Ratios and Proportions

⚖️ Ratios and Proportions

Part 1 of 5 — What Is a Ratio?

Topics in This Part

🔑 Key Concept: A ratio compares two quantities by division. "$3$ cats to $2$ dogs" tells you how many of one thing there are for every amount of another — and that comparison stays true no matter how many animals you have.

Three Ways to Write a Ratio

A ratio comparing $a$ to $b$ can be written three equivalent ways:

All three mean the same comparison. The fraction form is the most useful for calculating, because it lets you use everything you already know about fractions.

Order Matters

A ratio is an ordered comparison. If a fruit bowl has $5$ apples and $3$ oranges:

• apples to oranges $= 5 : 3$
• oranges to apples $= 3 : 5$

These are not the same — the first number always matches the first quantity named.

⚠️ Watch the order. "Boys to girls $= 4 : 7$" is different from "girls to boys $= 7 : 4$". Always match the numbers to the words in the same order.

Concept Check 🎯

Simplifying a Ratio

Because a ratio behaves like a fraction, you simplify it the same way: divide both parts by their greatest common factor (GCF).

Example: simplify $12 : 18$

The GCF of $12$ and $18$ is $6$:

$12 : 18 = \frac{12 \div 6}{18 \div 6} = \frac{2}{3} = 2 : 3$

So for every $2$ of the first thing there are $3$ of the second.

Example: simplify $20 : 4$

The GCF of $20$ and $4$ is $4$:

$20 : 4 = \frac{20 \div 4}{4 \div 4} = \frac{5}{1} = 5 : 1$

A ratio like $5 : 1$ means "five for every one."

💡 A ratio is in simplest form when the two numbers share no common factor other than $1$ — exactly like a fraction in lowest terms.

Simplify the Ratio 🧮

Write each ratio in simplest form. Enter your answer using a colon, like 2:3.

1) $9 : 15$ 2) $8 : 24$ 3) $30 : 12$

Putting Order and Simplifying Together

Real problems combine both skills: read the ratio in the right order, then simplify. You can even build a third ratio — comparing one part to the total (the sum of both parts).

For a group of $14$ boys and $16$ girls, the total is $14 + 16 = 30$, so "boys to total" $= 14 : 30 = 7 : 15$.

🔑 A part-to-total ratio compares one part to the whole group, while a part-to-part ratio compares the two parts to each other. Read the question carefully to know which one it wants.

Match the Words to the Ratio 🔽

A classroom has $14$ boys and $16$ girls. Choose the correct ratio for each description (simplest form).

Wrapping Up Part 1

You can now:

• write a ratio three ways: to, colon, and fraction,
• keep the order straight (first thing named goes first), and
• simplify by dividing both parts by the GCF.

🔑 A ratio is just a comparison-by-division. In Part 2 we add units to ratios and turn them into rates — the everyday tool behind speeds, prices, and recipes.

⚖️ Ratios and Proportions

Part 2 of 5 — Rates and Unit Rates

🔑 The Idea: A rate is a ratio that compares two quantities with different units — like miles and hours. A unit rate rewrites that comparison so the second quantity is exactly $1$ (miles per one hour).

Rates vs. Unit Rates

The little word "per" signals a unit rate: miles per hour, dollars per pound, words per minute.

Finding a Unit Rate

To find a unit rate, divide the first quantity by the second:

$\text{unit rate} = \frac{\text{first quantity}}{\text{second quantity}}$

Example: a car travels $120$ miles in $2$ hours

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

The car covers $60$ miles for each hour.

Concept Check 🎯

Why Unit Rates Are Useful: Comparing Prices

Unit rates let you compare options fairly. The smaller cost-per-item is the better buy.

Example: which is the better deal?

• Brand A: $6 for $4$ pounds
• Brand B: $10 for $8$ pounds

Find the price per pound for each:

Brand A: $\dfrac{6}{4} = 1.50$, i.e. $1.50 per lb

Brand B: $\dfrac{10}{8} = 1.25$, i.e. $1.25 per lb

Brand B costs less per pound ($1.25 < $1.50), so Brand B is the better buy.

💡 To compare two deals, always reduce each to the same unit rate (here, dollars per pound). Whichever number is lower wins.

Find the Unit Rate 🧮

Divide the first quantity by the second. Enter a number (decimals are fine).

1) $45 for $9$ shirts $\Rightarrow$ dollars per shirt $= \,?$ 2) $360$ words typed in $6$ minutes $\Rightarrow$ words per minute $= \,?$ 3) $7.50 for $5$ pounds of apples $\Rightarrow$ dollars per pound $= \,?$

Better Buy? 🔽

Two stores sell the same juice. Find each unit price, then pick the better deal.

⚖️ Ratios and Proportions

Part 3 of 5 — Proportions & Equivalent Ratios

🔑 What's a proportion? A proportion is an equation that says two ratios are equal, like $\dfrac{2}{3} = \dfrac{6}{9}$. Recognizing and building equal ratios is the heart of this whole topic.

Equivalent Ratios

Two ratios are equivalent if you can get one from the other by multiplying (or dividing) both parts by the same number.

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

So $2 : 3$, $4 : 6$, $8 : 12$, and $20 : 30$ are all the same ratio in different clothes.

⚠️ You must multiply or divide both numbers by the same value. Multiplying only the top changes the ratio.

