Understanding the Division Problem with Percentage Changes

The question asks us to find the new quotient when the original dividend and divisor are changed by specific percentages. We are given the details of the original division.

Defining Original and Changed Values

Let the original dividend be denoted by \(D\).

Let the original divisor be denoted by \(d\).

According to the problem, the original division had a quotient of 25 and a remainder of 0. This means:

\(D \div d = 25\)

Or, expressed differently:

\(D = 25d\)

Calculating the Mistaken Dividend

Pranjal took a new dividend, let's call it \(D'\), which was 10% more than the original dividend \(D\).

To find \(D'\), we add 10% of \(D\) to \(D\):

\(D' = D + 10\%\text{ of }D\)

\(D' = D + \frac{10}{100}D\)

\(D' = D + 0.10D\)

\(D' = 1.10D\)

Calculating the Mistaken Divisor

Pranjal took a new divisor, let's call it \(d'\), which was 25% more than the original divisor \(d\).

To find \(d'\), we add 25% of \(d\) to \(d\):

\(d' = d + 25\%\text{ of }d\)

\(d' = d + \frac{25}{100}d\)

\(d' = d + 0.25d\)

\(d' = 1.25d\)

Finding the New Quotient

Pranjal's division was using the new dividend \(D'\) and the new divisor \(d'\). The quotient he obtained, let's call it \(Q'\), is given by:

\(Q' = \frac{D'}{d'}\)

Now, we substitute the expressions for \(D'\) and \(d'\) in terms of \(D\) and \(d\):

\(Q' = \frac{1.10D}{1.25d}\)

We know from the original division that \(D = 25d\). Substitute this into the equation for \(Q'\):

\(Q' = \frac{1.10 \times (25d)}{1.25d}\)

Assuming the divisor \(d\) is not zero (which it must be for a division), we can cancel out \(d\) from the numerator and the denominator:

\(Q' = \frac{1.10 \times 25}{1.25}\)

Performing the Calculation

Now we calculate the value of the expression:

\(Q' = \frac{1.10 \times 25}{1.25}\)

\(Q' = \frac{27.5}{1.25}\)

To make the division easier, we can multiply both the numerator and the denominator by 100 to remove decimals:

\(Q' = \frac{27.5 \times 100}{1.25 \times 100}\)

\(Q' = \frac{2750}{125}\)

We can simplify this fraction. Both numbers are divisible by 25:

\(2750 \div 25 = 110\)

\(125 \div 25 = 5\)

So, the calculation becomes:

\(Q' = \frac{110}{5}\)

\(Q' = 22\)

The quotient that Pranjal obtained was 22.

Step-by-Step Calculation Summary

Revision Table: Key Concepts

Additional Information: How Percentage Changes Affect Division

When the dividend and divisor both change, the quotient changes based on the ratio of their changes. If the dividend increases by a larger percentage than the divisor, the quotient will likely increase. If the divisor increases by a larger percentage than the dividend, the quotient will likely decrease.

In this problem, the dividend increased by 10% (multiplied by 1.10) and the divisor increased by 25% (multiplied by 1.25). The new quotient is the old quotient multiplied by the ratio of the dividend change factor to the divisor change factor:

\(Q' = Q \times \frac{\text{Dividend Change Factor}}{\text{Divisor Change Factor}}\)

\(Q' = 25 \times \frac{1.10}{1.25}\)

\(Q' = 25 \times \frac{1.10}{1.25} = 25 \times \frac{110}{125} = 25 \times \frac{22 \times 5}{25 \times 5} = 25 \times \frac{22}{25} = 22\)

This confirms our previous calculation method.

Understanding how percentages affect calculations like division is crucial in quantitative aptitude problems. Always convert percentages to decimals or fractions before performing calculations.