Question

Out of a swarm of bees, one fifth settled on a blossom of Kadamba, one third on a flower of Silindhiri and three times the difference between these two numbers flew to the bloom of Kutaja. Only ten bees were then left in the swarm. What was the number of bees in the swarm? (Note; Kadamba, Silindhiri and Kutaja are flowering trees. The problem is from the incident Indian text on algebra.)

Answer 1

Hint: We can assume x to be the number of bees in the swarm. Then we can use the given relation to find the number of bees settled in each tree. Then we can take the sum of the bees in each tree and the number of bees left in the swarm and equate it to the total number of bees. Then we can solve for x in the obtained equation to get the number of bees in the swarm.

Complete step-by-step answer:
Let x be the total number of bees in the swarm.
It is given that, one fifth of the bees settled on a blossom of Kadamba. So, the number of bees settled in Kadamba is given by,
 ${B_1} = \dfrac{1}{5}x$
Then we have, one third of the bees settled on a flower of Silindhiri. So, the number of bees settled on flower of Silindhiri is given by,
 ${B_2} = \dfrac{1}{3}x$
It is also given that three times the difference between these two numbers flew to the bloom of Kutaja. So, the number of bees flew to the bloom of Kutaja is given by,
 ${B_3} = 3\left( {{B_2} - {B_1}} \right)$
On substituting the values, we get,
 $ \Rightarrow {B_3} = 3\left( {\dfrac{1}{3}x - \dfrac{1}{5}x} \right)$
On further simplification, we get,
 $ \Rightarrow {B_3} = 3x\left( {\dfrac{{5 - 3}}{{15}}} \right)$
 $ \Rightarrow {B_3} = \dfrac{2}{5}x$
It is given that only ten bees were than left in the swarm. So, we can write,
 ${B_4} = 10$
The sum of all the bees on different trees and the bees left in the swarm will be equal to the total number of bees. So, we can write it as an equation.
 $ \Rightarrow {B_1} + {B_2} + {B_3} + {B_4} = x$
On substituting the values, we get,
 $ \Rightarrow \dfrac{1}{5}x + \dfrac{1}{3}x + \dfrac{2}{5}x + 10 = x$
On finding the LCM of the fraction, we get,
 $ \Rightarrow \dfrac{{3x + 5x + 6x}}{{15}} + 10 = x$
On simplification and rearranging, we get,
 $ \Rightarrow x - \dfrac{{14x}}{{15}} = 10$
Again, we can find the LCM and subtract the numerators,
  $ \Rightarrow \dfrac{{15x - 14x}}{{15}} = 10$
  $ \Rightarrow \dfrac{x}{{15}} = 10$
On cross multiplication, we get,
 \[ \Rightarrow x = 10 \times 15\]
 \[ \Rightarrow x = 150\]
Therefore, the total number of bees in the swarm is 150.

Note: The method of forming mathematical equations from statements is known as mathematical modelling. We can assign any variable and, in this problem, only one variable is used. We must read the given statements carefully before writing them as equations. While calculating the difference, we must calculate it such that we subtract the lower value from the bigger value. The fraction with the highest denominator will have the least value. If we calculate the difference without keeping this in mind, our solution will go wrong. We can use the modulus of the difference, but it will make the problem more complex.