Solving Proportions

⚖️ Solving Proportions

Part 1 of 5 — Concept Introduction

Imagine you have a recipe that serves $4$ people, but tonight $8$ people are coming for dinner. How much of each ingredient do you need? Or you are looking at a map where $1$ inch stands for $50$ miles — how far apart are two cities that are $3$ inches apart on the map?

The math tool that answers all of these questions is the proportion.

What Is a Proportion?

A proportion is an equation that says two ratios are equal:

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

You read this as "$a$ is to $b$ as $c$ is to $d$." Both fractions describe the same relationship, just with different numbers.

Example: $\dfrac{1}{2} = \dfrac{3}{6}$ is a true proportion, because both fractions are equal to one-half.

Cross Multiplication ✖️

Most proportions have a missing number — usually written as $x$. To find it, we use a trick called cross multiplication.

When two ratios are equal, the diagonal products are equal too:

$\frac{a}{b} = \frac{c}{d} \quad \Rightarrow \quad a \times d = b \times c$

You multiply along each diagonal (corner to opposite corner) and set the two products equal.

The two diagonals form a little "X" across the equal sign — that is why it is called cross multiplication.

A First Worked Example 📝

Let's solve for $x$ in this proportion:

$\frac{3}{4} = \frac{x}{12}$

Step 1 — Cross multiply the diagonals. The $3$ pairs with the $12$, and the $4$ pairs with the $x$:

$3 \times 12 = 4 \times x$

Step 2 — Multiply the known numbers:

$36 = 4x$

Step 3 — Solve by dividing both sides by $4$:

$x = \frac{36}{4} = 9$

So $x = 9$. ✅

Check it! Substitute back in: $\dfrac{3}{4} = \dfrac{9}{12}$. Both fractions reduce to $\dfrac{3}{4}$, so our answer is correct.

Concept Check 🎯

Make sure you understand what cross multiplication does.

📝 Worked Examples

Part 2 of 5 — Worked Examples

Let's walk through two complete examples, one careful step at a time.

Example 1: Missing number on the bottom

$\frac{5}{x} = \frac{20}{12}$

Step 1 — Cross multiply the diagonals ($5$ with $12$, and $x$ with $20$):

$5 \times 12 = x \times 20$

Step 2 — Multiply the known numbers:

$60 = 20x$

Step 3 — Solve by dividing both sides by $20$:

$x = \frac{60}{20} = 3$

So $x = 3$. Check: $\dfrac{5}{3} = \dfrac{20}{12}$? Yes — $\dfrac{20}{12}$ reduces to $\dfrac{5}{3}$. ✅

Example 2: A bigger proportion

$\frac{x}{6} = \frac{15}{18}$

Step 1 — Cross multiply ($x$ with $18$, and $6$ with $15$):

$x \times 18 = 6 \times 15$

Step 2 — Multiply the right side:

$18x = 90$

Step 3 — Solve by dividing both sides by $18$:

$x = \frac{90}{18} = 5$

So $x = 5$. Check: $\dfrac{5}{6} = \dfrac{15}{18}$? Yes — $\dfrac{15}{18}$ reduces to $\dfrac{5}{6}$. ✅

The pattern is always the same: cross multiply, then divide to get $x$ alone.

Your Turn 🧮

Solve the proportion $\dfrac{4}{6} = \dfrac{x}{9}$ step by step.

1. Cross multiply the known numbers $4$ and $9$. What is $4 \times 9$? (Type a single number.)

2. Your equation is now $36 = 6x$. To solve, divide $36$ by $6$. What is $x$? (Type a single number.)

3. Check your answer: does $\dfrac{4}{6} = \dfrac{6}{9}$? Type yes or no.

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Cross multiply, then divide to find $x$. Work carefully and check each answer.

Fill In the Steps 🔍

Complete the solution to the proportion $\dfrac{6}{9} = \dfrac{x}{15}$.

🌍 Application & Word Problems

Part 4 of 5 — Real-World Problems

Proportions show up everywhere in real life. The trick to word problems is setting up the proportion correctly: keep the same kind of thing on the same level in both fractions.

Recipe Scaling 🍪

A cookie recipe uses $2$ cups of flour to make $24$ cookies. How much flour do you need to make $60$ cookies?

Step 1 — Set up the proportion, keeping flour over cookies on both sides:

$\frac{2 \text{ cups}}{24 \text{ cookies}} = \frac{x \text{ cups}}{60 \text{ cookies}}$

Step 2 — Cross multiply: $2 \times 60 = 24 \times x$, so $120 = 24x$.

Step 3 — Solve: $x = \dfrac{120}{24} = 5$ cups of flour. 🧁

Where Else Are Proportions Used?

• Scale drawings & maps — $1$ inch represents $50$ real miles
• Recipe conversions — doubling or tripling a recipe
• Unit conversions — changing feet to inches, or dollars to euros
• Similar figures — shapes with matching shape but different size

Tip: Always label your fractions (like "cups over cookies") so the matching units line up on top and bottom.

Map Distance Problem 🗺️

On a map, $1$ inch represents $50$ miles. Two cities are $4$ inches apart on the map. Set up and solve the proportion $\dfrac{1}{50} = \dfrac{4}{x}$ to find the real distance.

1. Cross multiply: what is $1 \times x$ equal to? Compute $50 \times 4$. (Type a single number.)

2. So $x =$ ? This is the real distance in miles. (Type a single number.)

3. If two other cities were $3$ inches apart, how many real miles is that ($3 \times 50$)? (Type a single number.)

Application Problem 📋

🏆 Review & Challenge

Part 5 of 5 — Review & Challenge

You made it! Here is everything about solving proportions in one place.

Quick Summary Table

Remember the 3 steps:

1. Cross multiply the diagonals and set the products equal.
2. Solve by dividing both sides so $x$ is by itself.
3. Check by substituting your answer back into the proportion.

Proportions help with maps, recipes, unit conversions, and similar figures. Now try these mixed challenge problems! 🎯

Challenge Round 🎯

These mix every idea together. Cross multiply, solve, and check carefully!