Ratios and proportions are fundamental mathematical concepts frequently used in daily life, from cooking and shopping to interpreting data. Here's a comprehensive guide:
1. What is a Ratio?
- Definition: A ratio compares the size of one part to another part.
- Common Formats:
- Colon format:
7:5
- Fraction format:
7/5
- Written in words: "7 to 5"
- Example:
- If there are 7 white boxes and 3 purple boxes:
- Ratio: Purple to white =
3:7.
- Fraction (purple to total): ( \frac{3}{10} ).
2. Reducing and Multiplying Ratios
- Reducing Ratios: Divide both parts by their greatest common factor.
- Example:
- Ratio
36:3 simplifies to 12:1 by dividing by 3.
- Multiplying Ratios: Scale up or down by multiplying both parts by the same number.
- Example:
- A ratio of
4:1 becomes 12:3 when multiplied by 3.
3. Scaling Ratios
- Scaling Up or Down: Multiply or divide both sides by the same factor.
- Applications: Maps, models, recipes.
- Example:
- Recipe for 12 cupcakes:
- Ingredients: 120g butter, 3 eggs.
- For 20 cupcakes: Scale ratio ( 12:20 ) = ( 3:5 ).
- Divide by 3 and multiply by 5:
- Butter: ( 120g \div 3 * 5 = 200g ).
- Eggs: ( 3 \div 3 * 5 = 5 ).
4. Proportion
- Definition: Compares a part to the whole, expressed as a fraction.
-
Example:
- If 3 out of 10 boxes are purple, the proportion is ( \frac{3}{10} ).
-
Use Case:
- Example:
- Pam's fish tank:
- Fish: 6 Tetra, 15 Minnow, 5 Platy, 4 Guppy (30 total).
- Proportion of Minnow: ( \frac{15}{30} = \frac{1}{2} ).
- Proportion of Platy: ( \frac{5}{30} = \frac{1}{6} ).
5. Key Tips and Applications
- Check Units Consistency:
- Ratios must have equivalent units (e.g., inches to inches, not inches to cm).
- Exceptions: Explicit unit conversions like "1 inch to 1 mile."
- Practical Uses:
- Converting currencies.
- Adjusting recipes for larger or smaller groups.
- Interpreting scale models or maps.
6. Ratios vs. Proportions
- Ratios: Compare parts to each other.
- Example:
3:7 (purple to white).
- Proportions: Compare part to whole.
- Example: ( \frac{3}{10} ) (purple to total).
Summing it up
Ratio and proportion help in comparing quantities.
These concepts are similar to fractions and are practical in everyday activities, like scaling recipes, comparing prices, or measuring.