Question

In an orchard, $\dfrac{1}{5}$ are orange trees, $\dfrac{3}{{13}}$ are mango trees and the rest are banana trees. If the banana trees are 148 in number, find the total number of trees in the orchard.

Answer 1

Hint:
Let x be the total number of trees in the orchard.
It is given that $\dfrac{1}{5}$ are orange trees, $\dfrac{3}{{13}}$ are mango trees and 148 are banana trees.
Then, the sum of $\dfrac{x}{5}$ orange trees, $\dfrac{{3x}}{{13}}$ mango trees and 148 banana trees will give the total number of trees in orchard i.e. x and find the value of x which gives the total number of trees in the orchard.

Complete step by step solution:
Let x be the total number of trees in the orchard.
It is given that $\dfrac{1}{5}$ are orange trees, $\dfrac{3}{{13}}$ are mango trees and 148 are banana trees.
Thus, number of orange trees $ = \dfrac{1}{5}x$ and the number of mango trees $ = \dfrac{3}{{13}}x$ .
Now, the sum of $\dfrac{x}{5}$ orange trees, $\dfrac{{3x}}{{13}}$ mango trees and 148 banana trees will give the total number of trees in orchard i.e. x.
  $ \Rightarrow \dfrac{x}{5} + \dfrac{{3x}}{{13}} + 148 = x$
Then, we will solve the above equation to get the total number of trees.
 $
   \Rightarrow \dfrac{{13x + 15x + \left( {65 \times 148} \right)}}{{65}} = x \\
   \Rightarrow 28x + 9620 = 65x \\
   \Rightarrow 65x - 28x = 9620 \\
   \Rightarrow 37x = 9620 \\
   \Rightarrow x = \dfrac{{9620}}{{37}} \Rightarrow x = 260 \\
 $
Thus, the total number of trees in the orchard is 260.
Now, the total number of orange trees $ = \dfrac{x}{5} = \dfrac{{260}}{5} = 52$ orange trees.
Also, number of mango trees $ = \dfrac{{3x}}{{13}} = \dfrac{{3\left( {260} \right)}}{{13}} = 3\left( {20} \right) = 60$ mango trees.

Note:
Alternate method:
Let x be the total number of trees in the orchard.
It is given that $\dfrac{1}{5}$ are orange trees, $\dfrac{3}{{13}}$ are mango trees and 148 are banana trees.
Thus, the number of orange trees $ = \dfrac{1}{5}x$ and the number of mango trees $ = \dfrac{3}{{13}}x$ .
The rest of 148 trees in the orchard are banana trees.
So, the number of banana trees can be given by (Total number of trees) – (Number of orange trees) – (Number of mango trees) = (Number of banana trees).
 $ \Rightarrow x - \dfrac{x}{5} - \dfrac{{3x}}{{13}} = 148$
Now, we will be taking LCM
 $
   \Rightarrow \dfrac{{65x - 13x - 15x}}{{65}} = 148 \\
   \Rightarrow 65x - 28x = 148 \times 65 \\
   \Rightarrow 37x = 9620 \\
   \Rightarrow x = \dfrac{{9620}}{{37}} \\
   \Rightarrow x = 260 \\
 $
Thus, the total number of trees in the orchard is 260.