Solving Proportions

⚖️ Solving Proportions

Part 1 of 5 — Ratios & Equivalent Fractions

Topics in This Part

🔑 Key Concept: A proportion is just an equation saying two ratios are equal. Before we can solve one, we need to be comfortable with ratios — that's Part 1.

What Is a Ratio?

A ratio compares two quantities. If a fruit bowl has $3$ apples and $4$ oranges, the ratio of apples to oranges is:

$3 \text{ to } 4 \qquad \text{or} \qquad 3 : 4 \qquad \text{or} \qquad \frac{3}{4}$

All three mean the same thing. In this lesson we'll almost always write ratios as fractions, because that makes them easy to solve.

⚠️ Order matters! The ratio of apples to oranges is $\dfrac{3}{4}$, but the ratio of oranges to apples is $\dfrac{4}{3}$. Always write the quantities in the order the problem names them.

Examples

Concept Check 🎯

Equivalent Ratios

Two ratios are equivalent if one is a scaled-up or scaled-down version of the other — exactly like equivalent fractions.

You get an equivalent ratio by multiplying (or dividing) both parts by the same number:

$\frac{2}{3} \;\overset{\times 4}{=}\; \frac{8}{12} \qquad\qquad \frac{10}{15} \;\overset{\div 5}{=}\; \frac{2}{3}$

🔑 Key Idea: Whatever you do to the top, you must do to the bottom. Multiplying both parts by the same number never changes the value of the ratio.

A Quick Picture

If $2$ pizzas feed $6$ people, then $4$ pizzas (double) feed $12$ people (double). The ratio $\dfrac{2}{6}$ and the ratio $\dfrac{4}{12}$ describe the same situation.

Build Equivalent Ratios 🧮

Fill in the missing number that makes the ratios equivalent.

1) $\dfrac{3}{5} = \dfrac{?}{20}$ 2) $\dfrac{2}{7} = \dfrac{10}{?}$ 3) $\dfrac{12}{18} = \dfrac{2}{?}$ (reduce first)

What Is a Proportion?

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

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

For example, $\dfrac{2}{3} = \dfrac{8}{12}$ is a proportion (both sides equal the same value). And $\dfrac{1}{2} = \dfrac{4}{8}$ is a proportion too.

But $\dfrac{2}{3} = \dfrac{4}{5}$ is not a proportion — those two ratios are not equal.

💡 Most proportion problems hide one of the four numbers and ask you to find it, like $\dfrac{2}{3} = \dfrac{x}{12}$. The next three parts are all about finding that missing value.

Is It a Proportion? 🔽

For each pair, decide whether the two ratios are equal.

⚖️ Solving Proportions

Part 2 of 5 — Cross Multiplication

🔑 The Big Tool: In any true proportion, the cross products are equal. This single fact turns every proportion into a simple equation you can solve.

The Cross-Products Rule

For any proportion:

$\frac{a}{b} = \frac{c}{d} \qquad\Longrightarrow\qquad a \cdot d = b \cdot c$

You multiply along each diagonal. The two diagonal products (the cross products) are always equal in a true proportion.

Why It Works

Start with $\dfrac{a}{b} = \dfrac{c}{d}$ and multiply both sides by $b \cdot d$:

$b \cdot d \cdot \frac{a}{b} = b \cdot d \cdot \frac{c}{d} \quad\Longrightarrow\quad d \cdot a = b \cdot c$

The $b$'s cancel on the left and the $d$'s cancel on the right — leaving the cross-products rule.

Quick Test

Is $\dfrac{3}{4} = \dfrac{9}{12}$ a true proportion? Check the cross products:

$3 \cdot 12 = 36 \qquad 4 \cdot 9 = 36$

They match, so yes — it's a true proportion. ✓

Concept Check 🎯

Solving for a Missing Value

To solve a proportion with an unknown, cross multiply, then undo the multiplication by dividing.

