Problem 23 - Algebra 2

Question

Simplify each complex fraction. \(\frac{1-\frac{1}{4}}{2-\frac{3}{5}}\)

Step-by-Step Solution

Answer

The simplified form of the complex fraction \(\frac{1-\frac{1}{4}}{2-\frac{3}{5}}\) is \(\frac{15}{28}\)

Step 1: Simplify the Fractions in the Numerator and Denominator

Let's start by simplifying the fractions within the numerator and the denominator. \n\nWe have, \n\nNumerator: \(1 - \frac{1}{4} = \frac{4-1}{4} = \frac{3}{4}\) \n\nand, \n\nDenominator: \(2 - \frac{3}{5} = \frac{10-3}{5} = \frac{7}{5}\)

Step 2: Calculate the Overall Fraction

Now, compute the overall fraction. We know that, when you divide by a fraction, it is the same as multiplying by the reciprocal of the fraction. \n\nSo, \(\frac{ \frac{3}{4} }{ \frac{7}{5} } = \frac{3}{4} * \frac{5}{7}\)

Step 3: Simplify the Final Fraction

Finally, multiply the numerators together to get the numerator of the answer, and multiply the denominators together to get the denominator of the answer.\n\nSo, \(\frac{3}{4} * \frac{5}{7} = \frac{15}{28}\)

Key Concepts

Numerator and Denominator Simplification

Simplifying complex fractions often begins with addressing the parts of the fraction inside, specifically the numerator and the denominator.
When dealing with expressions like \(1 - \frac{1}{4}\), it is essential to ensure both terms have a common denominator.
In this case, the whole number 1 can be converted as \(\frac{4}{4}\). Thus:

• Numerator: \(1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}\)

A similar process applies to the denominator. For expressions like \(2 - \frac{3}{5}\), convert the whole 2 into a fraction \(\frac{10}{5}\):

• Denominator: \(2 - \frac{3}{5} = \frac{10}{5} - \frac{3}{5} = \frac{7}{5}\)

This simplification is vital as it transforms a complex fraction into simpler terms, easing the next steps.

Fraction Multiplication

Once the numerator and denominator are simplified, the complex fraction is ready for multiplication.
In mathematical terms, dividing by a fraction involves multiplying by its reciprocal.
This means if you have \(\frac{3}{4}\) divided by \(\frac{7}{5}\), it can be rewritten as:

• Multiply: \(\frac{3}{4} \times \frac{5}{7}\)

This step is straightforward but requires attention to detail.
First, multiply the numerators: \(3 \times 5 = 15\).
Then, multiply the denominators: \(4 \times 7 = 28\).
So, the result of this multiplication yields \(\frac{15}{28}\).
This concise multiplication yields the simplified form of the complex fraction.

Reciprocal of a Fraction

The notion of a reciprocal is crucial in the operation of dividing by fractions.
A reciprocal simply flips a fraction's numerator and denominator positions.
For example, the reciprocal of \(\frac{7}{5}\) is \(\frac{5}{7}\).
When we say that dividing by a fraction is the same as multiplying by its reciprocal, this means instead of performing division operations, you multiply by the flipped fraction.
This concept simplifies complex arithmetic and is common when working with complex fractions.
In our exercise, replacing \(\frac{3}{4} \div \frac{7}{5}\) with \(\frac{3}{4} \times \frac{5}{7}\) is far more efficient and manageable.
Understanding reciprocals helps solve many algebraic expressions and is a fundamental element when dealing with fractions.