Ratios and Proportions, SAT math lesson

A bag holds red and blue marbles in the ratio 3 to 5, and there are 64 marbles in all. How many are red? The ratio does not tell you a count, it tells you a pattern. Every group of 8 marbles has 3 red and 5 blue, and 64 marbles make 8 such groups. Ratio questions on the SAT come down to finding that group size and scaling both parts by the same factor.

Key terms

Ratio

Definition A ratio compares two quantities by division. For positive quantities $a$ and $b$ , the ratio of $a$ to $b$ is written $a : b$ , ” $a$ to $b$ ” or $\frac{a}{b}$ , and its value is the number $\frac{a}{b}$ . Order matters, since the ratio of $b$ to $a$ is $\frac{b}{a}$ , a different number unless $a = b$ . When the two quantities have units, the ratio carries the unit of the first over the unit of the second.

In plain words A ratio says how many of one thing there are for each of another. The first quantity named goes first.

Example A class has 12 girls and 18 boys. The ratio of girls to boys is

12 : 18 = \frac{12}{18} = \frac{2}{3},

so there are 2 girls for every 3 boys. The ratio of boys to girls is $3 : 2$ .

Part-to-part and part-to-whole ratios

Definition If a whole is made of parts $a$ and $b$ , then $a : b$ is a part-to-part ratio and $a : (a + b)$ is a part-to-whole ratio. A part-to-whole ratio is the fraction of the whole that the part makes up. The two kinds have different denominators, $b$ for the first and $a + b$ for the second, so they are not interchangeable.

In plain words “3 red to 5 blue” compares the parts with each other. “3 out of 8” compares one part with the total.

Example Back to the bag with red to blue $3 : 5$ and $64$ marbles. The ratio has $3 + 5 = 8$ shares in all, and $64 \div 8 = 8$ , so each share stands for $8$ marbles. Red gets $3$ shares and blue gets $5$ ,

\begin{aligned} \text{red} &= 3 \cdot 8 = 24, \\[0.4em] \text{blue} &= 5 \cdot 8 = 40. \end{aligned}

The part-to-whole ratio for red is $\frac{3}{8}$ , and indeed $\frac{3}{8} \cdot 64 = 24$ . A common slip is to compute $\frac{3}{5} \cdot 64$ , which uses the part-to-part ratio as if it were a fraction of the whole.

Equivalent ratio

Definition Two ratios $a : b$ and $c : d$ of positive quantities are equivalent when they have the same value,

\frac{a}{b} = \frac{c}{d}.

This happens exactly when $c = ka$ and $d = kb$ for one positive number $k$ , the scale factor. Multiplying or dividing both parts of a ratio by the same positive number gives an equivalent ratio. Adding the same number to both parts usually does not.

In plain words Scale both parts by the same factor and the ratio stays the same. Scale only one part, or add instead of multiply, and it changes.

Example The ratios $3 : 5$ , $6 : 10$ and $24 : 40$ are equivalent, with scale factors $2$ and $8$ . Adding $1$ to both parts of $3 : 5$ gives $4 : 6$ , which simplifies to $2 : 3$ . Since $\frac{2}{3} \ne \frac{3}{5}$ , that is a different ratio.

Proportion

Definition A proportion is an equation stating that two ratios are equal,

\frac{a}{b} = \frac{c}{d}, \qquad b \ne 0, \ d \ne 0.

Multiplying both sides by $bd$ shows that it is equivalent to $ad = bc$ , which is called cross multiplication. To solve a proportion with one unknown, cross multiply and solve the resulting equation.

In plain words A proportion says two ratios match. Line up the same kinds of quantity in the same positions, then cross multiply.

Example A recipe uses $3$ cups of flour for $2$ cups of milk. How much milk goes with $7.5$ cups of flour? Keep flour on top on both sides,

\frac{3}{2} = \frac{7.5}{m}.

Cross multiplying gives $3m = 15$ , so $m = 5$ cups of milk. The scale factor is $7.5 \div 3 = 2.5$ , and $2 \cdot 2.5 = 5$ agrees.

