Learn Ratios and Proportions: Interactive Problems and Worked Examples
Use the question set below 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.
Answer the question set and review your mistakes at the end.
How this ratios and proportions practice works
1. Take the practice set: answer the questions below.
2. Open the lesson (optional): review the method with examples and quick checks.
3. Retry: return to the question set and apply what you reviewed.
What you will 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
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”?
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?
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 is 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.
Hint: Divide both numbers by their GCF (here it is 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: \(\,a\cdot d = b\cdot c\).
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?
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?
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.
