Ratios & Proportions
By the end of this lesson, you’ll be able to:
- Interpret ratios in fraction, colon, and “to” forms.
- Solve proportions using equivalent ratios and cross-multiplication.
- Apply ratios to real-world situations like recipes, scale models, and mixtures.
Key Ideas
A ratio compares two quantities.
Examples: \(3:5\), \(\frac{3}{5}\), “3 to 5”.Equivalent ratios show the same relationship:
\[ \frac{3}{5} = \frac{6}{10} = \frac{9}{15} \]
A proportion sets two ratios equal:
\[ \frac{a}{b} = \frac{c}{d} \]
To solve a proportion, use cross-multiplication:
\[ ad = bc \]
Ratios do not automatically give you the total.
A ratio of \(2:3\) means “2 of one thing for every 3 of another,” not necessarily 5 total unless stated.
Common Problem Types
Identifying and Simplifying Ratios
Example: Simplify 12:18.
Since \(\frac{12}{18} = \frac{2}{3}\), the simplified ratio is \(2:3\).
Solving Proportions
Example:
\[
\frac{x}{8} = \frac{3}{4}
\]
Cross-multiply: \(4x = 24\) → \(x = 6\).
Part–Whole vs Part–Part Ratios
Example: Boys:girls = \(3:5\), total students = 40
Total ratio parts = \(3 + 5 = 8\)
Each part = \(40/8 = 5\)
Boys = \(3 \times 5 = 15\)
Scale Models / Maps
Example: Scale \(1\text{ inch} : 20\text{ miles}\)
For 3 inches: \(1:20 = 3:x\) → \(x = 60\) miles.
Strategies
- Always determine if a ratio compares parts to parts or parts to whole.
- Use ratio tables when scaling up or down.
- Cross-multiply when solving proportions quickly.
- Be careful with units—ratios ignore them unless explicitly included.
Worked Examples
Example 1
Question: Cats:dogs = \(4:7\). There are 28 dogs. How many cats?
Solution: \(7 \to 28\) is ×4, so cats = \(4 \times 4 = 16\).
Example 2
Question:
\[
\frac{5}{x} = \frac{15}{18}
\]
Solution:
Cross-multiply: \(18 \cdot 5 = 90\) → \(x = 6\).
Example 3
Question: A recipe uses flour and sugar in a \(3:2\) ratio. If you have 12 cups of flour, how much sugar?
Solution: \(3:2 = 12:x\) → \(3x = 24\) → \(x = 8\) cups.
- Mixing up part-to-part and part-to-whole ratios.
- Cross-multiplying incorrectly or before setting up a valid proportion.
- Scaling only one part of a ratio instead of both parts.
- Ignoring units when comparing two ratios.
- Treating ratios as algebraic expressions without considering their context.
Practice Problems
- Simplify 45:60.
- If the ratio of red to blue marbles is \(2:5\) and there are 35 total marbles, how many are red?
- Solve:
\[ \frac{x}{12} = \frac{3}{4} \] - A blueprint uses a scale of \(1:50\). What actual length does 6 cm represent?
1.
GCF of 45 and 60 is 15.
\(45 \div 15 = 3\), \(60 \div 15 = 4\).
Answer: \(3:4\)
2.
Total ratio parts = \(2 + 5 = 7\).
Each part = \(35/7 = 5\).
Red = \(2 \times 5 = 10\).
Answer: 10 red marbles
3.
Cross-multiply: \(4x = 36\) → \(x = 9\).
Answer: \(x = 9\)
4.
Each 1 cm represents 50 cm in real life.
\(6 \times 50 = 300\) cm = 3 m.
Answer: 300 cm (3 m)
Summary
- Ratios compare quantities; proportions set two ratios equal.
- Simplify ratios by dividing both terms by their GCF.
- Cross-multiplication is the standard method for solving proportions.
- Part–whole vs part–part distinctions matter in word problems.
- Scale models rely on equivalent ratios to convert measurements.
- Ask: is this part–part or part–whole before setting up equations.
- Keep ratios in the same order across a proportion (e.g., red:blue = red:blue).
- Only cross-multiply after you’ve confirmed you truly have \(\frac{\text{ratio}}{\text{ratio}}\).
- When scaling, multiply both parts of the ratio by the same number.