From a group of 2x^{2} black bees, square root of half of the group went to a tree. Again eight-ninth of the bees went to the same tree. The remaining two got caught up in a fragrant lotus. How many bees were there in total?

#### Solution

Total numbers of black bees = 2x^{2 }

Half of the group = `1/2 xx 2x^2` = x^{2}

Square root of half of the group = `sqrt(x^2)` = x

Eight – ninth of the bees = `8/9 xx 2x^2 = (16x^2)/9`

Number of bees in the lotus = 2

By the given condition

`x + (16x^2)/9 + 2` = 2x^{2}

9x + 16x^{2} + 18 = 18x^{2 } ...(Multiply by 9)

18x^{2} – 16x^{2} – 9x – 18 = 0 ⇒ 2x^{2} – 9x – 18 = 0

2x^{2} – 12x + 3x – 18 = 0

2x(x – 6) + 3 (x – 6) = 0

(x – 6) (2x + 3) = 0

x – 6 = 0 or 2x + 3 = 0

x = 6 or 2x = – 3 ⇒ x = `(-3)/2` ...(number of bees will not be negative)

Total number of black bees = 2x^{2} = 2(6)^{2 }

= 72