The number of students in three classes is in the ratio 3:13:6. If 18 students…

Solution:

Let the initial numbers of students be 3x, 13x and 6x.

After adding 18 to each class the numbers become 3x + 18, 13x + 18 and 6x + 18. These are in the ratio 15 : 35 : 21, so there is a positive k such that:

1. 3x + 18 = 15k

2. 13x + 18 = 35k

3. 6x + 18 = 21k

Subtract the first equation from the second to eliminate the constant 18:

• (13x + 18) - (3x + 18) = 35k - 15k => 10x = 20k => x = 2k

Substitute x = 2k into the third equation:

• 6x + 18 = 21k => 6(2k) + 18 = 21k => 12k + 18 = 21k => 9k = 18 => k = 2

Then x = 2k = 4. The initial numbers are:

• 3x = 3 × 4 = 12

• 13x = 13 × 4 = 52

• 6x = 6 × 4 = 24

Total initially = 12 + 52 + 24 = 88

Answer: 88 students.