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

What is a ratio?

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.

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Summary

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

Simplifying and making equivalent ratios

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.

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Summary

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

Proportions and solving for an unknown

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: \(\,a\cdot d = b\cdot c\).

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Summary

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

Using a ratio to split 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 \(b = n\cdot\frac{T}{m+n}\).

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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 \(a:b:c\)

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.

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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 \(k\).

Rates, unit rates, and proportional relationships

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.

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Summary

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

Why ratios and proportions matter

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

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Quick check: equivalent ratios

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.