Table of Contents
Understanding Ratios
A ratio compares two quantities. It tells you how many times one quantity contains another, or how large one quantity is relative to another.
In this chapter, we focus on what ratios are, how to write them, and how to work with them. The idea of a ratio is used later when we talk about proportions and different kinds of variation, but here we stay with basic comparisons.
What a Ratio Represents
A ratio compares two numbers that measure the same kind of thing (or are at least being compared meaningfully).
Examples:
- 3 red apples and 2 green apples: the ratio of red to green apples is $3 : 2$.
- A class with 12 boys and 18 girls: the ratio of boys to girls is $12 : 18$.
- A map where $1$ cm represents $50$ km: the ratio of map distance to real distance is $1 : 50$.
A ratio does not tell you the actual amounts by itself; it tells you how they relate. The ratio $3 : 2$ could represent:
- 3 red apples and 2 green apples, or
- 6 red apples and 4 green apples, or
- 30 red apples and 20 green apples, and so on.
All of these have the same comparison between red and green.
Ways to Write Ratios
There are three common notations for a ratio comparing $a$ to $b$:
- Using a colon: $a : b$
- As a fraction: $\dfrac{a}{b}$ (read as “$a$ to $b$”)
- Using words: “$a$ to $b$”
For example, “5 to 8” can be written as:
- $5 : 8$
- $\dfrac{5}{8}$
- “5 to 8”
When we use the fraction form for a ratio, we are not necessarily talking about a part of a whole; we are talking about a comparison. For example, the ratio of dogs to cats could be $\dfrac{5}{3}$ even though “5/3 of a cat” does not make sense. The fraction notation is just a convenient way to work with the numbers.
Order matters:
- “boys to girls” $= 12 : 18$
- “girls to boys” $= 18 : 12$
These are different ratios.
Simplifying Ratios
Just as you simplify fractions, you can simplify ratios by dividing both parts by the same nonzero number (a common factor).
Example:
- A group has 12 boys and 18 girls.
- Ratio of boys to girls: $12 : 18$.
- The greatest common factor of 12 and 18 is 6.
- Divide both parts by 6:
$ : 18 = \dfrac{12}{6} : \dfrac{18}{6} = 2 : 3.$$
So the ratio of boys to girls simplifies to $2 : 3$. This means:
- For every $2$ boys, there are $3$ girls.
- The exact numbers may be larger, but they keep the same comparison.
If the greatest common factor is $1$, the ratio is already in simplest form.
Example:
- $7 : 9$ has no common factor greater than 1.
- $7 : 9$ is already simplified.
Using Fraction Form to Simplify
Since $a : b$ can be written as $\dfrac{a}{b}$, you can simplify a ratio by simplifying the fraction.
Example:
- Simplify $15 : 20$.
- Write as a fraction: $\dfrac{15}{20}$.
- Simplify the fraction by dividing top and bottom by 5:
$$\dfrac{15}{20} = \dfrac{3}{4}.$$ - So $15 : 20 = 3 : 4$.
Part-to-Part and Part-to-Whole Ratios
Ratios often compare:
- One part of a group to another part (part-to-part), or
- A part of a group to the whole group (part-to-whole).
Part-to-Part Ratios
These compare two different parts inside the same whole.
Example:
- A jar has 4 blue marbles and 6 red marbles.
- Ratio of blue to red: $4 : 6$ (simplifies to $2 : 3$).
- Ratio of red to blue: $6 : 4$ (simplifies to $3 : 2$).
These ratios do not directly say how many marbles in total, only how the colors compare.
Part-to-Whole Ratios
These compare one part to the total.
Using the same example:
- Total marbles $= 4 + 6 = 10$.
- Ratio of blue to total: $4 : 10$ (simplifies to $2 : 5$).
- Ratio of red to total: $6 : 10$ (simplifies to $3 : 5$).
Part-to-whole ratios are closely related to fractions and percentages, because they tell what fraction of the whole each part is.
