2230.51 – Jessamine Bush

Here is a problem proposed by ancient Hindu mathematicians: "The square root of half the number of bees in a swarm has flown out upon a jessamine bush. Besides these bees, one female bee flies about a male that is buzzing within a lotus flower into which he was allured in the night by its sweet odor and is now imprisoned in it. Aside from the bees mentioned so far, 8/9 of the whole swarm has remained behind. Tell me the number of the bees."

Solution

If $x$ is the number of bees in the swarm,

$\sqrt{\frac{x}{2}}$ are on the jessamine bush,

2 are at the lotus flower, and

$\frac{8x}{9}$ remained behind.

So $\sqrt{\frac{x}{2}}+\frac{8x}{9}+2=x$

$\sqrt{\frac{x}{2}}= \frac{x}{9}-2$

$\frac{x}{2} = (\frac{x}{9}-2)^2 = \frac{x^2}{81}-\frac{4x}{9}+4$

(x 162:) $\frac{162x}{2}=\frac{162x^2}{81}-\frac{4\cdot162x}{9}+4\cdot162$

$81x = 2x^2-72x + 648\

2x^2 -153x +648=0\

(2x-9)(x-72)=0$ by spectacular factoring or the quadratic formula

$x=72$ or $x = \frac{9}{2}$ (no)

So $x = 72$ .

Check: $\sqrt{\frac{x}{2}}+\frac{8x}{9}+2=6 + 64 +2 = 72.$