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. If a fruit bowl has $3$ apples and $5$ oranges, the ratio of apples to oranges is $3$ to $5$. Everything in this lesson grows from that one idea.

Three Ways to Write a Ratio

The same ratio can be written three different ways. For $3$ apples and $5$ oranges:

All three mean exactly the same comparison.

⚠️ Order matters! "Apples to oranges" is $3:5$, but "oranges to apples" is $5:3$. Always write the quantities in the order the problem names them.

💡 Part vs. whole. Sometimes a ratio compares a part to the total. With $3$ apples and $5$ oranges there are $8$ fruits total, so the ratio of apples to all fruit is $3:8$.

Concept Check 🎯

Equivalent Ratios

Two ratios are equivalent when one can be made from the other by multiplying (or dividing) both numbers by the same value.

$\frac{2}{3} = \frac{2 \cdot 2}{3 \cdot 2} = \frac{4}{6} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}$

So $2:3$, $4:6$, and $10:15$ are all the same ratio.

🔑 Golden Rule: Whatever you do to the top, do to the bottom. Multiplying $2:3$ by $4$ gives $8:12$ — still the same ratio because both sides grew by the same factor.

Building an Equivalent Ratio

Build Equivalent Ratios 🧮

Fill in the missing number so the two ratios are equivalent.

1) $\dfrac{2}{5} = \dfrac{?}{20}$ 2) $\dfrac{3}{4} = \dfrac{12}{?}$ 3) $\dfrac{6}{9} = \dfrac{2}{?}$

Simplifying to Lowest Terms

A ratio is in simplest form (lowest terms) when the two numbers share no common factor except $1$. To simplify, divide both numbers by their greatest common factor (GCF).

Example: Simplify $18:24$

The GCF of $18$ and $24$ is $6$:

$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

So $18:24$ simplifies to $3:4$.

Example: Simplify $20:50$

The GCF of $20$ and $50$ is $10$:

$\frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}$

💡 Why simplify? $3:4$ is much easier to picture and compare than $18:24$, even though they describe the same relationship.

Simplify Each Ratio 🔽

Choose the lowest-terms form of each ratio.

⚖️ Ratios and Rates

Part 2 of 5 — Rates and Unit Rates

🔑 The Idea: A rate is a special ratio that compares two quantities with different units — like miles and hours, or dollars and pounds. A unit rate shrinks that comparison down to "per $1$."

What Makes a Rate Special

A ratio like $3:5$ usually compares the same kind of thing (apples to oranges, both fruit). A rate compares different units:

A unit rate has a denominator of $1$ — "per one unit." To find it, divide the top by the bottom.

Example: $120$ miles in $2$ hours

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

💡 The word "per" signals a unit rate. Miles per hour, cost per pound, beats per minute — each means "for every $1$."

Concept Check 🎯

Find the Unit Rate 🧮

Divide the top quantity by the bottom to get the "per $1$" rate.

1) $45 for $5$ tickets $=$ $? per ticket 2) $360$ km in $4$ hours $=\,?$ km/h 3) $84$ heartbeats in $60$ seconds $=\,?$ beats per second (decimal is fine)

Unit Price and the "Best Buy"

A unit price is a unit rate for cost: dollars per single item or per unit of weight. It is the secret weapon for comparison shopping — the lower the unit price, the better the deal.

Example: Which is the better buy?

• Option A: $12$ granola bars for $4.80
• Option B: $8$ granola bars for $3.60

Find each unit price by dividing cost ÷ quantity:

$\text{A: } \frac{4.80}{12} = 0.40 \text{ per bar} \qquad \text{B: } \frac{3.60}{8} = 0.45 \text{ per bar}$

So A is $0.40 per bar and B is $0.45 per bar. Option A costs less per bar ($0.40 < $0.45), so A is the better buy.

⚠️ Bigger package $\ne$ always cheaper. Always reduce to the per-unit price before deciding — that's the only fair comparison.

Best Buy 🎯

⚖️ Ratios and Rates

Part 3 of 5 — Proportions

🔑 The Idea: A proportion is an equation that says two ratios are equal, like $\dfrac{2}{3} = \dfrac{8}{12}$. When one number is missing, we can solve for it — and that unlocks almost every real-world ratio problem.

Cross-Multiplication

In any true proportion, the cross products are equal:

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

You multiply diagonally — top-left times bottom-right equals top-right times bottom-left.

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

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

The cross products match ($36 = 36$), so yes, it's a true proportion. ✓

When they don't match

Is $\dfrac{2}{5} = \dfrac{3}{7}$? Cross products: $2 \cdot 7 = 14$ and $5 \cdot 3 = 15$. Since $14 \ne 15$, it is not a proportion.

Concept Check 🎯

Solving a Proportion for the Unknown

To find a missing value, cross-multiply, then divide.

