Understanding the Percentage Problem in Examination Marks

This problem involves calculating the percentage difference in marks obtained by different students based on given relationships. We are given how the marks of B, A, C, and D relate to each other in terms of percentages. Our goal is to find out by what percentage D's marks are more than A's marks.

Defining Variables for Student Marks

Let's represent the marks obtained by each student with a variable:

• Let A be the marks obtained by student A.
• Let B be the marks obtained by student B.
• Let C be the marks obtained by student C.
• Let D be the marks obtained by student D.

Translating Percentage Relationships into Equations

We are given the following information:

1. B obtained 20% more marks than A.
   This can be written as: \( B = A + 20\% \text{ of } A = A + \frac{20}{100}A = A + 0.20A = 1.20A \)
2. A obtained 10% less marks than C.
   This can be written as: \( A = C - 10\% \text{ of } C = C - \frac{10}{100}C = C - 0.10C = 0.90C \)
3. D obtained 20% more marks than C.
   This can be written as: \( D = C + 20\% \text{ of } C = C + \frac{20}{100}C = C + 0.20C = 1.20C \)

We are asked to find the percentage by which D's marks are more than A's marks. This percentage difference is calculated using the formula:

\( \text{Percentage Difference} = \frac{\text{Marks of D} - \text{Marks of A}}{\text{Marks of A}} \times 100 \)
or \( \text{Percentage Difference} = \left(\frac{D - A}{A}\right) \times 100 \)
or \( \text{Percentage Difference} = \left(\frac{D}{A} - 1\right) \times 100 \)

Calculating the Ratio of D's Marks to A's Marks

To find the percentage difference, we need to express D and A in terms of a common reference, or find the ratio D/A. From the given equations, we have:

• From equation 2: \( A = 0.90C \)
• From equation 3: \( D = 1.20C \)

We can express C in terms of A from equation 2:

\( C = \frac{A}{0.90} \)

Now substitute this expression for C into the equation for D:

\( D = 1.20 \times \left(\frac{A}{0.90}\right) \)

Simplify the coefficient:

\( D = \frac{1.20}{0.90} \times A \)
\( D = \frac{120}{90} \times A \)
\( D = \frac{12}{9} \times A \)
\( D = \frac{4}{3} \times A \)

So, the ratio of D's marks to A's marks is \( \frac{D}{A} = \frac{4}{3} \).

Finding the Percentage Increase

Now we use the percentage difference formula:

\( \text{Percentage Difference} = \left(\frac{D}{A} - 1\right) \times 100 \)

Substitute \( \frac{D}{A} = \frac{4}{3} \):

\( \text{Percentage Difference} = \left(\frac{4}{3} - 1\right) \times 100 \)
\( \text{Percentage Difference} = \left(\frac{4}{3} - \frac{3}{3}\right) \times 100 \)
\( \text{Percentage Difference} = \left(\frac{4-3}{3}\right) \times 100 \)
\( \text{Percentage Difference} = \left(\frac{1}{3}\right) \times 100 \)
\( \text{Percentage Difference} = \frac{100}{3} \)
\( \text{Percentage Difference} = 33\frac{1}{3} \% \)

The marks obtained by D are \( 33\frac{1}{3}\% \) more than the marks obtained by A.

Summary of Calculations

Derived Relationship between D and A:

Percentage increase of D over A:

Conclusion on Percentage Difference

Based on the calculations, D obtained \( 33\frac{1}{3}\% \) more marks than A.

Revision Table: Key Percentage Concepts

Additional Information: Working with Percentages

Working with percentages often involves converting them to decimals or fractions for calculations. A percentage represents a fraction out of 100. For example, 20% is equivalent to \( \frac{20}{100} = 0.20 \) or \( \frac{1}{5} \). 10% is \( \frac{10}{100} = 0.10 \) or \( \frac{1}{10} \).

When one quantity is a certain percentage more than another, you add that percentage of the base quantity to the base quantity. If it's a percentage less, you subtract. For example, if B has 20% more marks than A, and A has marks 'A', then the increase is \( 20\% \) of A, which is \( 0.20A \). So, B's marks are \( A + 0.20A = 1.20A \). If A has 10% less marks than C, and C has marks 'C', then the decrease is \( 10\% \) of C, which is \( 0.10C \). So, A's marks are \( C - 0.10C = 0.90C \).

The percentage difference between two quantities, say Q1 and Q2, relative to Q1 is \( \frac{Q2 - Q1}{Q1} \times 100\% \). If Q2 > Q1, this gives a positive percentage increase. If Q2 < Q1, it gives a negative percentage change (a decrease).