Write these ratio in its simplest form 200:180
The Correct Answer and Explanation is:
To simplify the ratio (200:180), we can follow these steps:
- Identify the numbers in the ratio: Here, we have (200) and (180).
- Find the greatest common divisor (GCD): The GCD of two numbers is the largest number that can divide both of them without leaving a remainder. We can find the GCD using the following methods:
- Listing Factors:
- Factors of (200): (1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200)
- Factors of (180): (1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180)
- The common factors are (1, 2, 4, 5, 10, 20). The largest of these is (20).
- Prime Factorization:
- (200 = 2^3 \times 5^2)
- (180 = 2^2 \times 3^2 \times 5)
- The GCD is found by taking the lowest power of each prime factor common to both factorizations:
- For (2), the minimum power is (2^2).
- For (5), the minimum power is (5^1).
- The prime (3) is not present in (200).
- Thus, GCD = (2^2 \times 5^1 = 4 \times 5 = 20).
- Divide both numbers by the GCD: To simplify the ratio, we divide both terms by their GCD ((20)):
[
\frac{200}{20} = 10 \quad \text{and} \quad \frac{180}{20} = 9
] - Write the simplified ratio: Therefore, the ratio (200:180) simplified is:
[
10:9
]
Conclusion
The simplest form of the ratio (200:180) is (10:9). This process demonstrates the importance of finding the GCD for simplifying ratios effectively. Simplifying ratios helps in comparing quantities more easily and is commonly used in various fields, including mathematics, cooking, and finance. The ability to simplify ratios is also essential in real-world applications, such as determining proportions in recipes or assessing the relationship between quantities in data analysis. By practicing this method, you can become proficient at simplifying ratios and enhancing your mathematical skills.