A number was divided by 9, instead of being multiplied by 9. As a result of this, there was an error in the answer. What is the percentage difference (correct to two decimal places) in the answer due to this miscalculation?
In this problem, we need to find the percentage difference in an answer when a number was mistakenly divided by 9 instead of being multiplied by 9. This kind of miscalculation can lead to a significant error, and understanding how to quantify it is important.
Percentage Difference Miscalculation Explained
Let's assume the original number is represented by the variable x. We will first determine the correct result and then the incorrect result due to the miscalculation.
Correct Answer Calculation
The problem states that the number was supposed to be multiplied by 9. Therefore, the correct answer, let's call it Correct Value, would be:
Correct Value $$ = \text{Number} \times 9 $$
Correct Value $$ = x \times 9 = 9x $$
Incorrect Answer due to Division
Instead of multiplication, the number was divided by 9. So, the incorrect answer, let's call it Incorrect Value, would be:
Incorrect Value $$ = \text{Number} \div 9 $$
Incorrect Value $$ = x \div 9 = \frac{x}{9} $$
Calculating the Difference in Answers
To find the error, we need to calculate the difference between the Correct Value and the Incorrect Value. This difference tells us how much the miscalculation affected the result.
Difference $$ = \text{Correct Value} - \text{Incorrect Value} $$
Difference $$ = 9x - \frac{x}{9} $$
To subtract these, we find a common denominator, which is 9:
Difference $$ = \frac{9x \times 9}{9} - \frac{x}{9} $$
Difference $$ = \frac{81x - x}{9} $$
Difference $$ = \frac{80x}{9} $$
Determining Percentage Difference
The percentage difference is calculated by dividing the Difference by the Correct Value and then multiplying by 100%. This shows the error as a proportion of what the answer should have been.
Percentage Difference $$ = \frac{\text{Difference}}{\text{Correct Value}} \times 100\% $$
Substitute the values we found:
Percentage Difference $$ = \frac{\frac{80x}{9}}{9x} \times 100\% $$
To simplify the fraction, remember that dividing by $9x$ is the same as multiplying by $\frac{1}{9x}$:
Percentage Difference $$ = \frac{80x}{9 \times 9x} \times 100\% $$
Percentage Difference $$ = \frac{80x}{81x} \times 100\% $$
The x terms cancel out:
Percentage Difference $$ = \frac{80}{81} \times 100\% $$
Now, perform the division and multiplication:
Percentage Difference $$ \approx 0.98765432 \times 100\% $$
Percentage Difference $$ \approx 98.765432\% $$
Rounding to Two Decimal Places
The question asks for the percentage difference corrected to two decimal places. We look at the third decimal place. If it is 5 or greater, we round up the second decimal place.
The third decimal place is 5. So, we round up 76 to 77.
Percentage Difference $$ \approx 98.77\% $$
Therefore, the percentage difference in the answer due to this miscalculation is approximately 98.77%.
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