Lesson 1: Ratios

New Concept

A ratio is a powerful tool for multiplicative comparison. This lesson explores how to use division to compare the relative sizes of two quantities, expressing them as fractions, decimals, or rates to analyze their relationship.

What’s next

Coming up, you'll tackle interactive examples and practice cards to master calculating and simplifying ratios and rates.

Property

We can think of additive and multiplicative comparison as different ways to evaluate the relationship between two quantities. Additive comparison gives us the actual difference between two quantities, found using subtraction. Multiplicative comparison, found using division, takes into account their relative sizes. For example, for two quantities $a$ and $b$:
Additive Comparison: $a - b$
Multiplicative Comparison: $\frac{a}{b}$

Examples

• A giraffe is 18 feet tall and a person is 6 feet tall. Multiplicatively, the giraffe is $\frac{18}{6} = 3$ times as tall as the person. Additively, the giraffe is $18 - 6 = 12$ feet taller.
• A phone costs 800 dollars and a case costs 40 dollars. The phone is $\frac{800}{40} = 20$ times more expensive than the case.
• A luxury car costs 50,100 dollars, while the standard model costs 50,000 dollars. The multiplicative comparison is $\frac{50100}{50000} = 1.002$, showing the price difference is very small relative to the total cost.

Explanation

Additive comparison tells you the simple difference, like '10 more'. Multiplicative comparison shows a relative difference, like 'twice as big'. This is often more meaningful for understanding the scale of the comparison, not just the amount.

Property

A ratio shows the relative sizes of two values by dividing one value into the other. The ratio of $a$ to $b$ is sometimes denoted by $a : b$, and it is computed as the quotient $\frac{a}{b}$.

Examples

• A classroom has 24 students and 2 teachers. The ratio of students to teachers is $\frac{24}{2}$, which simplifies to $\frac{12}{1}$ or 12 to 1.
• A recipe calls for 3 cups of flour and 2 cups of sugar. The ratio of flour to sugar is 3 to 2, or $\frac{3}{2}$.
• In a parking lot, there are 50 cars and 15 trucks. The ratio of cars to trucks is $\frac{50}{15}$, which simplifies to $\frac{10}{3}$. For every 10 cars, there are 3 trucks.

Explanation

A ratio is a way to compare two quantities by division. Think of it as a recipe: 'for every $a$ of this, you have $b$ of that'. Simplifying the ratio makes this relationship easier to understand at a glance.

Property

Any ratio can be expressed as a decimal fraction by dividing the numerator by the denominator. We often use decimal form when comparing two different ratios or when the original numbers are decimals.

Examples

• A company has 9 million dollars in profits and 4 million dollars in expenses. The ratio of profit to expenses is $\frac{9}{4} = 2.25$.
• Team A won 15 out of 20 games, a ratio of $\frac{15}{20} = 0.75$. Team B won 14 out of 18 games, a ratio of $\frac{14}{18} \approx 0.778$. Team B has a slightly better winning ratio.
• A company's advertising budget is 4.2 million dollars and its research budget is 7.5 million dollars. The ratio of advertising to research is $\frac{4.2}{7.5} = 0.56$.

Explanation

Converting a ratio to a decimal gives you a single number that's easy to work with. It's perfect for comparing which of two ratios is larger, telling you exactly 'how many times bigger' one quantity is than another.

Property

A rate is a ratio that compares quantities with different units. For example, miles per gallon is a rate written as $\frac{\text{miles}}{\text{gallon}}$.
A unit rate is a rate that has been simplified so that the denominator is 1 unit.

Examples

• A car travels 240 miles on 8 gallons of gas. Its fuel efficiency is a unit rate: $\frac{240 \text{ miles}}{8 \text{ gallons}} = 30$ miles per gallon.
• An employee earns 120 dollars for 6 hours of work. Their pay rate is $\frac{120 \text{ dollars}}{6 \text{ hours}} = 20$ dollars per hour.
• A 16-ounce box of cereal costs 4 dollars. The unit price is $\frac{4 \text{ dollars}}{16 \text{ ounces}} = 0.25$ dollars per ounce, or 25 cents per ounce.

Explanation

A rate is a special ratio comparing different types of measurements, like distance and time. A unit rate makes it simple by telling you how much of the first quantity corresponds to just one unit of the second quantity.