Math Core

In every proportional relationship, one number does all the work: the number you multiply by to get from one quantity to the other. That number has a name, and once you find it you can answer almost any question about the relationship.

The number that stays the same

In the last lesson, the rice table had $\dfrac{y}{x} = 1.5$ in every row. That steady ratio is the heart of the relationship.

Definition

Constant of proportionality

In a proportional relationship between $x$ and $y$ , the constant of proportionality is the number $k$ that you multiply $x$ by to get $y$ :

k = \frac{y}{x}

It is the same for every pair of values. It is also the unit rate: the amount of $y$ for 1 of $x$ .

For the rice, $k = 1.5$ : you use 1.5 cups of water per cup of rice.

Finding k from a table

Worked example: Pay for hours worked

Marcus is paid the same amount for every hour he works.

hours, $x$	pay in dollars, $y$
3	45
5	75
8	120

Find the constant of proportionality and explain what it means.

Divide $y$ by $x$ in each row:

\frac{45}{3} = 15, \qquad \frac{75}{5} = 15, \qquad \frac{120}{8} = 15

So $k = 15$ . Marcus earns $15 per hour.

The constant doesn't have to be a whole number. It is often a fraction or a decimal.

Worked example: A fraction for k

A bread recipe always uses salt and flour in the same ratio.

cups of flour, $x$	teaspoons of salt, $y$
4	3
8	6
12	9

Find the constant of proportionality.

\frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{3}{4}

So $k = \dfrac{3}{4}$ . The recipe uses $\dfrac{3}{4}$ teaspoon of salt per cup of flour.

Finding the constant of proportionality

From a table: divide $y$ by $x$ in any row (check that every row gives the same result).
From a description: find the unit rate, the amount of $y$ for 1 of $x$ .
Always divide in the order " $y$ per $x$ ," and state what $k$ means with units.

Which quantity comes first?

The same situation has two constants, depending on which quantity you call $x$ and which you call $y$ .

Worked example: Miles and gallons

A car travels 160 miles on 5 gallons of gas. Distance is proportional to gas used.

Find the constant of proportionality for miles in terms of gallons.
Find the constant of proportionality for gallons in terms of miles.

Solutions.

Miles per gallon: $k = \dfrac{160}{5} = 32$ . The car goes 32 miles on each gallon.
Gallons per mile: $k = \dfrac{5}{160} = \dfrac{1}{32}$ . The car uses $\dfrac{1}{32}$ of a gallon for each mile.

The two constants are reciprocals: $32 \times \dfrac{1}{32} = 1$ .

Common mistake

"The constant of proportionality of $y$ to $x$ " or " $y$ in terms of $x$ " means $\dfrac{y}{x}$ , with $y$ on top. If you flip it, you get the reciprocal, which answers a different question. Read which quantity is "per" what before you divide.

When the amounts are fractions

Sometimes you only know one pair of values, and they are fractions. Divide just as you did with unit rates.

Worked example: Milk and oats

A granola recipe uses $\dfrac{2}{3}$ cup of milk for every $1\dfrac{1}{2}$ cups of oats. Find the constant of proportionality for milk in terms of oats.

Milk in terms of oats means milk $\div$ oats. Write $1\dfrac{1}{2} = \dfrac{3}{2}$ :

k = \frac{2}{3} \div \frac{3}{2} = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}

The recipe uses $\dfrac{4}{9}$ cup of milk per cup of oats.

Tip

Once you know $k$ , you can find any missing $y$ by multiplying: $y = k \times x$ . And you can find any missing $x$ by dividing: $x = y \div k$ . With $k = 15$ dollars per hour, 6 hours pays $15 \times 6 = 90$ dollars, and $150 takes $150 \div 15 = 10$ hours.