30 junior and 42 senior participants participate in a

Solution

Let the number of boys in junior be 'x'.

^xC2 = 105

Or, x(x - 1) ÷ 2 = 105

Or, x² - x - 210 = 0

Or, x² - 15x + 14x - 210 = 0

Or, x(x - 15) + 14(x - 15) = 0

Or, (x - 15)(x + 14) = 0

So, 'x' = 15 or 'x' = -14

But 'x' cannot be negative. So, 'x' = 15

Number of boys in junior = 15

Number of girls in junior = 30 - 15 = 15

Let the number of girls in senior be 'y'.

yC2 = 210

Or, y(y - 1) ÷ 2 = 210

Or, y² - y - 420 = 0

Or, y² - 21y + 20y - 420 = 0

Or, y(y - 21) + 20(y - 21) = 0

Or, (y - 21)(y + 20) = 0

So, 'y' = 21 or 'y' = -20

But 'y' cannot be negative. So, 'y' = 21

Number of girls in senior = 21

Number of boys in senior = 42 - 21 = 21

Required number of matches = 15 × 15 + 21 × 21 = 225 + 441 = 666

7n + 57 = 666

Or, 7n = 609

Or, n = 87