Table of Contents
Understanding Proportions
A proportion is a statement that two ratios are equal.
If you already know what a ratio is, a proportion simply compares two of them:
$$
\frac{a}{b} = \frac{c}{d}
$$
This equation is read “$a$ is to $b$ as $c$ is to $d$.”
In words, a proportion says that two comparisons are in the same relative size, even if the actual numbers are different.
Examples:
- $\dfrac{1}{2} = \dfrac{2}{4}$
- $\dfrac{3}{5} = \dfrac{9}{15}$
Both are true proportions because the two ratios on each side describe the same relationship.
A proportion can also be written with colons:
- $a:b = c:d$
- “$a$ is to $b$ as $c$ is to $d$.”
Cross-Multiplication in Proportions
When you have a true proportion
$$
\frac{a}{b} = \frac{c}{d},
$$
there is a very important relationship among the four numbers:
$$
a \cdot d = b \cdot c
$$
This is called cross-multiplication (or the cross-product rule).
- The product of the extremes equals the product of the means.
- In $\dfrac{a}{b} = \dfrac{c}{d}$, the extremes are $a$ and $d$; the means are $b$ and $c$.
Example:
$$
\frac{2}{3} = \frac{8}{12}
$$
Check with cross-multiplication:
- $2 \cdot 12 = 24$
- $3 \cdot 8 = 24$
The products match, so the proportion is true.
Cross-multiplication lets you:
- Check if a proportion is true.
- Solve for an unknown in a proportion.
Solving Proportions for an Unknown
A very common use of proportions is to find a missing number when three of the four numbers are known.
Unknown in the numerator
Example:
$$
\frac{x}{5} = \frac{6}{15}
$$
Cross-multiply:
$$
x \cdot 15 = 5 \cdot 6
$$
$$
15x = 30
$$
Now solve for $x$:
$$
x = \frac{30}{15} = 2
$$
Check:
$$
\frac{2}{5} = \frac{6}{15}
$$
Both equal $\dfrac{2}{5} = 0.4$ and $\dfrac{6}{15} = 0.4$, so it works.
Unknown in the denominator
Example:
$$
\frac{4}{x} = \frac{2}{5}
$$
Cross-multiply:
$$
4 \cdot 5 = 2 \cdot x
$$
$$
20 = 2x
$$
Solve:
$$
x = \frac{20}{2} = 10
$$
Check:
$$
\frac{4}{10} = \frac{2}{5}
$$
Both are equal (both simplify to $\dfrac{2}{5}$).
Unknown on both sides
Example:
$$
\frac{3}{x} = \frac{y}{10}
$$
Cross-multiply:
$$
3 \cdot 10 = x \cdot y
$$
$$
30 = xy
$$
This relationship links $x$ and $y$. If you know one, you can find the other.
For example, if $x = 5$, then $30 = 5y$, so $y = 6$.
Checking Whether a Proportion Is True
To decide if an equation of the form
$$
\frac{a}{b} = \frac{c}{d}
$$
is a true proportion:
- Cross-multiply: compute $a \cdot d$ and $b \cdot c$.
- Compare the products.
- If they are equal, the proportion is true.
- If they are not equal, it is false.
Example:
Is
$$
\frac{5}{8} = \frac{15}{24}
$$
a true proportion?
Cross-multiply:
- $5 \cdot 24 = 120$
- $8 \cdot 15 = 120$
They are equal, so this is a true proportion.
Another example:
Is
$$
\frac{2}{7} = \frac{5}{14}
$$
a proportion?
Cross-multiply:
- $2 \cdot 14 = 28$
- $7 \cdot 5 = 35$
Not equal, so this is not a true proportion.
Equivalent Ratios and Proportions
A proportion is one way of expressing that two ratios are equivalent.
If
$$
\frac{a}{b} = \frac{c}{d},
$$
then:
- You can get $c$ by multiplying $a$ by the same number that turns $b$ into $d$.
