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If \(x:y = 8:5\) and \(x = 24\), what is \(y\)?

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Explanation: Scale factor = \(24/8=3\); y = \(5×3=15\).

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Ratios & Proportions

Ratios and Proportions Practice Quiz with a Step-by-Step Interactive Lesson

Use the quiz at the top of the page to practice ratios and proportions (simplifying ratios, finding equivalent ratios, solving proportions, and answering real-world ratio word problems). If you want a refresher, click Start lesson to open a step-by-step guide.

How this ratios and proportions practice works

1. Take the quiz: answer the questions at the top of the page.
2. Open the lesson (optional): review the method with examples and quick checks.
3. Retry: return to the quiz and apply what you reviewed.

What you’ll learn in the ratios and proportions lesson

Meaning & vocabulary

What a ratio means (a comparison)
Common forms: \(a:b\), “\(a\) to \(b\)”, and \(\frac{a}{b}\)
Terms, part-to-part, and part-to-whole

Equivalent ratios

Simplify ratios using the greatest common factor
Make equivalent ratios by scaling up/down
Use ratio tables and “same multiplier” thinking

Proportions & missing values

What a proportion is: two equal ratios
Solve for an unknown using cross products or scaling
Check reasonableness (does the answer match the ratio?)

Real-world applications

Unit rates (per 1) and constant scaling
Scale factor, maps, and scale drawings
Recipes, speed, unit price, and measurement conversions

Back to the quiz

When you’re ready, return to the quiz at the top of the page and continue practicing.

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Ratios &
Proportions

Step-by-step guide

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Ratios & Proportions Lesson

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Lesson Overview

Lesson overview

Purpose: Understand ratios and proportions, build fluency with equivalent ratios, and learn reliable steps for solving missing-value and word problems.

Success criteria

Explain a ratio as a comparison using \(a:b\), “\(a\) to \(b\)”, or \(\frac{a}{b}\).
Identify part-to-part and part-to-whole ratios.
Simplify a ratio to lowest terms using the greatest common factor.
Create equivalent ratios by multiplying/dividing both terms by the same number.
Solve a proportion for a missing value using scaling or cross products.
Solve ratio problems with a total by using “total parts” and a scale factor.
Use unit rates and scale factors in real contexts (recipes, maps, speed, unit price).

Key vocabulary

Ratio: a comparison of two quantities by division.
Term: each number in a ratio (in \(a:b\), \(a\) and \(b\) are the terms).
Equivalent ratios: ratios that represent the same relationship (for example, \(2:3\) and \(4:6\)).
Proportion: an equation that states two ratios are equal.
Unit rate: a rate with a denominator of 1 (for example, 60 km per 1 hour).

Quick pre-check

Pre-check 1: Which ratio represents “4 to 7”?

\(4:7\) \(7:4\) \(4+7\) \(4\times 7\)

Hint: Order matters. “4 to 7” starts with 4.

Pre-check 2: If the ratio of cats to dogs is \(2:3\) and there are \(6\) cats, how many dogs are there?

Hint: To go from 2 cats to 6 cats, multiply by 3. Do the same to the dogs: \(3\times 3=9\).

Understanding Ratios

What is a ratio?

Learning goal: Interpret ratios correctly and choose the right order for a ratio (what compares to what).

Key idea

A ratio compares two quantities by division. You will see ratios written in three common forms: \(a:b\), “\(a\) to \(b\)”, and \(\frac{a}{b}\). The order matters: \(2:5\) is not the same as \(5:2\).

Part-to-part vs. part-to-whole

A ratio can compare two parts (part-to-part) or a part to the total (part-to-whole). Always read the problem carefully to know which ratio is being asked.

Worked example

Example: A bag has 8 red marbles and 12 blue marbles.

Red:blue \(= 8:12\). Simplify by dividing both terms by 4: \(8:12 = 2:3\).
Red:total \(= 8:(8+12)=8:20\). Simplify: \(8:20 = 2:5\).

Try it

Try it 1: A class has 10 girls and 15 boys. What is the girls:boys ratio in simplest form?

