To work out a ratio in its simplest form, you need to divide all parts of the ratio by their highest common factor (HCF) until no further division is possible. This process makes the ratio easier to understand and compare.
Understanding Ratios and Simplification
A ratio expresses the proportional relationship between two or more quantities. Simplifying a ratio means finding an equivalent ratio where the numbers are as small as possible, while still maintaining the same proportion. It's similar to simplifying fractions.
The Step-by-Step Process to Simplify Ratios
Simplifying a ratio boils down to one key step: identifying and dividing by the Highest Common Factor (HCF).
1. Identify All Parts of the Ratio
First, look at all the numbers in your ratio. Ratios can have two parts (e.g., 4:2) or more (e.g., 6:9:12).
2. Find the Highest Common Factor (HCF)
The HCF is the largest number that divides into all parts of the ratio without leaving a remainder.
- Method 1: Listing Factors
- List all factors for each number in the ratio.
- Identify the largest factor that appears in all lists.
- Method 2: Prime Factorization
- Find the prime factorization for each number.
- Multiply the common prime factors (raised to the lowest power they appear in any factorization).
3. Divide Each Part by the HCF
Once you've found the HCF, divide every number in the ratio by that HCF.
4. Check for Further Simplification
After dividing, ensure that the new numbers have no common factors other than 1. If they do, you might have missed the true HCF or need to repeat the process. The ratio is in its simplest form when the HCF of its parts is 1.
Practical Examples of Ratio Simplification
Let's walk through some examples to illustrate the process.
Example 1: Simplifying a Two-Part Ratio (4:2)
Consider the ratio 4:2.
- Parts: 4 and 2.
- Find HCF:
- Factors of 4: 1, 2, 4
- Factors of 2: 1, 2
- The highest common factor is 2.
- Divide by HCF:
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1
- Simplified Ratio: 2:1
This means for every 2 units of the first quantity, there is 1 unit of the second quantity.
Example 2: Simplifying a Two-Part Ratio (12:18)
Let's simplify 12:18.
- Parts: 12 and 18.
- Find HCF:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- The highest common factor is 6.
- Divide by HCF:
- 12 ÷ 6 = 2
- 18 ÷ 6 = 3
- Simplified Ratio: 2:3
Example 3: Simplifying a Three-Part Ratio (6:9:12)
When a ratio has more than two parts, the principle remains the same.
- Parts: 6, 9, and 12.
- Find HCF:
- Factors of 6: 1, 2, 3, 6
- Factors of 9: 1, 3, 9
- Factors of 12: 1, 2, 3, 4, 6, 12
- The highest common factor that appears in all three lists is 3.
- Divide by HCF:
- 6 ÷ 3 = 2
- 9 ÷ 3 = 3
- 12 ÷ 3 = 4
- Simplified Ratio: 2:3:4
Table of Simplified Ratios
| Original Ratio | Parts | HCF | Simplified Ratio |
|---|---|---|---|
| 4:2 | 4, 2 | 2 | 2:1 |
| 12:18 | 12, 18 | 6 | 2:3 |
| 6:9:12 | 6, 9, 12 | 3 | 2:3:4 |
| 10:25 | 10, 25 | 5 | 2:5 |
Why is Simplifying Ratios Important?
- Clarity: Simplified ratios are easier to comprehend and interpret. For example, 1:2 is much clearer than 50:100.
- Comparison: They facilitate easier comparison between different ratios.
- Standard Practice: It's a standard mathematical practice, ensuring consistency in calculations and problem-solving.
- Problem Solving: Simplified ratios are often necessary for further calculations, such as finding specific quantities when given a total.
Tips for Effective Ratio Simplification
- Whole Numbers First: If your ratio contains decimals or fractions (e.g., 0.5:2 or 1/2:1/4), convert them into whole numbers before finding the HCF. You can do this by multiplying all parts by a common multiple.
- Example: For 0.5:2, multiply both by 2 to get 1:4.
- Example: For 1/2:1/4, multiply both by 4 to get 2:1.
- Step-by-Step Reduction: If finding the HCF directly is difficult for larger numbers, you can repeatedly divide all parts by any common factor you find until no further division is possible.
- Example: For 24:36, you could divide by 2 (12:18), then by 2 again (6:9), then by 3 (2:3).
By consistently applying the method of dividing by the highest common factor, you can effectively simplify any ratio into its most concise form. For more detailed explanations on factors and multiples, you can explore resources like Khan Academy or BBC Bitesize.
