Solving Proportions

⚖️ Solving Proportions

Part 1 of 5 — Ratios & What a Proportion Is

Topics in This Part

🔑 Key Concept: A proportion is just an equation that says two ratios are equal. Before we can solve one, we need to know what a ratio is and when two of them match — that's Part 1.

What Is a Ratio?

A ratio compares two quantities by division. The ratio "3 to 4" can be written three ways:

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

In this lesson we almost always use the fraction form $\dfrac{3}{4}$, because that's the form we can manipulate algebraically.

Examples

💡 Order matters. The ratio of flour to water $\left(\dfrac{2}{3}\right)$ is not the same as the ratio of water to flour $\left(\dfrac{3}{2}\right)$. Always keep the same quantity on top in both ratios of a proportion.

Equivalent Ratios

Two ratios are equivalent if one is the other scaled up or down — multiply or divide both parts by the same nonzero number.

$\frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}$

So $\dfrac{2}{3}$ and $\dfrac{8}{12}$ are equivalent ratios. They simplify to the same lowest-terms fraction.

🔑 A proportion is two equivalent ratios set equal, like $\dfrac{2}{3} = \dfrac{8}{12}$.

Fill the Missing Part 🧮

Each pair is equivalent. Find the missing number by scaling top and bottom by the same factor.

1) $\dfrac{2}{3} = \dfrac{?}{15}$ (denominator went $3 \to 15$) 2) $\dfrac{4}{5} = \dfrac{12}{?}$ (numerator went $4 \to 12$) 3) $\dfrac{6}{10} = \dfrac{3}{?}$ (simplify: numerator went $6 \to 3$)

Concept Check 🎯

Scaling top and bottom by the same factor is the heartbeat of every proportion. Use that idea to pick out equivalent ratios and translate a word phrase into a ratio.

Is the Proportion True?

A proportion like $\dfrac{a}{b} = \dfrac{c}{d}$ is true when the two ratios really are equal. The fastest test is to simplify both sides and compare.

Example: Is $\dfrac{4}{6} = \dfrac{6}{9}$ true?

Simplify each side:

$\frac{4}{6} = \frac{2}{3} \qquad \frac{6}{9} = \frac{2}{3}$

Both reduce to $\dfrac{2}{3}$, so the proportion is true. ✓

Example: Is $\dfrac{3}{4} = \dfrac{5}{6}$ true?

$\dfrac{3}{4} = 0.75$ but $\dfrac{5}{6} \approx 0.833$. They're not equal, so this proportion is false.

⚠️ Don't be fooled by "close" numbers. $\dfrac{3}{4}$ and $\dfrac{5}{6}$ look similar but are not equal.

True or False? 🔽

Decide whether each proportion is true by simplifying both ratios.

Wrapping Up Part 1

You now know:

• A ratio compares two numbers, usually written as a fraction.
• Equivalent ratios scale by the same factor on top and bottom.
• A proportion sets two ratios equal, and it's true when both reduce to the same fraction.

In Part 2 we'll handle the real goal: when one number is missing, we solve for it using cross products.

⚖️ Solving Proportions

Part 2 of 5 — The Cross-Products Method

🔑 The Idea: In any true proportion $\dfrac{a}{b} = \dfrac{c}{d}$, the cross products are equal: $a \cdot d = b \cdot c$. This one fact lets us solve for any missing value.

Why Cross Products Work

Start with a proportion and multiply both sides by both denominators:

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

Multiply both sides by $b \cdot d$:

$\frac{a}{b}\cdot bd = \frac{c}{d}\cdot bd \;\Longrightarrow\; a\,d = b\,c$

The $b$'s cancel on the left and the $d$'s cancel on the right, leaving the cross-multiply rule:

$\boxed{\;a \cdot d = b \cdot c\;}$

💡 Picture it as an "X". Multiply along each diagonal of $\dfrac{a}{b} = \dfrac{c}{d}$: top-left × bottom-right equals bottom-left × top-right.

