Write the fraction 24/40 in its simplest form.
The Correct Answer and Explanation is :
To simplify the fraction ( \frac{24}{40} ), we need to find the greatest common divisor (GCD) of the numerator (24) and the denominator (40). The GCD is the largest number that can divide both numbers without leaving a remainder.
Step 1: Find the GCD
- List the Factors:
- The factors of 24 are: ( 1, 2, 3, 4, 6, 8, 12, 24 )
- The factors of 40 are: ( 1, 2, 4, 5, 8, 10, 20, 40 )
- Identify the Common Factors:
- The common factors of 24 and 40 are: ( 1, 2, 4, 8 )
- Determine the GCD:
- The greatest of these common factors is ( 8 ), which is the GCD.
Step 2: Divide Both the Numerator and Denominator by the GCD
Now that we have the GCD, we can simplify the fraction:
[
\frac{24 \div 8}{40 \div 8} = \frac{3}{5}
]
Conclusion
The simplest form of the fraction ( \frac{24}{40} ) is ( \frac{3}{5} ).
Explanation
Simplifying fractions is essential for mathematical clarity and understanding. It helps in reducing numbers to their most basic form, making calculations easier. In this case, simplifying ( \frac{24}{40} ) to ( \frac{3}{5} ) shows that both the numerator and the denominator can be reduced by the same factor, reflecting the underlying proportional relationship.
Additionally, this process is useful in various applications, such as in measurements or in problems involving ratios. For example, if you are cooking and need to adjust a recipe, using simplified fractions can make it easier to scale up or down.
Ultimately, simplifying fractions not only helps in performing calculations but also in interpreting and communicating quantitative information more effectively. Understanding how to find the GCD and reduce fractions is a fundamental skill in mathematics that can be applied across different fields, including science, finance, and engineering.