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

1. Cross-multiply: $\;4 \cdot x = 12 \cdot 3$
2. Simplify: $\;4x = 36$
3. Divide by $4$: $\;x = 9$

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

Example: Solve $\dfrac{6}{x} = \dfrac{9}{15}$

$9 \cdot x = 6 \cdot 15 \;\Rightarrow\; 9x = 90 \;\Rightarrow\; x = 10$

💡 The unknown can be anywhere — top or bottom, left or right. Cross-multiplication handles all four positions the same way.

Solve the Proportion 🧮

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

1) $\dfrac{x}{10} = \dfrac{6}{5}$ 2) $\dfrac{4}{x} = \dfrac{12}{21}$ 3) $\dfrac{7}{8} = \dfrac{x}{24}$

Setting Up a Proportion from Words

The hardest part of a word problem is setting it up. The trick: keep the same units lined up — top with top, bottom with bottom.

Example: $3$ pencils cost $1.20. How much do $7$ pencils cost?

Set up "pencils over dollars" on both sides:

$\frac{3 \text{ pencils}}{1.20 \text{ dollars}} = \frac{7 \text{ pencils}}{x \text{ dollars}}$

Cross-multiply: $\;3x = 1.20 \cdot 7 = 8.40$, so $x = 8.40 \div 3 = 2.80$, giving $2.80.

⚠️ Line up the units! If pencils are on top on the left, pencils must be on top on the right. Flipping one side gives the wrong answer.

Set It Up and Solve 🔽

If $4$ notebooks cost $10, how much do $10$ notebooks cost?

⚖️ Ratios and Rates

Part 4 of 5 — Scaling, Maps, Recipes & Speed

🔑 Big Payoff: Ratios and proportions power real life — scaling recipes up or down, reading map distances, mixing colors, and connecting distance, rate, and time.

Scaling Recipes Up and Down

To scale a recipe, multiply every ingredient by the same scale factor. Doubling means $\times 2$; halving means $\times \frac{1}{2}$.

Example: A recipe for $4$ servings uses $6$ cups of flour. How much for $10$ servings?

Set up a proportion (servings over cups):

$\frac{4 \text{ servings}}{6 \text{ cups}} = \frac{10 \text{ servings}}{x \text{ cups}}$

$4x = 6 \cdot 10 = 60 \;\Rightarrow\; x = 15 \text{ cups}$

💡 Shortcut with unit rate: $6 \div 4 = 1.5$ cups per serving, so $10$ servings need $10 \times 1.5 = 15$ cups — same answer.

Concept Check 🎯

Map Scales

A map scale is a ratio: map distance to real distance. A scale of "$1$ inch $= 50$ miles" means every inch on paper stands for $50$ real miles.

Example: The scale is $1$ inch $= 50$ miles. Two cities are $3.5$ inches apart on the map. How far apart are they really?

$\frac{1 \text{ inch}}{50 \text{ miles}} = \frac{3.5 \text{ inches}}{x \text{ miles}}$

$1 \cdot x = 50 \cdot 3.5 \;\Rightarrow\; x = 175 \text{ miles}$

💡 With a "$1$ to something" scale you can skip cross-multiplication: just multiply the map distance by the scale. $3.5 \times 50 = 175$ miles.

Map & Scale Practice 🧮

1) Scale $1$ in $= 25$ mi. A road is $4$ inches long on the map. Real length $=\,?$ miles. 2) Scale $1$ cm $= 8$ km. Two towns are $6$ cm apart. Real distance $=\,?$ km. 3) A model car is built at scale $1:18$. The model is $9$ inches long. The real car is $\,?$ inches long.

Distance, Rate, and Time

Speed is a rate, so the three quantities are tied together:

$\text{distance} = \text{rate} \times \text{time} \qquad \text{rate} = \frac{\text{distance}}{\text{time}} \qquad \text{time} = \frac{\text{distance}}{\text{rate}}$

Example: A train goes $240$ miles in $4$ hours. At that rate, how far in $7$ hours?

First the unit rate: $240 \div 4 = 60$ mph. Then $60 \times 7 = 420$ miles.

Example: How long to drive $300$ miles at $50$ mph?

$\text{time} = \frac{300}{50} = 6 \text{ hours}$

⚠️ Match the units. If the rate is miles per hour, time must be in hours — not minutes.

Speed Check 🎯

⚖️ Ratios and Rates

Part 5 of 5 — Mixed Practice & Mastery Check

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

Quick Reference

⚠️ Two reminders that catch most mistakes: keep the order of a ratio consistent, and line up matching units (top with top) when you build a proportion.

Mixed Practice 🧮

1) Simplify the ratio $24:36$ to lowest terms. Enter as a fraction like a/b. 2) $7.50 for $6$ apples is how many dollars per apple? (decimal) 3) Solve $\dfrac{x}{18} = \dfrac{5}{6}$ for $x$.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.