Write the following ratio in its simplest form: 64:18.
The Correct Answer and Explanation is :
To simplify the ratio 64:18, we need to find the greatest common divisor (GCD) of both numbers and then divide both by this GCD.
Step 1: Find the GCD
- Prime Factorization:
- 64 can be broken down into prime factors:
[
64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6
] - 18 can be factored as:
[
18 = 2 \times 3 \times 3 = 2^1 \times 3^2
]
- Identifying Common Factors:
- The only prime factor common to both numbers is 2. The lowest power of this common factor is (2^1).
- GCD Calculation:
- Thus, the GCD of 64 and 18 is 2.
Step 2: Simplify the Ratio
Now, we divide both terms of the ratio by the GCD:
[
\frac{64}{2} = 32
]
[
\frac{18}{2} = 9
]
Result
So, the simplified form of the ratio 64:18 is:
[
32:9
]
Explanation of Ratios
A ratio compares two quantities, showing the relative size of one quantity to another. Ratios can be expressed in various forms: as fractions, with a colon, or with the word “to.” Simplifying a ratio makes it easier to understand and compare, especially when working with larger numbers. The process involves identifying the GCD to find a common scale that both quantities can be divided by without losing the relationship between them.
Understanding ratios is particularly important in fields like mathematics, finance, and everyday situations where comparisons are made. For example, in cooking, adjusting recipes often requires understanding ratios to maintain the correct proportions of ingredients. In business, ratios help analyze financial performance and efficiency. Thus, simplifying ratios not only aids in clearer communication but also facilitates calculations and decision-making across various contexts.