Calculating Marks in an Examination

This problem involves calculating the actual marks obtained by students based on their percentages and the sum of their marks in the same examination. We are given the percentages obtained by Avanish and Kapil and the total sum of their marks.

Let's break down the information given:

• Avanish's marks percentage: 78%
• Kapil's marks percentage: 64%
• Sum of marks obtained by Kapil and Avanish: 923

We need to find the marks obtained by Kapil.

Step-by-Step Solution to Find Kapil's Marks

To solve this, let's assume the total marks for the examination is 'T'. The marks obtained by each student can be expressed as a percentage of the total marks.

Step 1: Express marks in terms of total marks (T).

• Avanish's marks = 78% of T = $\frac{78}{100} \times T = 0.78T$
• Kapil's marks = 64% of T = $\frac{64}{100} \times T = 0.64T$

Step 2: Use the given sum of marks to form an equation.

The sum of the marks obtained by Kapil and Avanish is 923. So, we can write the equation:

Avanish's marks + Kapil's marks = 923

$0.78T + 0.64T = 923$

Step 3: Solve the equation for the total marks (T).

Combine the terms with T:

$(0.78 + 0.64)T = 923$

$1.42T = 923$

Now, isolate T by dividing 923 by 1.42:

$T = \frac{923}{1.42}$

To make the division easier, we can remove the decimal by multiplying both the numerator and the denominator by 100:

$T = \frac{923 \times 100}{1.42 \times 100} = \frac{92300}{142}$

Let's perform the division:

$\frac{92300}{142} = 650$

So, the total marks for the examination is 650.

Step 4: Calculate Kapil's marks.

Kapil's marks are 64% of the total marks (T).

Kapil's marks = 64% of 650

Kapil's marks = $\frac{64}{100} \times 650$

Kapil's marks = $0.64 \times 650$

Kapil's marks = $416$

So, the marks obtained by Kapil in the examination is 416.

Let's quickly verify Avanish's marks to check the sum:

Avanish's marks = 78% of 650

Avanish's marks = $\frac{78}{100} \times 650$

Avanish's marks = $0.78 \times 650$

Avanish's marks = $507$

Sum of marks = Kapil's marks + Avanish's marks = $416 + 507 = 923$. This matches the information given in the question.

Final Answer

The marks obtained by Kapil in the examination are 416.

Revision Table - Percentage and Marks Calculations

Additional Information - Solving Percentage Problems

Percentage problems are common in examinations and real-life scenarios. Understanding how percentages relate to a total value is key. When dealing with percentages of the same total, you can often work with the percentages directly or find the total value first.

• What is a Percentage? A percentage is a number or ratio expressed as a fraction of 100. The symbol "%" is used to indicate percentage. For example, 78% means 78 out of 100.
• Converting Percentage: To use a percentage in calculations, convert it to a decimal (divide by 100) or a fraction (write over 100).
• Sum of Percentages: If two parts make up a whole, their percentages (of the same whole) add up to the total percentage. In this problem, Avanish's marks and Kapil's marks are parts of the total examination marks. Their percentages refer to this same total. However, the sum of their *percentage values* (78% + 64% = 142%) represents 142% of the total marks, which corresponds to the sum of their actual marks (923). This is why $1.42T = 923$.
• Checking your answer: Always check if your calculated values make sense in the context of the problem. Does Kapil's score (416) seem reasonable given he got 64%? Is the total score (650) plausible? Do the individual scores add up to the given sum?

This problem demonstrates a typical application of percentages where an unknown total needs to be found before calculating individual values.