Calculating Percentage Error in Fraction Subtraction

This problem requires us to find the actual answer to a mathematical operation involving fractions and unity, and then calculate the percentage error in a given answer. We need to perform the following steps:

1. Calculate the sum of the two fractions.
2. Subtract the sum from unity (which is 1).
3. Convert the result to a decimal to find the true answer.
4. Calculate the absolute error between the true answer and the boy's answer.
5. Calculate the percentage error.

Step 1: Find the Sum of the Fractions

The fractions are \(\frac{1}{4}\) and \(\frac{1}{5}\). To add them, we need a common denominator.

The least common multiple (LCM) of 4 and 5 is 20.

So, we convert the fractions:

• \(\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}\)
• \(\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}\)

Now, add the converted fractions:

Sum = \(\frac{5}{20} + \frac{4}{20} = \frac{5 + 4}{20} = \frac{9}{20}\)

Step 2: Subtract the Sum from Unity

Unity means 1. We need to subtract the sum \(\frac{9}{20}\) from 1.

\(1 - \frac{9}{20}\)

We can write 1 as a fraction with a denominator of 20: \(1 = \frac{20}{20}\).

So, the actual value before converting to decimal is:

Value = \(\frac{20}{20} - \frac{9}{20} = \frac{20 - 9}{20} = \frac{11}{20}\)

Step 3: Express the Actual Answer in Decimals

The true answer is \(\frac{11}{20}\). To convert this fraction to a decimal, we can divide the numerator by the denominator or multiply the numerator and denominator by a number that makes the denominator a power of 10.

Multiply by 5 to make the denominator 100:

True Answer = \(\frac{11 \times 5}{20 \times 5} = \frac{55}{100} = 0.55\)

So, the true answer to the question is 0.55.

Step 4: Calculate the Error

The boy's answer was 0.45.

The true answer is 0.55.

The error is the difference between the true value and the measured value (the boy's answer). We take the absolute difference.

Error = |True Answer - Boy's Answer|

Error = \(|0.55 - 0.45| = |0.10| = 0.10\)

Step 5: Calculate the Percentage Error

The formula for percentage error is:

\(\text{Percentage Error} = \left( \frac{\text{Error}}{\text{True Value}} \times 100 \right)\%\)

Substitute the values:

True Value = 0.55

Error = 0.10

Percentage Error = \(\left( \frac{0.10}{0.55} \times 100 \right)\%\)

We can write the decimals as fractions:

\(0.10 = \frac{10}{100}\) and \(0.55 = \frac{55}{100}\)

Percentage Error = \(\left( \frac{\frac{10}{100}}{\frac{55}{100}} \times 100 \right)\%\)

Percentage Error = \(\left( \frac{10}{100} \times \frac{100}{55} \times 100 \right)\%\)

Percentage Error = \(\left( \frac{10}{55} \times 100 \right)\%\)

Simplify the fraction \(\frac{10}{55}\) by dividing both numerator and denominator by 5:

\(\frac{10 \div 5}{55 \div 5} = \frac{2}{11}\)

Percentage Error = \(\left( \frac{2}{11} \times 100 \right)\%\)

Percentage Error = \(\left( \frac{200}{11} \right)\%\)

Thus, the percentage of error in his answer was \(\left( \frac{200}{11} \right)\%\).

Revision Table: Key Calculations

Additional Information: Understanding Percentage Error

Percentage error is a way to quantify how close a measured value is to the true value. It is often used in experiments and measurements.

• The formula is \(\text{Percentage Error} = \left| \frac{\text{Measured Value} - \text{True Value}}{\text{True Value}} \right| \times 100\%\). Some definitions use absolute value in the numerator to ensure the percentage error is always positive, indicating the magnitude of the error regardless of whether the measurement was too high or too low.
• A smaller percentage error means the measured value is closer to the true value.
• In this problem, the boy's answer (0.45) was different from the true answer (0.55), leading to a percentage error.
• Understanding fraction operations, subtraction from unity, decimal conversions, and the percentage error formula are key concepts highlighted in this problem.