a candidates scores an aggregate of 60% marks, scoring an average of 56% in 4 of the papers and 68% in the others. how many papers were these totally?

Let’s find the total number of papers by breaking down the given information and setting up an equation.

### Step-by-Step Solution:

1. **Define Variables:**
   - Let the total number of papers be \( n \).
   - Let the number of papers in which the candidate scored 56% be \( x \).
   - Since the candidate scored 56% in 4 papers, we have \( x = 4 \).
   - Let the number of papers in which the candidate scored 68% be \( y \).
   - So, the total number of papers is \( n = x + y \).

2. **Aggregate and Average Calculations:**
   - The aggregate score of all papers is 60%.
   - The total score from the 4 papers (56% papers) is \( 56 \times x = 56 \times 4 = 224\% \).
   - The total score from the other \( y \) papers (68% papers) is \( 68 \times y \).
   - The combined aggregate is \( 224\% + 68y\% \).

3. **Using the Aggregate Percentage:**
   - The average aggregate score for all \( n \) papers is 60%.
   - So, the total score for all papers can be written as:

     \[
     \text{Aggregate} = 60 \times n = 60 \times (x + y) = 60 \times (4 + y)
     \]

4. **Set Up the Equation:**
   - Equating the two expressions for total scores, we get:

     \[
     224 + 68y = 60 \times (4 + y)
     \]

   - Simplify the equation:

     \[
     224 + 68y = 240 + 60y
     \]

5. **Solve for \( y \):**
   - Rearrange to find \( y \):

     \[
     68y - 60y = 240 - 224
     \]

     \[
     8y = 16
     \]

     \[
     y = 2
     \]

6. **Find Total Number of Papers:**
   - The total number of papers \( n \) is:

     \[
     n = x + y = 4 + 2 = 6
     \]

Thus, there are **6 papers in total**.

Would you like more details or have any questions?

Here are some related questions:

1. What would be the aggregate if the scores in the 68% papers increased to 72%?
2. If the candidate scored 50% in one of the 56% papers, how does this affect the aggregate?
3. How many papers would there be if the average score in the 68% papers was 70% instead?
4. If the average score in the 4 papers decreased by 2%, what would be the new aggregate?
5. What if the candidate needed a 65% aggregate; how many more 68% papers would be required?

**Tip:** Always double-check the variables and equations set up based on problem conditions to ensure correct calculations.

Learn how to determine the total number of papers based on aggregate and average scores in a percentage calculation problem. This problem involves finding the number of papers where a candidate scored different percentages and achieving a specified aggregate score. Suitable for high school students.

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