Unit rate

Definition A rate is a ratio of two quantities measured in different units. A unit rate is a rate expressed per one unit of the second quantity. If an amount $A$ of one quantity corresponds to an amount $B$ of another, with $B \ne 0$ , the unit rate is

\frac{A}{B},

measured in units of the first quantity per one unit of the second.

In plain words A unit rate answers “how much for one?” It is the easiest way to compare two offers or to scale up.

Example Store one sells $6$ notebooks for $9$ dollars and store two sells $8$ for $11.60$ dollars. In dollars per notebook the unit prices are

\frac{9}{6} = 1.50, \qquad \frac{11.60}{8} = 1.45,

so store two is cheaper per notebook. At store one, $14$ notebooks would cost $14 \cdot 1.50 = 21$ dollars.

Direct proportion

Definition A quantity $y$ is directly proportional to a quantity $x$ when there is a constant $k \ne 0$ with

y = kx

for every value of $x$ in the situation being modeled. The number $k$ is the constant of proportionality. Equivalently, the ratio $\frac{y}{x}$ equals the same number $k$ for every nonzero $x$ in that domain, and $y = 0$ when $x = 0$ . The graph is part of a line through the origin. A linear relationship $y = kx + b$ with $b \ne 0$ is not a direct proportion, since $\frac{y}{x}$ is not constant.

In plain words Double one quantity and the other doubles too. The test is a constant ratio, not a constant difference.

Example Decide whether each table describes a direct proportion.

$x$	$2$	$5$	$8$
$y$	$7$	$17.5$	$28$

Here $\frac{7}{2} = \frac{17.5}{5} = \frac{28}{8} = 3.5$ , so $y = 3.5x$ fits every column. This is a direct proportion with $k = 3.5$ .

$x$	$2$	$5$	$8$
$y$	$7$	$16$	$25$

Here $y$ rises by $9$ each time $x$ rises by $3$ , so the data are linear, $y = 3x + 1$ . But $\frac{7}{2} = 3.5$ while $\frac{16}{5} = 3.2$ , so the ratio is not constant and this is not a direct proportion.

A model is a direct proportion only within the values it describes. A taxi fare of $2.50$ dollars per mile with no base fee is proportional to the distance for distances the taxi actually drives, but it says nothing about negative miles.

Check yourself

A recipe calls for $3$ cups of flour for every $2$ cups of milk. How many cups of milk are needed with $10.5$ cups of flour?

Show the answer

Set up the proportion with flour on top,

\frac{3}{2} = \frac{10.5}{m}.

Cross multiplying gives $3m = 21$ , so $m = 7$ cups of milk.

In a club the ratio of boys to girls is $4$ to $5$ , and the club has $36$ members. How many members are girls? (A) $16$ (B) $20$ (C) $45$ (D) $180$

Show the answer

The answer is (B). Each group of $4 + 5 = 9$ members has $5$ girls, and $36 \div 9 = 4$ groups, so there are $5 \cdot 4 = 20$ girls. Choice (A) is the number of boys. Choice (C) treats $36$ as the number of boys, and choice (D) multiplies $36$ by $5$ without dividing by the total of $9$ shares.

Which set of points could lie on the graph of a direct proportion $y = kx$ ? (A) $(1, 4)$ , $(2, 6)$ , $(3, 8)$ (B) $(1, 3)$ , $(2, 6)$ , $(5, 15)$ (C) $(2, 5)$ , $(4, 10)$ , $(6, 16)$ (D) $(1, 2)$ , $(2, 4)$ , $(3, 8)$

Show the answer

The answer is (B), since $\frac{3}{1} = \frac{6}{2} = \frac{15}{5} = 3$ . Choice (A) has a constant difference but its ratios $4$ , $3$ and $\frac{8}{3}$ differ, so it is linear with $y = 2x + 2$ but not proportional. Choices (C) and (D) start with a constant ratio and break it at the last point.

Ratios and Proportions

Theory, part 3

Practice, part 4

Key terms

Ratio

Part-to-part and part-to-whole ratios

Equivalent ratio

Proportion

Unit rate

Direct proportion

Check yourself

How this lesson connects

Builds on

Used later in