In a plantation, the ratio of the number of apple trees to the number of mango trees was 6 : 11. A fire destroyed 78 of the apple trees and the ratio of apple trees to the number of mango trees became 1 : 4. How many trees were left in the plantation after the fire?
a/m = 6/11 so m = 11 a / 6
(a-78) / m = 1/4 so m = 4 a - 312
4 a - 312 = 11 a/6
312 = 4 a - 11 a / 6
312 = (24 - 11) a/6 = 13 a/6
a = 144
m = 11 a /6 = 264
so after fire
n = 264 + 144 - 78 = 330
or
number of apple trees = 6x
number of mango trees = 11x
(6x-78) : 11x = 1 : 4
24x - 312 = 11x
13x = 312
x = 24
trees left = 6x + 11x - 78 = 144+264-78 = 330
To solve this problem, we can set up a system of equations using the given information.
Let's assume the number of apple trees before the fire is 6x, and the number of mango trees before the fire is 11x.
After the fire, the number of apple trees becomes 6x - 78, and the ratio of apple trees to mango trees becomes 1 : 4. This means the number of mango trees after the fire is 4 times the number of apple trees.
We can set up the equation:
(6x - 78) / (11x) = 1 / 4
To solve this equation, we can cross-multiply:
4 * (6x - 78) = 11x
Simplifying, we get:
24x - 312 = 11x
Now, let's isolate the x term:
24x - 11x = 312
13x = 312
Dividing both sides by 13, we find:
x = 24
Now that we know the value of x, we can find the number of trees remaining. Substituting x = 24 into our equations, we get:
Number of apple trees before the fire = 6x = 6 * 24 = 144
Number of mango trees before the fire = 11x = 11 * 24 = 264
Number of apple trees after the fire = 6x - 78 = 144 - 78 = 66
Number of mango trees after the fire = 4 * number of apple trees = 4 * 66 = 264
Therefore, there are 66 apple trees and 264 mango trees left in the plantation after the fire.