The Cross-Products Test

How can you check whether two ratios are equal without simplifying both? Use cross products.

For $\dfrac{a}{b} = \dfrac{c}{d}$, the cross products are $a \cdot d$ and $b \cdot c$.

🔑 The rule: $\dfrac{a}{b} = \dfrac{c}{d}$ is true exactly when $a \cdot d = b \cdot c$ (the cross products are equal).

Example: is $\dfrac{3}{4} = \dfrac{9}{12}$?

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

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

Example: is $\dfrac{2}{5} = \dfrac{6}{14}$?

$2 \cdot 14 = 28 \qquad 5 \cdot 6 = 30$

$28 \ne 30$, so no — these ratios are not equal.

Concept Check 🎯

Build Equivalent Ratios 🧮

Fill in the missing number so the ratios are equivalent.

1) $\dfrac{2}{7} = \dfrac{?}{21}$ 2) $\dfrac{5}{8} = \dfrac{15}{?}$ 3) $\dfrac{4}{9} = \dfrac{?}{36}$

Back to Cross Products

Building equivalent ratios is great when you can see the scale factor. But sometimes the numbers don't share an obvious multiplier — that's when cross products are the fastest check.

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

If the two cross products are equal, the proportion is true; if not, it's false. No simplifying required.

💡 Cross products work even with messy numbers like $\dfrac{17}{23}$ — that's exactly why this test is so powerful.

True or Not? 🔽

Use cross products to decide whether each proportion is true.

⚖️ Ratios and Proportions

Part 4 of 5 — Solving Proportions for the Unknown

🔑 The Payoff: When a proportion has a missing value, cross products turn it into a simple one-step equation you can solve. This is the single most useful skill in the whole topic.

Solving with Cross Multiplication

To solve a proportion like $\dfrac{x}{6} = \dfrac{4}{3}$:

1. Cross multiply to clear the fractions: $x \cdot 3 = 6 \cdot 4$
2. Simplify: $3x = 24$
3. Divide both sides by the number next to $x$: $x = \dfrac{24}{3} = 8$

$\frac{x}{6} = \frac{4}{3} \;\Rightarrow\; 3x = 24 \;\Rightarrow\; x = 8$

✅ Check: $\dfrac{8}{6} = \dfrac{4}{3}$? Cross products: $8 \cdot 3 = 24$ and $6 \cdot 4 = 24$. ✓

Example: the unknown on the bottom

$\frac{5}{n} = \frac{15}{12}$

Cross multiply: $5 \cdot 12 = 15 \cdot n$, so $60 = 15n$, giving $n = \dfrac{60}{15} = 4$.

Concept Check 🎯

Setting Up a Proportion from Words

The trick is to keep the same kind of quantity in matching positions — top with top, bottom with bottom.

Example: scaling a recipe

A recipe uses $2$ cups of flour for every $3$ cups of milk. How much flour for $12$ cups of milk?

Put flour on top and milk on the bottom in both ratios:

$\frac{\text{flour}}{\text{milk}}: \quad \frac{2}{3} = \frac{x}{12}$

Cross multiply: $3x = 24$, so $x = 8$. You need $8$ cups of flour.

💡 The golden rule of setup: whatever unit is on top in the first ratio must be on top in the second ratio too. Mixing them up is the #1 word-problem mistake.

Solve the Proportion 🧮

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

1) $\dfrac{x}{4} = \dfrac{9}{6} \Rightarrow x = \,?$ 2) $\dfrac{7}{n} = \dfrac{21}{15} \Rightarrow n = \,?$ 3) $\dfrac{a}{20} = \dfrac{3}{5} \Rightarrow a = \,?$

Word Problems 🧮

Set up a proportion, then solve. Enter just the number.

1) If $3$ notebooks cost $12, how much do $7$ notebooks cost? (dollars) 2) A map uses $1$ inch for every $25$ miles. How many miles is $4$ inches? (miles)

⚖️ Ratios and Proportions

Part 5 of 5 — Applications & Mastery Check

You can now write ratios, simplify them, find unit rates, recognize proportions, and solve for unknowns. Let's apply all of it to real situations — and then prove your mastery.

Scale Drawings & Similar Shapes

A scale is a ratio that compares a drawing to real life, like $1 \text{ cm} : 50 \text{ km}$ on a map.

Example: a model car

A model car is built at a scale of $1 : 18$ — every $1$ inch on the model is $18$ inches on the real car. If the model is $9$ inches long, how long is the real car?

$\frac{1}{18} = \frac{9}{x} \;\Rightarrow\; x = 18 \cdot 9 = 162 \text{ inches}$

The real car is $162$ inches long.

💡 Scale problems are just proportions in disguise: model : real $=$ model : real, with units kept in matching positions.

Scale & Application Practice 🧮

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

1) A blueprint uses $1$ inch for every $4$ feet. A wall is $20$ feet long. How many inches is it on the blueprint? (inches) 2) $5$ workers paint a fence in the same way, finishing $3$ fences in a day. At that rate, how many fences would $5$ workers finish in $4$ days? (fences)

Pick the Right Setup 🔽

A train travels $240$ miles in $4$ hours. Choose the correct value for each.

Quick Reference

⚠️ When setting up a word problem, always line up matching units: top with top, bottom with bottom.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.