🎯⭐ INTERACTIVE LESSON

Solving Proportions

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Solving Proportions - Complete Interactive Lesson

Part 1: ⚖️ 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:

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

A First Worked Example 📝

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

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

Concept Check 🎯

Make sure you understand what cross multiplication does.

Part 2: 📝 Worked Examples

📝 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

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

Part 3: 🧭 Guided Practice

🧭 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 $\dfrac{6}{9} = \dfrac{x}{15}$ .

Part 4: 🌍 Application & Word Problems

🌍 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:

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

Part 5: Review & Challenge

🏆 Review & Challenge

Part 5 of 5 — Review & Challenge

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

Quick Summary Table

Idea	What It Means	Example
Proportion	Two equal ratios	$\frac{a}{b} = \frac{c}{d}$

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

Step	What You Do
1. Set up	Write the proportion $\frac{a}{b} = \frac{c}{d}$
2. Cross multiply	Multiply the diagonals: $a \times d$ and $b \times c$
3. Set equal	Write $a \times d = b \times c$
4. Solve	Divide both sides to get $x$ by itself

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

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:

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

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

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

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:

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 — reduces to . ✅

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.

Cross multiply the known numbers $4$ and $9$ . What is $4 \times 9$ ? (Type a single number.)
Your equation is now $36 = 6x$ . To solve, divide $36$ by $6$ . What is $x$ ? (Type a single number.)
Check your answer: does $\dfrac{4}{6} = \dfrac{6}{9}$ ? Type yes or no.

\frac{2 cups}{24 cookies} = \frac{x cups}{60 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.

Cross multiply: what is $1 \times x$ equal to? Compute $50 \times 4$ . (Type a single number.)
So $x =$ ? This is the real distance in miles. (Type a single number.)
If two other cities were $3$ inches apart, how many real miles is that ()? (Type a single number.)

Application Problem 📋

Remember the 3 steps:

Cross multiply the diagonals and set the products equal.
Solve by dividing both sides so $x$ is by itself.
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!