Ratios and Rates

⚖️ Ratios and Rates

Part 1 of 5 — What Is a Ratio?

Topics in This Part

🔑 Key Concept: A ratio compares two quantities by division. It tells you how much of one thing there is for every amount of another — the foundation for rates, proportions, percents, and slope.

The Meaning of a Ratio

A ratio compares two quantities. If a fruit bowl has $6$ apples and $4$ oranges, the ratio of apples to oranges is

$6 \text{ to } 4 \quad=\quad 6 : 4 \quad=\quad \frac{6}{4}.$

All three notations mean the same thing. Because a ratio is really a fraction, you can simplify it by dividing both numbers by a common factor:

$\frac{6}{4} = \frac{6 \div 2}{4 \div 2} = \frac{3}{2} \quad\Longrightarrow\quad 3 : 2.$

So for every $3$ apples there are $2$ oranges.

⚠️ Order matters. "Apples to oranges" is $6:4$, but "oranges to apples" is $4:6 = 2:3$. Write the quantities in the order the problem names them.

Part-to-Part vs. Part-to-Whole

There are two flavors of ratio, and mixing them up is the most common early mistake.

The total is the sum of all parts: $6 + 4 = 10$ pieces of fruit.

💡 A part-to-whole ratio is exactly what becomes a fraction of the whole and later a percent. Here apples are $\dfrac{6}{10} = \dfrac{3}{5} = 60\%$ of the fruit.

Concept Check 🎯

Simplify Each Ratio 🧮

Write each ratio in simplest form. Enter your answer as a fraction like 3/2 (first number over second number).

1) $8 : 12 = \,?$ 2) $15 : 25 = \,?$ 3) $9 : 9 = \,?$ (yes, ratios can be $1:1$)

Equivalent Ratios

Two ratios are equivalent when one simplifies to the other — exactly like equivalent fractions. You can multiply or divide both numbers by the same value:

$2 : 3 \;=\; 4 : 6 \;=\; 6 : 9 \;=\; 20 : 30.$

You cannot add or subtract the same number from both — that changes the relationship.

🔑 Key Idea: Scaling both parts by the same factor keeps a ratio equivalent. This is the engine behind every proportion you'll set up in Part 3.

Spot the Equivalent Ratio 🔽

Each ratio below is scaled from $2:3$. Pick the value that keeps it equivalent.

⚖️ Ratios and Rates

Part 2 of 5 — Rates & Unit Rates

🔑 The Idea: A rate is a ratio that compares two quantities with different units (miles and hours, dollars and pounds). A unit rate scales it down to "per one" — the single most useful number for comparing deals and speeds.

Rates and Unit Rates

A rate compares quantities measured in different units:

$\frac{120 \text{ miles}}{2 \text{ hours}}, \qquad \frac{\$6}{3 \text{ pounds}}, \qquad \frac{15 \text{ pages}}{5 \text{ minutes}}.$

A unit rate has a denominator of $1$. To find it, divide the top by the bottom:

$\frac{120 \text{ miles}}{2 \text{ hours}} = \frac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ mph}.$

$\frac{\$6}{3 \text{ lb}} = \frac{\$2}{1 \text{ lb}} = \$2 \text{ per pound}.$

💡 The word "per" signals a unit rate: miles per hour, dollars per pound, words per minute. "Per" always means divide by that unit.

Find the Unit Rate 🧮

Divide the top quantity by the bottom quantity. Enter just the number.

1) $150$ miles in $3$ hours $= \,?$ miles per hour 2) $12 for $4$ notebooks $= \,?$ dollars per notebook 3) $90$ words in $2$ minutes $= \,?$ words per minute

Unit Rates Compare Deals

Unit rates let you compare options that come in different sizes — the "better buy."

Example: Which juice is cheaper?

• Brand A: $3.00 for $24$ oz $\;\Rightarrow\; 3.00 \div 24 = 0.125$, i.e. $0.125 per oz
• Brand B: $4.50 for $40$ oz $\;\Rightarrow\; 4.50 \div 40 = 0.1125$, i.e. $0.1125 per oz

Brand B costs less per ounce ($0.1125 $<$ $0.125), so Brand B is the better buy.

⚠️ Always compare the same unit. A lower total price is not always the better deal — compare price per ounce (or per item), never just the sticker price.

Concept Check 🎯

Match the Unit Rate 🔽

For each rate, choose the correct unit rate.

⚖️ Ratios and Rates

Part 3 of 5 — Proportions & Cross-Multiplication

🔑 Why it works: A proportion is an equation stating that two ratios are equal. Cross-multiplication turns it into a simple linear equation you can solve for the unknown.