\(10:15\) \(15:10\) \(2:3\) \(3:2\)

Hint: Simplify \(10:15\) by dividing both terms by 5.

Try it 2: If the ratio of apples to oranges is \(1:2\) and there are \(4\) apples, how many oranges are there?

Hint: If 1 apple matches 2 oranges, then 4 apples match \(4\times 2=8\) oranges.

Summary

A ratio compares two quantities, and order matters.
Ratios can be part-to-part or part-to-whole depending on what’s asked.

Equivalent Ratios

Simplifying and making equivalent ratios

Learning goal: Simplify ratios to lowest terms and build equivalent ratios by scaling both terms.

Key idea

You simplify a ratio the same way you simplify a fraction: divide both terms by their greatest common factor (GCF). To make an equivalent ratio, multiply (or divide) both terms by the same nonzero number.

Worked example

Example: Simplify \(35:50\)

The GCF of 35 and 50 is 5.
Divide both terms by 5: \(35:50 = 7:10\).
So, the ratio in lowest terms is \(7:10\).

Try it

Try it 1: Simplify the ratio \(81:54\) to lowest terms.

\(3:2\) \(2:3\) \(9:6\) \(27:18\)

Hint: Divide both numbers by their GCF (here it’s 27).

Try it 2: If \(x:y = 4:5\) and \(x = 16\), what is \(y\)?

Hint: \(4\to 16\) is \(\times 4\). Do the same to 5: \(5\times 4=20\).

Summary

Simplify a ratio by dividing both terms by the GCF.
Equivalent ratios come from multiplying/dividing both terms by the same number.

Proportions

Proportions and solving for an unknown

Learning goal: Set up a proportion and solve missing-value problems accurately.

Key idea

A proportion is an equation that says two ratios are equal: \(\frac{a}{b} = \frac{c}{d}\) (with b≠ 0 and d≠ 0). One reliable method is cross multiplication: \(\,a\cdot d = b\cdot c\).

Worked example

Example: Solve \(\frac{3}{5} = \frac{x}{20}\)

Cross multiply: \(3\cdot 20 = 5\cdot x\).
\(60 = 5x\). Divide by 5: \(x=12\).

Try it

Try it 1: Solve the proportion \(\frac{6}{x} = \frac{3}{4}\). What is \(x\)?

Hint: Cross multiply: \(6\cdot 4 = 3\cdot x\).

Try it 2: Solve the proportion \(\frac{9}{12} = \frac{x}{16}\). What is \(x\)?

Hint: Simplify \(\frac{9}{12}\) first, then scale to a denominator of 16.

Summary

A proportion states that two ratios are equal.
Cross multiplication ( \(a\cdot d=b\cdot c\) ) helps you solve for the unknown.

Ratios with Totals

Using a ratio to split a total

Learning goal: Use the “total parts” method to find each amount when you know a ratio and a total.

Key idea

If \(a:b = m:n\) and the total is \(T\), then the total number of “parts” is \(m+n\). Each part is \(\frac{T}{m+n}\). Then: \(a = m\cdot\frac{T}{m+n}\) and \(b = n\cdot\frac{T}{m+n}\).

Worked example

Example: Cars:bikes \(= 2:5\), total \(=21\)

Total parts: \(2+5=7\).
Each part: \(21\div 7=3\).
Bikes: \(5\times 3=15\). Cars: \(2\times 3=6\).

Try it

Try it 1: If the ratio of cars to bikes is \(2:5\) and the total is \(21\), how many are bikes?

Hint: Add the ratio parts \(2+5\), then divide the total by that sum.

Worked solution

Total parts \(=2+5=7\).
Each part \(=21\div 7=3\).
Bikes \(=5\times 3=15\).

Try it 2: If \(a:b=1:4\) and \(a+b=10\), what is \(a\)?

Hint: Total parts \(=1+4=5\). Each part \(=10\div 5\).

Summary

When you know a ratio and a total, add the ratio parts first.
Divide the total by the number of parts, then multiply to find each amount.

