Question Details

Some students (only boys and girls) from different schools appeared for an Olympiad exam. 20% of the boys and 15% of the girls failed the exam. The number of boys who passed the exam was 70 more than that of the girls who passed the exam. A total of 90 students failed. Find the number of students that appeared for the exam.

Options

420

400

500

350

Show Answer

Correct Answer :

Option C

500

Solution :

The correct option is 500.

Let us solve the problem step-by-step by setting up a system of linear equations.

Let the total number of boys who appeared for the exam be $B$ and the total number of girls who appeared for the exam be $G$ .

Step 1: Express the number of students who failed in terms of $B$ and $G$ .
According to the problem, 20% of the boys and 15% of the girls failed the exam. The total number of students who failed is 90.
This gives us our first equation:
$0.20 B + 0.15 G = 90$
To simplify this equation, we can multiply the entire equation by 100 to clear the decimals:
$20 B + 15 G = 9000$
We can divide this equation by 5 to simplify it further:
$4 B + 3 G = 1800$ --- (Equation 1)

Step 2: Express the number of students who passed in terms of $B$ and $G$ .
If 20% of the boys failed, then the percentage of boys who passed is:
$100 % - 20 % = 80 %$
So, the number of boys who passed is $0.80 B$ .
Similarly, if 15% of the girls failed, then the percentage of girls who passed is:
$100 % - 15 % = 85 %$
So, the number of girls who passed is $0.85 G$ .

Step 3: Set up the second equation.
The problem states that the number of boys who passed was 70 more than the number of girls who passed.
This gives us:
$0.80 B - 0.85 G = 70$
Multiply the entire equation by 100 to clear the decimals:
$80 B - 85 G = 7000$
Divide by 5 to simplify:
$16 B - 17 G = 1400$ --- (Equation 2)

Step 4: Solve the system of linear equations.
We have:
1) $4 B + 3 G = 1800$
2) $16 B - 17 G = 1400$
Let us multiply Equation 1 by 4 so that the coefficient of $B$ matches the coefficient in Equation 2:
$4 \times (4 B + 3 G) = 4 \times 1800$
$16 B + 12 G = 7200$ --- (Equation 3)
Now, subtract Equation 2 from Equation 3:
$(16 B + 12 G) - (16 B - 17 G) = 7200 - 1400$
$29 G = 5800$
$G = \frac{5800}{29}$
$G = 200$

Substitute $G = 200$ back into Equation 1 to find $B$ :
$4 B + 3 (200) = 1800$
$4 B + 600 = 1800$
$4 B = 1200$
$B = 300$

Step 5: Find the total number of students.
The total number of students who appeared for the exam is the sum of the boys and the girls:
$Total Students = B + G = 300 + 200 = 500$
Therefore, the total number of students that appeared for the exam is 500.