Worked Example: $\dfrac{x}{12} = \dfrac{3}{4}$

Step 1 — Cross multiply: $4 \cdot x = 12 \cdot 3$

Step 2 — Simplify: $4x = 36$

Step 3 — Divide both sides by $4$: $x = \frac{36}{4} = 9$

✅ Check: $\dfrac{9}{12} = \dfrac{3}{4}$? Cross products: $9 \cdot 4 = 36$ and $12 \cdot 3 = 36$. ✓

Worked Example: $\dfrac{7}{x} = \dfrac{14}{10}$

Here the variable is on the bottom — cross multiplication still works exactly the same way.

$7 \cdot 10 = x \cdot 14 \quad\Longrightarrow\quad 70 = 14x$

Divide both sides by $14$:

$x = \frac{70}{14} = 5$

💡 Tip: No matter where the unknown sits — top, bottom, left, or right — cross multiply first, then divide by the number multiplying the variable. The method never changes.

Order the Steps 🔽

You're solving $\dfrac{x}{8} = \dfrac{6}{4}$. Choose what happens at each stage.

Two-Step Recap

Every basic proportion is now a two-step problem:

1. Cross multiply to clear the fractions.
2. Divide both sides by the number multiplying $x$.

✅ Habit to build: after you find $x$, plug it back in and check that the cross products match. It catches almost every arithmetic slip.

Now try a few on your own.

Solve Each Proportion 🧮

Cross multiply, then divide. Enter the value of $x$.

1) $\dfrac{x}{6} = \dfrac{5}{3} \;\Rightarrow\; x = \,?$ 2) $\dfrac{4}{x} = \dfrac{8}{14} \;\Rightarrow\; x = \,?$ 3) $\dfrac{9}{15} = \dfrac{x}{5} \;\Rightarrow\; x = \,?$

⚖️ Solving Proportions

Part 3 of 5 — Trickier Proportions

🔑 Level Up: Real proportions don't always come out to whole numbers. In this part we handle decimal answers, fractions, and a variable that appears inside a sum like $(x + 2)$.

When the Answer Isn't a Whole Number

Nothing changes about the method — you just may end with a decimal or a fraction.

Worked Example: $\dfrac{x}{4} = \dfrac{7}{8}$

$8 \cdot x = 4 \cdot 7 \quad\Longrightarrow\quad 8x = 28 \quad\Longrightarrow\quad x = \frac{28}{8} = 3.5$

Worked Example: $\dfrac{5}{6} = \dfrac{x}{4}$

$5 \cdot 4 = 6 \cdot x \quad\Longrightarrow\quad 20 = 6x \quad\Longrightarrow\quad x = \frac{20}{6} = \frac{10}{3}$

💡 An answer like $\dfrac{10}{3}$ is perfectly fine — leave it as a reduced fraction (or write it as $\approx 3.33$). Both are correct.

Decimal & Fraction Answers 🧮

Solve. Decimals or reduced fractions are both fine.

1) $\dfrac{x}{10} = \dfrac{3}{4} \;\Rightarrow\; x = \,?$ 2) $\dfrac{3}{8} = \dfrac{x}{12} \;\Rightarrow\; x = \,?$

A Variable Inside a Sum

Sometimes the unknown is part of an expression like $(x + 2)$. Cross multiply first, then distribute and solve like a normal equation.

Worked Example: $\dfrac{x + 2}{5} = \dfrac{4}{2}$

Step 1 — Cross multiply: $2 \cdot (x + 2) = 5 \cdot 4$

Step 2 — Distribute the $2$: $2x + 4 = 20$

Step 3 — Subtract $4$ from both sides: $2x = 16$

Step 4 — Divide by $2$: $x = 8$

✅ Check: $\dfrac{8 + 2}{5} = \dfrac{10}{5} = 2$, and $\dfrac{4}{2} = 2$. ✓

Order the Steps 🔽

You're solving $\dfrac{x + 1}{6} = \dfrac{5}{3}$. Choose what happens at each stage.

