided by, start text, t, o, t, a, l, space, o, r, a, n, g, e, s, end text, end fraction on both sides of the equation.
From there, we can use one of the methods for solving equations.
812=x2481224=x242416=x\begin{aligned} \dfrac{8}{12} &= \dfrac{x}{24}\\\\ \dfrac{8}{12}\cdot 24 &= \dfrac{x}{24}\cdot 24\\\\ 16 &= x \end{aligned}
Solving the equation can take an extra step if the unknown is in the denominator of the equation. Suppose we had set up the equation using the rate start fraction, start text, t, o, t, a, l, space, o, r, a, n, g, e, s, end text, divided by, start text, r, i, p, e, space, o, r, a, n, g, e, s, end text, end fraction instead.
128=24x128x=24xx128x=24812128x=81224x=16\begin{aligned} \dfrac{12}{8} &= \dfrac{24}{x}\\\\ \dfrac{12}{8}\cdot x &= \dfrac{24}{x}\cdot x\\\\ \dfrac{12}{8}x&= 24\\\\ \dfrac{8}{12}\cdot \dfrac{12}{8}x &= \dfrac{8}{12} \cdot 24\\\\ x &= 16 \end{aligned}
Try it yourself with our Solving proportions exercise.

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