Question

1. \\(dfrac{dfrac{25}{4}}{dfrac{1}{5} - dfrac{4}{25}}\\) 25) \\(dfrac{dfrac{a}{25} - dfrac{a}{5}}{a}\\)

Explanation:

To solve these two fraction - related problems, we will follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and the rules of fraction arithmetic.

Problem 1: \(\frac{\frac{25}{4}}{\frac{1}{5}-\frac{4}{25}}\)

Step 1: Simplify the denominator

First, we need to find a common denominator for the fractions in the denominator. The common denominator of 5 and 25 is 25.
\(\frac{1}{5}-\frac{4}{25}=\frac{1\times5}{5\times5}-\frac{4}{25}=\frac{5}{25}-\frac{4}{25}\)
According to the rule of subtracting fractions with the same denominator \( \frac{a}{c}-\frac{b}{c}=\frac{a - b}{c}\) (\(c
eq0\)), we have:
\(\frac{5}{25}-\frac{4}{25}=\frac{5 - 4}{25}=\frac{1}{25}\)

Step 2: Divide the numerator by the simplified denominator

Now our expression becomes \(\frac{\frac{25}{4}}{\frac{1}{25}}\). When dividing by a fraction, we multiply by its reciprocal. So \(\frac{\frac{25}{4}}{\frac{1}{25}}=\frac{25}{4}\times25\)
\(\frac{25}{4}\times25=\frac{25\times25}{4}=\frac{625}{4} = 156.25\)

Problem 2: \(\frac{\frac{a}{25}-\frac{a}{5}}{a}\)

Step 1: Simplify the numerator

First, find a common denominator for the fractions in the numerator. The common denominator of 25 and 5 is 25.
\(\frac{a}{25}-\frac{a}{5}=\frac{a}{25}-\frac{a\times5}{5\times5}=\frac{a}{25}-\frac{5a}{25}\)
Using the rule of subtracting fractions with the same denominator \( \frac{a}{c}-\frac{b}{c}=\frac{a - b}{c}\) (\(c
eq0\)), we get:
\(\frac{a}{25}-\frac{5a}{25}=\frac{a-5a}{25}=\frac{- 4a}{25}\)

Step 2: Divide the simplified numerator by \(a\)

Our expression is now \(\frac{\frac{-4a}{25}}{a}\). When dividing by \(a\) (assuming \(a
eq0\)), we can rewrite it as \(\frac{-4a}{25}\times\frac{1}{a}\)
The \(a\) in the numerator and the \(a\) in the denominator cancel out (since \(a
eq0\)), and we are left with \(\frac{-4}{25}\)

Final Answers

For the first problem \(\frac{\frac{25}{4}}{\frac{1}{5}-\frac{4}{25}}\), the answer is \(\boldsymbol{\frac{625}{4}}\) (or \(156.25\)).
For the second problem \(\frac{\frac{a}{25}-\frac{a}{5}}{a}\), the answer is \(\boldsymbol{-\frac{4}{25}}\) (where \(a
eq0\)).

Answer:

To solve these two fraction - related problems, we will follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and the rules of fraction arithmetic.

Problem 1: \(\frac{\frac{25}{4}}{\frac{1}{5}-\frac{4}{25}}\)

Step 1: Simplify the denominator

First, we need to find a common denominator for the fractions in the denominator. The common denominator of 5 and 25 is 25.
\(\frac{1}{5}-\frac{4}{25}=\frac{1\times5}{5\times5}-\frac{4}{25}=\frac{5}{25}-\frac{4}{25}\)
According to the rule of subtracting fractions with the same denominator \( \frac{a}{c}-\frac{b}{c}=\frac{a - b}{c}\) (\(c
eq0\)), we have:
\(\frac{5}{25}-\frac{4}{25}=\frac{5 - 4}{25}=\frac{1}{25}\)

Step 2: Divide the numerator by the simplified denominator

Now our expression becomes \(\frac{\frac{25}{4}}{\frac{1}{25}}\). When dividing by a fraction, we multiply by its reciprocal. So \(\frac{\frac{25}{4}}{\frac{1}{25}}=\frac{25}{4}\times25\)
\(\frac{25}{4}\times25=\frac{25\times25}{4}=\frac{625}{4} = 156.25\)

Problem 2: \(\frac{\frac{a}{25}-\frac{a}{5}}{a}\)

Step 1: Simplify the numerator

First, find a common denominator for the fractions in the numerator. The common denominator of 25 and 5 is 25.
\(\frac{a}{25}-\frac{a}{5}=\frac{a}{25}-\frac{a\times5}{5\times5}=\frac{a}{25}-\frac{5a}{25}\)
Using the rule of subtracting fractions with the same denominator \( \frac{a}{c}-\frac{b}{c}=\frac{a - b}{c}\) (\(c
eq0\)), we get:
\(\frac{a}{25}-\frac{5a}{25}=\frac{a-5a}{25}=\frac{- 4a}{25}\)

Step 2: Divide the simplified numerator by \(a\)

Our expression is now \(\frac{\frac{-4a}{25}}{a}\). When dividing by \(a\) (assuming \(a
eq0\)), we can rewrite it as \(\frac{-4a}{25}\times\frac{1}{a}\)
The \(a\) in the numerator and the \(a\) in the denominator cancel out (since \(a
eq0\)), and we are left with \(\frac{-4}{25}\)

Final Answers

For the first problem \(\frac{\frac{25}{4}}{\frac{1}{5}-\frac{4}{25}}\), the answer is \(\boldsymbol{\frac{625}{4}}\) (or \(156.25\)).
For the second problem \(\frac{\frac{a}{25}-\frac{a}{5}}{a}\), the answer is \(\boldsymbol{-\frac{4}{25}}\) (where \(a
eq0\)).