Simplifying Ratios
What is a Ratio and Why Simplify It?
A ratio is a way to compare two or more quantities, showing the relative size of one quantity to another. It tells us how much of one thing there is compared to another. For example, if a smoothie recipe calls for 2 apples and 1 banana, the ratio of apples to bananas is 2:1. We say this as "2 to 1".
But why do we need to simplify them? Imagine a recipe that requires 12 cups of flour and 8 cups of milk. The ratio is 12:8. While this is accurate, it's not the simplest way to understand the relationship. Simplifying a ratio makes it easier to grasp, communicate, and use for calculations. A simplified ratio expresses the same relationship using the smallest possible whole numbers, just like simplifying a fraction. The ratio 12:8 simplifies to 3:2, which is much clearer.
The Golden Rule: Dividing by the Highest Common Factor (HCF)
The core process of simplifying a ratio relies on one key mathematical tool: the Highest Common Factor (HCF)[1], also known as the Greatest Common Divisor (GCD). The HCF of two or more numbers is the largest number that divides evenly into all of them without leaving a remainder.
The golden rule for simplifying a ratio is: Divide all parts of the ratio by their Highest Common Factor (HCF).
Let's break down the process into simple, repeatable steps using the ratio 18:24:
- Identify the factors of each number in the ratio.
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Find the common factors shared by both numbers.
- Common factors: 1, 2, 3, 6
- Select the Highest Common Factor (HCF).
- HCF: 6
- Divide both parts of the ratio by the HCF.
- $18 ÷ 6 = 3$
- $24 ÷ 6 = 4$
- Write the simplified ratio.
- The simplified form of 18:24 is 3:4.
Simplifying Ratios with More Than Two Numbers
Ratios can compare more than two quantities! The process for simplifying them is exactly the same; you just need to find the HCF that is common to all parts of the ratio.
Example: Simplify the ratio 12:18:24.
- Find the factors of each number.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Identify the common factors of all three numbers.
- Common factors: 1, 2, 3, 6
- The Highest Common Factor is 6.
- Divide all three parts by 6.
- $12 ÷ 6 = 2$
- $18 ÷ 6 = 3$
- $24 ÷ 6 = 4$
- The simplified ratio is 2:3:4.
Simplifying Ratios with Decimal or Fraction Parts
Sometimes, ratios aren't given as nice, neat whole numbers. They might include decimals or fractions. The goal remains the same: eliminate the decimals or fractions to get a ratio of whole numbers.
For Decimals: Multiply both parts by 10, 100, or whatever power of 10 is needed to make them whole numbers. Then simplify as usual.
Example: Simplify 1.5 : 2.5.
Multiply both parts by 10: 15 : 25.
Find the HCF of 15 and 25, which is 5.
Divide both parts by 5: 3 : 5.
For Fractions: Multiply both parts by the Lowest Common Multiple (LCM)[2] of the denominators to clear the fractions. Then simplify.
Example: Simplify $\frac{1}{2} : \frac{2}{3}$.
The denominators are 2 and 3. The LCM is 6.
Multiply both parts by 6: $(\frac{1}{2} \times 6) : (\frac{2}{3} \times 6) = 3 : 4$.
The ratio is already simplified: 3:4.
Ratio Simplification in Action: Real-World Scenarios
Simplifying ratios is not just a math class exercise; it's a skill used daily in various fields. Here are some concrete examples:
| Scenario | Original Ratio | Simplified Ratio | Why It's Useful |
|---|---|---|---|
| Cooking & Baking | A pancake recipe uses 6 cups of flour and 4 cups of milk. | 3:2 (flour to milk) | Easily scale the recipe up or down. For every 3 scoops of flour, you need 2 scoops of milk. |
| Maps & Scale Models | A map scale says 100,000 cm on the ground is 1 cm on the map. | 1:100,000 | This is the standard, simplified way to express map scale, making it universal and easy to understand. |
| Chemistry | A molecule of water, H_2_O, has 2 hydrogen atoms and 1 oxygen atom. | 2:1 (H to O) | This is the empirical formula, the simplest whole-number ratio of atoms in a compound. |
| Mixing Paint | To make a specific green, you mix 200 mL of blue and 300 mL of yellow. | 2:3 (blue to yellow) | You can mix any amount correctly as long as you maintain the 2:3 ratio. |
Common Mistakes and Important Questions
Q: Is the simplified ratio 2:3 the same as the fraction 2/3?
No, they are related but not the same. The ratio 2:3 compares two parts to make a whole. It means "for every 2 of one thing, there are 3 of another." The fraction $\frac{2}{3}$, however, represents a part of a single whole. If you have a total of 5 items in a 2:3 ratio, then $\frac{2}{5}$ are the first item and $\frac{3}{5}$ are the second.
Q: What if the HCF is 1? Does that mean I did something wrong?
Not at all! If the Highest Common Factor of the numbers in a ratio is 1, it means the ratio is already in its simplest form. For example, the ratio 3:4 is already simplified because the only common factor of 3 and 4 is 1. This is the final, correct answer.
Q: I have a ratio with units, like 300g : 2kg. How do I simplify that?
This is a very common and important situation. You must convert the quantities to the same unit before simplifying. In this case, convert 2 kg to grams (2000 g). Now the ratio is 300g : 2000g. You can now drop the units and simplify 300 : 2000. The HCF is 100, so the simplified ratio is 3 : 20.
Simplifying ratios is a fundamental mathematical skill that brings clarity and efficiency to how we express relationships between quantities. By mastering the simple process of finding the Highest Common Factor and dividing all parts of the ratio, we transform complex comparisons into their most basic and understandable form. This skill is not confined to the classroom; it is essential for tasks ranging from adapting a recipe and reading a map to understanding chemical formulas. Remember, a simplified ratio maintains the original relationship while making it universally accessible and easy to work with, forming a cornerstone for more advanced topics in mathematics and science.
Footnote
[1] Highest Common Factor (HCF): The largest whole number that divides exactly into two or more given numbers without leaving a remainder. For example, the HCF of 16 and 24 is 8.
[2] Lowest Common Multiple (LCM): The smallest whole number that is a multiple of two or more given numbers. For example, the LCM of 4 and 6 is 12.