- That is, there exists some number $k$ such that:
$$
c = k \cdot a \quad \text{and} \quad d = k \cdot b.
$$
Example:
$$
\frac{3}{4} = \frac{9}{12}
$$
Here $k = 3$:
- $9 = 3 \cdot 3$
- $12 = 3 \cdot 4$
This shows $3:4$ and $9:12$ are equivalent ratios, and the equation between them is a proportion.
You can also see this by simplifying:
$$
\frac{9}{12} = \frac{3}{4}
$$
So
$$
\frac{3}{4} = \frac{3}{4},
$$
which is clearly true.
Solving Practical Problems with Proportions
Proportions are often used to scale things up or down while keeping the same ratio. This includes situations like:
- Maps and scale drawings
- Recipes
- Conversions between units
- Constant speed or unit price (details of these applications can be explored more deeply elsewhere; here we focus on the proportion itself).
In each case, you usually:
- Identify the known ratio.
- Set up a new ratio using the unknown quantity.
- Write a proportion saying these two ratios are equal.
- Use cross-multiplication to solve.
Example: Simple scaling
A drawing shows that $1$ cm represents $5$ km in real life.
What real distance does $7$ cm represent?
Known ratio (drawing to real):
$$
\frac{1 \text{ cm}}{5 \text{ km}}
$$
New ratio with unknown:
$$
\frac{7 \text{ cm}}{x \text{ km}}
$$
Set up a proportion:
$$
\frac{1}{5} = \frac{7}{x}
$$
Cross-multiply:
$$
1 \cdot x = 5 \cdot 7
$$
$$
x = 35
$$
So $7$ cm on the drawing corresponds to $35$ km in reality.
Example: Recipe scaling
A recipe uses $2$ cups of flour to make $8$ servings. How many cups are needed for $12$ servings, assuming you keep the same proportions?
Known ratio:
$$
\frac{2 \text{ cups}}{8 \text{ servings}}
$$
New ratio:
$$
\frac{x \text{ cups}}{12 \text{ servings}}
$$
Proportion:
$$
\frac{2}{8} = \frac{x}{12}
$$
Cross-multiply:
$$
2 \cdot 12 = 8 \cdot x
$$
$$
24 = 8x
$$
$$
x = \frac{24}{8} = 3
$$
So you need $3$ cups of flour for $12$ servings.
Setting Up Proportions Carefully
A common source of mistakes is mismatching the order of numbers when forming a proportion.
To avoid errors:
- Be consistent about what goes on top and what goes on the bottom.
- Either:
- Put the same type of quantity in the numerators on both sides, and the other type in the denominators, or
- Use labels to make sure you’re matching correctly.
Example (correct setup):
A car travels $120$ km in $3$ hours. How far in $5$ hours at the same speed?
Ratio of distance to time:
$$
\frac{120 \text{ km}}{3 \text{ h}} = \frac{x \text{ km}}{5 \text{ h}}
$$
Both numerators are distances, both denominators are times. Now solve:
$$
120 \cdot 5 = 3 \cdot x
$$
$$
600 = 3x
$$
$$
x = 200
$$
So the car travels $200$ km in $5$ hours at the same speed.
If you accidentally flip one of the ratios (for example, put hours on top on one side and kilometers on top on the other), the proportion will not match the situation and you’ll get a wrong answer.
Being careful in setting up the proportion is just as important as solving it correctly.
Summary of Key Ideas About Proportions
- A proportion states that two ratios are equal: $\dfrac{a}{b} = \dfrac{c}{d}$.
- In a true proportion,
$$
a \cdot d = b \cdot c
$$
(cross-products are equal). - You can solve for an unknown in a proportion using cross-multiplication.
- You can check if a statement is a true proportion by comparing cross-products.
- Proportions describe equivalent ratios and are widely used to scale quantities up or down while keeping the same relative relationship.