Concept Check 🎯

Solving a Proportion — The Three Steps

To solve a proportion for the unknown:

1. Cross-multiply to clear the fractions.
2. Simplify each side.
3. Divide to isolate the variable.

Worked Example: $\dfrac{x}{4} = \dfrac{3}{8}$

$x \cdot 8 = 4 \cdot 3 \;\Longrightarrow\; 8x = 12 \;\Longrightarrow\; x = \frac{12}{8} = \frac{3}{2} = 1.5$

✅ Check: $\dfrac{1.5}{4} = 0.375$ and $\dfrac{3}{8} = 0.375$ ✓

Worked Example: $\dfrac{6}{n} = \dfrac{2}{5}$

The variable is in the denominator — that's fine, the method is identical:

$6 \cdot 5 = n \cdot 2 \;\Longrightarrow\; 30 = 2n \;\Longrightarrow\; n = \frac{30}{2} = 15$

✅ Check: $\dfrac{6}{15} = \dfrac{2}{5}$ — both reduce to $0.4$ ✓

Worked Example: $\dfrac{7}{10} = \dfrac{x}{4}$

$7 \cdot 4 = 10 \cdot x \;\Longrightarrow\; 28 = 10x \;\Longrightarrow\; x = \frac{28}{10} = 2.8$

💡 Answers don't have to be whole numbers — fractions and decimals are perfectly normal.

Order the Steps 🔽

You're solving $\dfrac{x}{6} = \dfrac{10}{15}$. Choose what happens at each stage.

Your Turn to Drive

You've seen the variable on top, on the bottom, and the whole three-step routine. Now run the full cross-multiply → simplify → divide process yourself on three fresh proportions.

Solve It 🧮

Solve each proportion for the variable. (Decimals or fractions are fine.)

1) $\dfrac{x}{9} = \dfrac{4}{6} \;\Rightarrow\; x = \,?$ 2) $\dfrac{5}{n} = \dfrac{1}{4} \;\Rightarrow\; n = \,?$ 3) $\dfrac{3}{8} = \dfrac{w}{20} \;\Rightarrow\; w = \,?$

⚖️ Solving Proportions

Part 3 of 5 — Variables in Binomials & Both Sides

🔑 Why it matters: Real proportions often have a whole expression like $(x+2)$ in a numerator or denominator. Cross products still work — you just distribute and solve the linear equation that pops out.

When a Variable Is Inside a Binomial

Cross-multiply exactly as before, then distribute and solve.

Worked Example: $\dfrac{x+2}{5} = \dfrac{3}{1}$

$\frac{x+2}{5} = \frac{3}{1} \;\Longrightarrow\; 1\,(x+2) = 5 \cdot 3$

$x + 2 = 15 \;\Longrightarrow\; x = 13$

✅ Check: $\dfrac{13+2}{5} = \dfrac{15}{5} = 3$ ✓

Worked Example: $\dfrac{x+1}{4} = \dfrac{x}{3}$

A variable appears on both sides. Cross-multiply, distribute, then collect the $x$-terms:

$3\,(x+1) = 4\,(x) \;\Longrightarrow\; 3x + 3 = 4x$

Subtract $3x$ from both sides:

$3 = 4x - 3x = x \;\Longrightarrow\; x = 3$

✅ Check: $\dfrac{3+1}{4} = \dfrac{4}{4} = 1$ and $\dfrac{3}{3} = 1$ ✓

⚠️ Distribute carefully. $3(x+1) = 3x + 3$, not $3x + 1$. Forgetting to multiply the second term is the #1 error here.

Concept Check 🎯

Worked Example: $\dfrac{2x}{9} = \dfrac{6}{27}$

When a coefficient sits on the variable, divide it off at the end:

$2x \cdot 27 = 9 \cdot 6 \;\Longrightarrow\; 54x = 54 \;\Longrightarrow\; x = 1$

Worked Example: $\dfrac{x}{x+6} = \dfrac{2}{5}$

$5x = 2(x+6) \;\Longrightarrow\; 5x = 2x + 12$

$5x - 2x = 12 \;\Longrightarrow\; 3x = 12 \;\Longrightarrow\; x = 4$

✅ Check: $\dfrac{4}{4+6} = \dfrac{4}{10} = \dfrac{2}{5}$ ✓

Build the Equation 🔽

You're solving $\dfrac{x+3}{8} = \dfrac{x}{4}$. Fill in each stage.

Put It All Together

These next three mix the patterns from this part: a binomial on top, a variable on both sides, and a coefficient on the variable. Cross-multiply first, then distribute wherever there's a binomial.

Solve It 🧮

Solve each proportion for $x$. (Whole numbers here.)

