Table of Contents
Understanding Ratios and Proportions
This chapter focuses on two very closely related ideas: ratios and proportions. Both are about comparing quantities rather than looking at them in isolation. You will see them in recipes, maps, photos, speed, prices, and many other parts of daily life.
Because this is a parent chapter, later subsections will go into more detail about each topic. Here we build the overall picture: what these ideas are, how they are connected, and why they matter.
Ratios: Comparing Two (or More) Quantities
A ratio compares how much of one quantity there is relative to another.
If you have 2 apples and 3 oranges, the ratio of apples to oranges is
$$
2:3
$$
You can read this as “2 to 3”.
Ratios can be written in several equivalent ways:
- Using a colon: $2:3$
- As a fraction: $\dfrac{2}{3}$ (when the second quantity is not zero)
- In words: “2 to 3”
The key idea is that a ratio does not just tell you a raw amount, it tells you how two amounts relate. For example, if a class has 10 boys and 15 girls, the ratio of boys to girls is
$$
10:15
$$
This ratio can be simplified by dividing both parts by the same nonzero number (here, 5):
$$
10:15 = 2:3
$$
The simplified ratio $2:3$ tells you that for every 2 boys, there are 3 girls. Whether the class is small or large, as long as this relationship holds, the ratio stays $2:3$.
Ratios can compare:
- Part to part: boys to girls, red marbles to blue marbles
- Part to whole: boys to total students, red marbles to all marbles
The exact way you state the ratio depends on what you are comparing.
Proportions: When Two Ratios Match
A proportion is a statement that two ratios are equal.
For example, the statement
$$
\frac{2}{3} = \frac{4}{6}
$$
is a proportion. In words: “2 is to 3 as 4 is to 6.”
Both sides describe the same comparison, just with different actual numbers.
In general, a proportion has the form
$$
\frac{a}{b} = \frac{c}{d},
$$
where $b \neq 0$ and $d \neq 0$. This says “$a$ is to $b$ as $c$ is to $d$.”
If two ratios form a true proportion, the fractions represent the same value. That means each can be reduced or changed to look like the other by multiplying or dividing numerator and denominator by the same nonzero number.
For example, the ratios $1:4$ and $3:12$ form a proportion:
$$
\frac{1}{4} = \frac{3}{12}
$$
since you can get from $1$ to $3$ and from $4$ to $12$ by multiplying by $3$.
The Link Between Ratios and Proportions
Ratios and proportions are directly connected:
- A ratio is one comparison: “this to that.”
- A proportion compares two ratios and says they are the same kind of comparison.
So whenever you see proportions, there are always ratios inside them.
If you know two ratios are in proportion, you know the relationship between the quantities is staying consistent. For example, if a recipe uses 2 cups of flour for every 1 cup of sugar, then any larger or smaller version of the recipe should keep the ratio $2:1$. Saying that
$$
\frac{2}{1} = \frac{6}{3}
$$
is a proportion means the larger recipe (6 cups of flour, 3 cups of sugar) keeps the same taste because the ratio is the same.
Constant Ratio and Scaling
One important idea that connects ratios and proportions is scaling. If two ratios form a proportion, one is just a scaled-up (or scaled-down) version of the other.
For example:
- Start with $2:5$.
- Multiply both numbers by $4$ to get $8:20$:
$$
\frac{2}{5} = \frac{8}{20}.
$$
Here, $8:20$ is the same comparison as $2:5$, but each quantity is 4 times larger. This is exactly what a proportion represents: the same ratio at a different scale.
This idea of keeping a constant ratio while changing the scale is behind many practical uses of proportions, such as:
- Enlarging or shrinking a picture without changing its shape
- Making a larger or smaller batch of a recipe
- Adjusting a map scale (for example, 1 cm represents 5 km)
- Working with speeds (for example, 60 km in 1 hour is the same speed as 120 km in 2 hours)
Later sections on direct and inverse variation will build on this idea of scaling.
Proportions and Multiplicative Relationships
When two ratios form a proportion, there is a multiplicative relationship between the quantities, not just an additive one.
For example, in the proportion
$$
\frac{2}{3} = \frac{4}{6},
$$
we see that:
- $2$ is multiplied by $2$ to get $4$, and
- $3$ is multiplied by $2$ to get $6$.
The same factor ($2$) is used for both parts of the ratio. This is what preserves the ratio and makes the proportion true.
This focus on multiplication is what makes proportions powerful for solving problems. If you know:
- three of the four numbers in a proportion, and
- that the relationship is multiplicative,
you can usually find the missing number by thinking in terms of “What should I multiply or divide by?” The later sections on proportions and direct variation will use this idea in detail.
Everyday Uses of Ratios and Proportions
Ratios and proportions show up in many common situations:
- Recipes: If a drink mix uses 1 cup of syrup for every 4 cups of water, this is a ratio of $1:4$. Doubling the recipe while keeping the same taste means using a proportion to keep the ratio the same.
- Maps and scale drawings: A map might say “1 cm represents 10 km.” This is a ratio of map distance to real distance, and using it involves setting up proportions.
- Prices and unit rates: Comparing “$4 for 2 liters” to “$6 for 3 liters” is comparing the ratios $4:2$ and $6:3$ and checking if they form a proportion.
- Speed: Saying “60 km in 1 hour” gives a ratio of distance to time. If you keep the same speed, any new distance and time pair should form a proportion with $60:1$.
As you go into the subsections on ratios, proportions, direct variation, and inverse variation, you will build on this chapter’s ideas: comparing quantities, keeping relationships constant through scaling, and using those relationships to solve practical problems.