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.

Given:

20% of the boys and 15% of the girls failed the exam.

Total number of students who failed = 90

Calculation:

The percentage of boys who passed = (100 - 20)% = 80%

The percentage of girls who passed = (100 - 15)% = 85%

Let, the number of appeared boys = x

The number of appeared girls = y

So, 80x/100 - 85y/100 = 70

⇒ 5 (16x - 17y) = 7000

⇒ 16x - 17y = 1400   .....(1)

Also, 20x/100 + 15y/100 = 90

⇒ 5 (4x + 3y) = 9000

⇒ 4x + 3y = 1800   .....(2)

Multiplying 4 to equation (2),

16x + 12y = 7200   .....(3)

Subtracting equation (2) from equation (3),

16x + 12y - 16x + 17y = 7200 - 1400

⇒ 29y = 5800

⇒ y = 5800/29 = 200

∴ The number of girls appeared = 200

Putting y = 200 in equation (2),

4x + 3 × 200 = 1800

⇒ 4x = 1800 - 600 = 1200

⇒ x = 1200/4 = 300

∴ The number of boys who appeared = 300

The total number of students who appeared = 300 + 200 = 500

∴ The number of students that appeared for the exam is 500

Alternate Method Calculation:-

Let the Number of boys and Girls who appeared in exam be x and y respectively,

So, according to question-

Number of boys passed in exam = [(100 - 20) × x]/100 = 0.80x

Number of girls passed in exam = [(100 - 15) × y]/100 = 0.85y

Condition (1) -  

⇒ 0.80x = 0.85y + 70

⇒ 0.80x - 0.85y = 70    ...(i)

Condition (2) -

Total failed students = 90

⇒ 0.20x + 0.15y = 90    ...(ii)

From eqⁿ (1) - [4 × eqⁿ (ii)]

⇒ (-0.85y) - 0.60y = 70 - 360 

⇒ 1.45y = 290

⇒ y = 200

Put this value in eqⁿ (i)

⇒ 0.80x = 0.85 × 200 + 70

⇒ 0.80x = 170 + 70 = 240

⇒ x = 300

So, total number of students appeared in exam = 300 + 200 = 500.