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

⚖️ Solving Proportions

Part 4 of 5 — Word Problems & Real Applications

🔑 Big Payoff: Proportions are everywhere — recipes, maps, unit prices, similar figures, and unit conversions. The skill is setting up the proportion with matching units lined up top-to-top and bottom-to-bottom.

Setting Up a Word Problem

The golden rule: keep the same kind of quantity in the same position in both ratios.

$\frac{\text{miles}}{\text{hours}} = \frac{\text{miles}}{\text{hours}}$

Worked Example: Unit Rate

A car travels $150$ miles in $3$ hours. At that rate, how far does it go in $5$ hours?

Let $d$ be the distance. Line up miles over hours on both sides:

$\frac{150}{3} = \frac{d}{5}$

$150 \cdot 5 = 3 \cdot d \;\Longrightarrow\; 750 = 3d \;\Longrightarrow\; d = 250 \text{ miles}$

✅ Check: $150$ miles in $3$ hours is $50$ mph; $50 \times 5 = 250$ miles ✓

Worked Example: Recipe Scaling

A recipe needs $2$ cups of sugar for every $5$ cups of flour. How much sugar pairs with $20$ cups of flour?

$\frac{\text{sugar}}{\text{flour}} = \frac{2}{5} = \frac{s}{20}$

$2 \cdot 20 = 5 \cdot s \;\Longrightarrow\; 40 = 5s \;\Longrightarrow\; s = 8 \text{ cups of sugar}$

Worked Example: Map Scale

On a map, $1$ inch represents $25$ miles. Two cities are $3.5$ inches apart. How far apart are they really?

$\frac{1 \text{ in}}{25 \text{ mi}} = \frac{3.5 \text{ in}}{m} \;\Longrightarrow\; 1 \cdot m = 25 \cdot 3.5 \;\Longrightarrow\; m = 87.5 \text{ miles}$

💡 Tip: Put the unknown where it naturally belongs (here, the real-world miles), and the "$1$ inch = $25$ miles" rate becomes one ratio.

Set It Up 🎯

Similar Figures

Similar figures have the same shape but different sizes — their corresponding sides are proportional. Match up corresponding sides and solve.

Worked Example

Two similar triangles: the small one has sides $3$ and $4$; the large one has the corresponding sides $9$ and $x$.

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

🔑 Shadow trick: A $6$-ft person casts a $4$-ft shadow while a tree casts a $20$-ft shadow at the same time. Then $\dfrac{6}{4} = \dfrac{t}{20}$, giving a tree height $t = 30$ ft.

Solve the Similar-Figure Problem 🔽

The shadow problem: a $6$-ft person casts a $4$-ft shadow; at the same time a tree casts a $20$-ft shadow. Find the tree's height $t$.

Real-World Round

Now tackle three on your own: a cost problem, a map-scale problem, and a similar-figures problem. For each one, set up the ratio with matching units lined up, then cross-multiply and solve.

Word Problems 🧮

Set up a proportion and solve. Enter just the number.

1) $3$ apples cost $2. How many dollars do $12$ apples cost? (dollars) 2) A map uses $1$ inch $= 30$ miles. How many miles is $4$ inches? (miles) 3) Two similar rectangles: small is $5$ by $8$; the large one's matching side to the $5$ is $15$. Find the side matching the $8$.

⚖️ Solving Proportions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) recognize and test proportions, (2) solve with cross products, (3) handle variables in binomials and on both sides, and (4) set up real-world word problems. Let's put it all together.

Quick Reference

⚠️ Top mistakes to avoid: flipping a ratio's order, forgetting to distribute across a binomial, and mixing units (miles over hours on one side, hours over miles on the other).

Mixed Practice 🎯

Two More to Warm Up

Before the Exit Quiz, solve these two — one with the variable in the denominator, one with it on top. Remember decimals and fractions are valid answers.

One More Set 🧮

Solve each. (Decimals/fractions ok.)

1) $\dfrac{8}{n} = \dfrac{2}{3} \;\Rightarrow\; n = \,?$ 2) $\dfrac{x}{5} = \dfrac{7}{2} \;\Rightarrow\; x = \,?$

You're Ready

That's every tool: test a proportion, cross-multiply, distribute through binomials, and model word problems. Finish strong with the three-question Exit Quiz below.

Exit Quiz ✅

Answer all three to finish the lesson.