Understanding the Fraction Problem

This problem involves a positive fraction where a mistake was made by using its reciprocal instead of the fraction itself. This error caused the value to decrease by a specific percentage. We need to find the original value of the fraction.

Setting up the Math Equation

Let the original positive fraction be \(x\). Since the value was reduced when the reciprocal was used, the original fraction must be greater than its reciprocal. For a positive fraction, this means \(x > \frac{1}{x}\), which implies \(x^2 > 1\), so \(x > 1\).

The reciprocal of the fraction is \(\frac{1}{x}\).

The problem states that the value got reduced by \(\frac{175}{4}\%\). The percentage reduction is calculated with respect to the original value.

Percentage Reduction = \(\left( \frac{\text{Original Value} - \text{Mistyped Value}}{\text{Original Value}} \right) \times 100\)

In this case:

Original Value = \(x\)

Mistyped Value (Reciprocal) = \(\frac{1}{x}\)

Percentage Reduction = \(\frac{175}{4}\%\)

So, we can write the equation:

$$\frac{175}{4} = \left( \frac{x - \frac{1}{x}}{x} \right) \times 100$$

Solving for the Original Fraction

Now, let's solve the equation for \(x\).

Divide both sides by 100:

$$\frac{175}{4 \times 100} = \frac{x - \frac{1}{x}}{x}$$

$$\frac{175}{400} = \frac{\frac{x^2 - 1}{x}}{x}$$

Simplify the fraction \(\frac{175}{400}\). Both numerator and denominator are divisible by 25.

$$175 \div 25 = 7$$

$$400 \div 25 = 16$$

So, the equation becomes:

$$\frac{7}{16} = \frac{x^2 - 1}{x^2}$$

Now, cross-multiply:

$$7 \times x^2 = 16 \times (x^2 - 1)$$

$$7x^2 = 16x^2 - 16$$

Rearrange the terms to solve for \(x^2\):

$$16 = 16x^2 - 7x^2$$

$$16 = 9x^2$$

$$x^2 = \frac{16}{9}$$

Since \(x\) is a positive fraction, we take the positive square root:

$$x = \sqrt{\frac{16}{9}}$$

$$x = \frac{4}{3}$$

Verification

Let's check if the original fraction \(x = \frac{4}{3}\) results in the given percentage reduction when its reciprocal is used.

Original fraction: \(\frac{4}{3}\)

Reciprocal: \(\frac{3}{4}\)

Reduction amount: \(\frac{4}{3} - \frac{3}{4} = \frac{16 - 9}{12} = \frac{7}{12}\)

Percentage reduction: \(\left( \frac{\text{Reduction amount}}{\text{Original value}} \right) \times 100 = \left( \frac{\frac{7}{12}}{\frac{4}{3}} \right) \times 100\)

$$= \left( \frac{7}{12} \times \frac{3}{4} \right) \times 100$$

$$= \left( \frac{7}{4 \times 4} \right) \times 100$$

$$= \frac{7}{16} \times 100 = \frac{700}{16}$$

Let's calculate \(\frac{700}{16}\):

The calculated percentage reduction is 43.75%, which is equal to \(\frac{175}{4}\%\). This confirms our answer is correct.

Conclusion

The value of the original fraction was \(\frac{4}{3}\).

Revision Table: Fraction and Reciprocal Concepts

Additional Information on Percentage Calculations

Understanding percentage changes is crucial in many quantitative problems. A percentage change can be either an increase or a decrease.

If a value changes from \(V_{original}\) to \(V_{new}\):

• Absolute Change = \(V_{new} - V_{original}\)
• Percentage Change = \(\left( \frac{V_{new} - V_{original}}{V_{original}} \right) \times 100\)

If \(V_{new} > V_{original}\), the percentage change is positive, indicating a percentage increase.

If \(V_{new} < V_{original}\), the percentage change is negative, indicating a percentage decrease or reduction. The percentage reduction is often stated as the absolute value of this negative percentage change.

In this problem, the new value (\(\frac{1}{x}\)) was less than the original value (\(x\)), leading to a percentage reduction.

The problem used the term "reduced by \(\frac{175}{4}\%\)", which directly gives us the percentage reduction amount to use in the formula.