Don't Forget to Distribute

The single most common error with a sum in the numerator is forgetting to multiply every term inside the parentheses.

$2(x + 3) = 2x + 6 \qquad \color{red}{\text{not}} \qquad 2x + 3$

So when you cross multiply $\dfrac{x+3}{5} = \dfrac{4}{1}$ you get $1 \cdot (x+3) = 5 \cdot 4$, then $x + 3 = 20$, then $x = 17$.

⚠️ After distributing, you've got an ordinary two-step equation: undo the addition first, then undo the multiplication.

Concept Check 🎯

⚖️ Solving Proportions

Part 4 of 5 — Word Problems & Real Applications

🔑 Big Payoff: Proportions power recipes, maps, unit prices, scale models, and similar shapes. The key skill is setting up the proportion correctly — keeping like units lined up.

Setting Up a Word Problem

The golden rule: keep the same kind of quantity in the same position (top with top, bottom with bottom).

Worked Example — Recipe

A recipe uses $2$ cups of flour to make $12$ cookies. How much flour for $30$ cookies?

Put flour on top, cookies on bottom on both sides:

$\frac{\text{flour}}{\text{cookies}}:\quad \frac{2}{12} = \frac{x}{30}$

Cross multiply: $\;2 \cdot 30 = 12 \cdot x \Rightarrow 60 = 12x \Rightarrow x = 5$.

So you need $5$ cups of flour.

⚠️ Common mistake: flipping one side. If flour is on top on the left, it must be on top on the right too. Mixing the order gives a wrong answer.

Set It Up 🔽

A car drives $150$ miles on $5$ gallons of gas. How far on $8$ gallons? Build the proportion.

Unit Rates & Best Buy

A unit rate is a ratio with $1$ on the bottom — like price per one item or miles per one hour. Proportions let you compare deals.

Worked Example — Better Buy

• Store A: $6 for $4$ granola bars
• Store B: $9 for $6$ granola bars

Find each price per bar:

A: $\frac{6}{4} = 1.50$ per bar, i.e. $1.50 per bar. B: $\frac{9}{6} = 1.50$ per bar, i.e. $1.50 per bar.

They're actually the same unit price! Both are $\dfrac{3}{2}$ — equivalent ratios.

💡 Why unit rates help: once everything is "per $1$," you can compare directly. The smaller per-unit price is the better deal.

Real-World Proportions 🧮

Set up a proportion and solve.

1) On a map, $2$ inches $= 50$ miles. How many miles is $5$ inches? $\,?$ (miles) 2) If $3$ apples cost $1.20, how much do $7$ apples cost? $\,?$ (dollars) 3) A printer prints $8$ pages in $10$ seconds. How many pages in $25$ seconds? $\,?$ (pages)

Bonus: Similar Figures

Similar shapes have the same form but different sizes. Their matching sides are proportional, so you can find a missing side with a proportion.

If two triangles are similar and the small one has sides $3$ and $4$ matching the large one's sides $9$ and $x$:

$\frac{3}{9} = \frac{4}{x} \quad\Longrightarrow\quad 3x = 36 \quad\Longrightarrow\quad x = 12$

🔑 Setup rule for similar figures: match corresponding sides. Small-triangle side over large-triangle side, on both sides of the equation.

Application Check 🎯

⚖️ Solving Proportions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) write and compare ratios, (2) cross multiply, (3) solve for any missing value (even decimals and sums), and (4) set up word problems. Let's put it all together.

Quick Reference

⚠️ Two reminders: order matters in a ratio, and in a word problem the same unit must go in the same position on both sides.

Mixed Practice 🎯

One More Set 🧮

1) $\dfrac{x + 3}{4} = \dfrac{5}{2} \;\Rightarrow\; x = \,?$ 2) $15$ feet of rope costs $6. What does $25$ feet cost? $\,?$ (dollars)

Exit Quiz ✅

Answer all three to finish the lesson.