Three-Term Ratios

Three-term ratios \(a:b:c\)

Learning goal: Use a scale factor to solve problems with three quantities in a ratio.

Key idea

A three-term ratio \(a:b:c = p:q:r\) means there is a scale factor \(k\) so that: \(a=pk\), \(b=qk\), and \(c=rk\). If you know one value (or a difference or a total), you can find \(k\) and then find the others.

Worked example

Example: If \(a:b:c=2:3:4\) and \(a=6\), find \(b\) and \(c\).

Since \(a=2k\) and \(a=6\), we have \(2k=6\) so \(k=3\).
Then \(b=3k=3\times 3=9\) and \(c=4k=4\times 3=12\).

Try it

Try it 1: If \(a:b:c=2:3:4\) and \(a=10\), what is \(c\)?

Hint: If \(a=2k\) and \(a=10\), then \(k=5\). So \(c=4k\).

Try it 2: If \(a:b:c=1:2:4\) and \(c-a=24\), what is \(a\)?

Hint: \(a=k\) and \(c=4k\). So \(c-a=3k\).

Summary

In \(a:b:c=p:q:r\), each value is the ratio term times the same scale factor \(k\).
Use the given information (one value, a total, or a difference) to find \(k\).

Unit Rates

Rates, unit rates, and proportional relationships

Learning goal: Find a unit rate and use proportional reasoning to scale up or down.

Key idea

A rate is a ratio that compares quantities with different units (for example, kilometers and hours). A unit rate tells the amount “per 1” unit. When two quantities are proportional, they change by the same scale factor.

Worked example

Example: A car travels 180 km in 3 hours. What is the speed in km per hour?

Unit rate \(=\frac{180}{3}=60\).
Answer: The speed is 60 km per hour.

Try it

Try it 1: A recipe uses 4 cups of flour for 16 muffins. How many cups of flour are needed for 20 muffins?

Hint: Simplify \(4:16\) to \(1:4\). Then 20 muffins need \(20\div 4=5\) cups.

Try it 2: In a proportional relationship, if one quantity doubles, what happens to the other quantity?

It doubles. It halves. It increases by 2. It stays the same.

Hint: “Proportional” means the ratio between the quantities stays constant.

Summary

A unit rate tells you the amount per 1 unit.
Proportional relationships scale by the same factor (double, triple, halve, etc.).

Applications

Why ratios and proportions matter

Learning goal: Connect ratios and proportions to real-life scaling and decision-making — and build intuition for checking answers.

Where you use ratios and proportions

Recipes: scale ingredients up or down while keeping the same taste.
Maps and scale drawings: convert a drawing distance to a real distance using a scale factor.
Unit price: compare “cost per 1 item” to find the best deal.
Science and health: concentrations (like mg per mL) and mixtures.
Probability: ratios describe chances (for example, favorable outcomes to total outcomes).

Worked example: map scale

Example: A map uses a scale of 1 cm to 5 km. Two towns are 7 cm apart on the map.

Each centimeter represents 5 km.
Real distance \(=7\times 5=35\) km.
Answer: The towns are 35 km apart.

Try it

Try it 1: A map uses a scale of 1 cm to 5 km. Two cities are 9 cm apart on the map. How many kilometers apart are they?

Hint: Multiply the map distance by 5 km per cm.

Quick check: equivalent ratios

Try it 2: Which pair of ratios are equivalent?

\(2:3\) and \(4:6\) \(3:4\) and \(6:10\) \(5:8\) and \(10:12\) \(1:2\) and \(2:5\)

Hint: Equivalent ratios simplify to the same lowest-terms ratio.

Final recap

A ratio is a comparison. Write it as \(a:b\), “\(a\) to \(b\)”, or \(\frac{a}{b}\).
Simplify ratios using the GCF, and build equivalent ratios by scaling both terms.
A proportion is an equation of two equal ratios; cross multiplication can solve missing values.
When a ratio and a total are given, use the “total parts” method to split the total.
Unit rates and scale factors help with real-world problems like recipes, maps, speed, and unit price.

Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the skill.