Equivalent Ratios
Two ratios are equivalent if they express the same comparison, even if the numbers are different.
You can create equivalent ratios by multiplying or dividing both parts by the same nonzero number.
Example:
Start with $2 : 3$.
- Multiply both parts by 2: $4 : 6$.
- Multiply both parts by 5: $10 : 15$.
- Divide both parts of $8 : 12$ by 4 to get $2 : 3$.
So $2 : 3$, $4 : 6$, $8 : 12$, and $10 : 15$ are all equivalent.
Using fractions:
- $2 : 3$ corresponds to $\dfrac{2}{3}$.
- $4 : 6$ corresponds to $\dfrac{4}{6}$.
- $\dfrac{2}{3} = \dfrac{4}{6}$, so the ratios are equivalent.
Ratios With Units
Sometimes the two quantities in a ratio have different units. This is common in “rates,” such as speed or price per item, but here we will just describe them as ratios with units.
Examples:
- Speed: 60 kilometers in 2 hours.
- Ratio of distance to time: $60 \text{ km} : 2 \text{ hours}$.
- Divide both parts by 2: $30 \text{ km} : 1 \text{ hour}$.
- This is often said as “30 km per hour,” written $30 \text{ km/h}$.
- Price: \$12 for 3 notebooks.
- Ratio of cost to notebooks: \$12 : 3 notebooks.
- Divide both parts by 3: \$4 : 1 notebook.
- This can be read as “\$4 per notebook.”
In each case, you use the same idea: simplify by dividing both parts by the same number.
Writing Ratios From Descriptions
You will often be given a description and asked to write a ratio.
Key steps:
- Identify what is being compared.
- Note the order of comparison.
- Write the ratio in the requested form (colon, fraction, or words).
- Simplify if asked.
Example 1:
- “In a bag, there are 5 green balls and 7 yellow balls. Write the ratio of green to yellow.”
- Comparing green to yellow: $5 : 7$.
- Already simplified.
Example 2:
- “At a park, there are 9 dogs and 6 cats. Write the ratio of cats to total animals.”
- Total animals $= 9 + 6 = 15$.
- Ratio of cats to total: $6 : 15$.
- Simplify by dividing both parts by 3: $2 : 5$.
Example 3:
- “A recipe uses 2 cups of sugar and 5 cups of flour. Write the ratio of sugar to flour as a fraction.”
- Sugar to flour: $2 : 5$.
- As a fraction: $\dfrac{2}{5}$.
Scaling Ratios
You might need to “scale up” or “scale down” a ratio to match a situation with larger or smaller quantities, while keeping the same comparison.
To scale up, multiply both parts by the same number.
To scale down, divide both parts by the same number.
Example:
- A drink mix uses a water to syrup ratio of $4 : 1$.
- This could mean:
- 4 cups water and 1 cup syrup,
- 8 cups water and 2 cups syrup (multiply both parts by 2),
- 12 cups water and 3 cups syrup (multiply both parts by 3),
and so on. - All of these keep the same taste because the ratio stays $4 : 1$.
If you are given actual amounts and asked whether they match a given ratio, you can:
- Simplify the given amounts and see if the simplified ratio matches, or
- See if one set can be obtained by multiplying both parts of the other set by the same number.
Example:
- Does 9 red and 12 blue objects match the ratio $3 : 4$?
- Simplify $9 : 12$ by dividing both parts by 3: $3 : 4$.
- Yes, $9 : 12$ matches $3 : 4$.
Common Pitfalls With Ratios
- Reversing the order: “A to B” is not the same as “B to A”. Always follow the order given in the problem.
- Forgetting the total when needed: If the ratio is “part to whole,” you must calculate the total correctly.
- Not simplifying when asked: A ratio can often be simplified by dividing both parts by a common factor.
- Mixing units carelessly: In ratios with units, make sure you are comparing meaningful quantities (for example, cost to items, distance to time) and keep the units consistent.
Understanding these basics of ratios prepares you for working with proportions, where you compare relationships between two ratios.