What Is a Proportion?

A proportion sets two ratios equal:

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

Two ratios form a true proportion exactly when their cross products are equal:

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

Quick check: Is $\dfrac{3}{4} = \dfrac{9}{12}$?

Cross-multiply: $3 \cdot 12 = 36$ and $4 \cdot 9 = 36$. The cross products match, so yes, it's a true proportion.

🔑 The cross-product rule ($a d = b c$) is just "multiply both sides by $b$ and by $d$." It's the workhorse for solving any proportion.

Concept Check 🎯

Solving a Proportion for the Unknown

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

1. Cross-multiply: $4x = 8 \cdot 3 = 24$
2. Divide both sides by $4$: $x = \dfrac{24}{4} = 6$

✅ Check: $\dfrac{6}{8} = \dfrac{3}{4}$ ✓ (both reduce to $\frac{3}{4}$).

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

1. Cross-multiply: $2n = 5 \cdot 6 = 30$
2. Divide by $2$: $n = 15$

✅ Check: $\dfrac{5}{15} = \dfrac{1}{3}$ and $\dfrac{2}{6} = \dfrac{1}{3}$ ✓

Order the Steps 🔽

You're solving $\dfrac{x}{6} = \dfrac{10}{4}$. Choose what happens at each stage.

Solve Each Proportion 🧮

Cross-multiply and solve for the variable. Enter just the number.

1) $\dfrac{x}{9} = \dfrac{4}{3} \;\Rightarrow\; x = \,?$ 2) $\dfrac{6}{n} = \dfrac{3}{8} \;\Rightarrow\; n = \,?$ 3) $\dfrac{7}{10} = \dfrac{y}{40} \;\Rightarrow\; y = \,?$

⚖️ Ratios and Rates

Part 4 of 5 — Real-World Applications

🔑 Big Payoff: Proportions solve a huge slice of everyday math — scaling recipes, reading maps, converting units, finding speeds, and computing similar-figure lengths. The skill is setting up the proportion correctly.

Setting Up a Word-Problem Proportion

The golden rule: keep the same units lined up — top with top, bottom with bottom.

Example: Recipe scaling

A recipe needs $2$ cups of flour for $3$ servings. How much flour for $12$ servings?

$\frac{2 \text{ cups}}{3 \text{ servings}} = \frac{x \text{ cups}}{12 \text{ servings}}$

Cross-multiply: $3x = 2 \cdot 12 = 24$, so $x = 8$ cups.

⚠️ Label your units. Putting "cups" on top in both ratios (and "servings" on the bottom in both) is what guarantees a correct setup. Mismatched units give a wrong answer.

Concept Check 🎯

Apply a Proportion 🧮

Set up a proportion (units lined up!) and solve. Enter just the number.

1) $4$ apples cost $3. How much do $12$ apples cost (in dollars)? 2) A car goes $60$ miles in $1$ hour. How far in $3.5$ hours? (miles) 3) A scale model is $1\,\text{cm} = 4\,\text{m}$. A wall is $20\,\text{m}$ tall. How tall is the model (cm)?

Percents Are Just Proportions

A percent is a ratio out of $100$. To solve a percent problem, use the proportion

$\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}.$

Example: What is $15\%$ of $80$?

$\frac{x}{80} = \frac{15}{100} \;\Rightarrow\; 100x = 80 \cdot 15 = 1200 \;\Rightarrow\; x = 12.$

So $15\%$ of $80$ is $12$.

💡 The same template answers all three percent questions: find the part, find the whole, or find the percent — just leave the unknown as the variable.

Percent Proportions 🔽

Use $\dfrac{\text{part}}{\text{whole}} = \dfrac{\text{percent}}{100}$ to choose each answer.

⚖️ Ratios and Rates

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) write and simplify ratios, (2) compute unit rates and find the better buy, (3) solve proportions by cross-multiplying, and (4) apply proportions to recipes, maps, and percents. Let's put it together.

Quick Reference

⚠️ Remember: line up like units (top with top), keep ratio order as written, and compare unit rates — not totals — when judging a deal.

Mixed Practice 🎯

Mixed Skills Drill 🧮

Use the right tool for each. Enter just the number.

1) Simplify $18:24$ — enter as a fraction like 3/4. 2) A bus travels $180$ miles in $4$ hours. Unit rate $= \,?$ miles per hour. 3) Solve $\dfrac{x}{12} = \dfrac{5}{4} \;\Rightarrow\; x = \,?$

Choose the Result 🔽

One quick choice from each topic in this lesson.

Exit Quiz ✅

Answer all three to finish the lesson.