Write $\frac{2}{5}-\frac{3}{5 k}$ as a single fraction in its simplest form.
Answer (2)
Find the common denominator: The common denominator for the two fractions is 5 k .
Rewrite the fractions with the common denominator: 5 2 = 5 k 2 k .
Combine the fractions: 5 k 2 k − 5 k 3 = 5 k 2 k − 3 .
Simplify the fraction: The fraction is already in its simplest form, so the final answer is 5 k 2 k − 3 .
Explanation
Problem Analysis We are given the expression 5 2 − 5 k 3 and we want to combine these two fractions into a single fraction in its simplest form.
Finding the Common Denominator To combine the fractions, we need to find a common denominator. In this case, the common denominator is 5 k .
Rewriting the First Fraction We rewrite the first fraction with the common denominator 5 k . To do this, we multiply the numerator and denominator of 5 2 by k to get 5 k 2 k .
Combining the Fractions Now we can rewrite the original expression as a single fraction: 5 2 − 5 k 3 = 5 k 2 k − 5 k 3 = 5 k 2 k − 3 .
Simplifying the Fraction We need to check if the numerator and the denominator have any common factors that can be cancelled to simplify the fraction. The numerator is 2 k − 3 and the denominator is 5 k . Let's consider some values of k . If k = 1 , then 2 k − 3 = 2 ( 1 ) − 3 = − 1 and 5 k = 5 ( 1 ) = 5 . The greatest common divisor of − 1 and 5 is 1 . If k = 2 , then 2 k − 3 = 2 ( 2 ) − 3 = 1 and 5 k = 5 ( 2 ) = 10 . The greatest common divisor of 1 and 10 is 1 . If k = 3 , then 2 k − 3 = 2 ( 3 ) − 3 = 3 and 5 k = 5 ( 3 ) = 15 . The greatest common divisor of 3 and 15 is 3 . So, 5 ( 3 ) 2 ( 3 ) − 3 = 15 3 = 5 1 . However, the question asks to write the expression in its simplest form in general , not for a specific value of k . In general, 2 k − 3 and 5 k do not have any common factors, so the fraction is already in its simplest form.
Final Answer Therefore, the expression 5 2 − 5 k 3 as a single fraction in its simplest form is 5 k 2 k − 3 .
Examples
In a factory, two machines produce parts. The first machine produces 5 2 of the total parts needed, while the second machine produces 5 k 3 of the total parts, where k depends on the machine's speed setting. To find the fraction of the total parts produced by both machines combined, you need to subtract the fractions and express the result as a single fraction. This helps in determining the overall production rate and managing the inventory effectively. For example, if k = 2 , then the second machine produces 10 3 of the parts. Together, the two machines produce 5 2 + 10 3 = 10 4 + 10 3 = 10 7 of the parts. If we want to know the difference, we calculate 5 2 − 10 3 = 10 4 − 10 3 = 10 1 .
To combine 5 2 and 5 k 3 into a single fraction, use the common denominator 5 k to rewrite the fractions, resulting in 5 k 2 k − 3 . This fraction is already in its simplest form. Thus, the final answer is 5 k 2 k − 3 .
;
