A farmer planted only apples and pears on his orchard. There are 120 more pears than apples. 1/4 of the apples and 2/13 of the pears were rotten. If 1/5 of the fruits were rotten, how many fruits are there in the orchard?
p = a + 120
a/4 rotten
2p/13 rotten
a/4 + 2p/13 = 1/5 ( p+a ) or a/4-a/5 = p/5 - 2p/13
a/20 = 13 p / 65 - 10p/65 = 3 p/65
65 a/20 = 3 p
65 a = 60 p
p = 13 a /12
and we know
p = a + 120
a +120 = 13 a/12
12 a/12 + 120 = 13 a/12
a/12 = 120
a = 1440
p = 1560
sum = 3000
Let's assume the number of apples in the orchard is A.
The number of pears would be A + 120 because there are 120 more pears than apples.
The total number of fruits in the orchard would be A + (A + 120) = 2A + 120.
Now, let's calculate the number of rotten fruits.
The number of rotten apples would be 1/4 of A, which is A/4.
The number of rotten pears would be 2/13 of (A + 120), which is (2/13) * (A + 120).
The total number of rotten fruits would be A/4 + (2/13) * (A + 120).
We know that 1/5 of the fruits were rotten, so we can write the equation:
(A/4 + (2/13) * (A + 120)) = 1/5 * (2A + 120).
Let's solve this equation step by step:
Multiplying both sides of the equation by 5 to eliminate the fraction:
5 * (A/4 + (2/13) * (A + 120)) = 5 * (1/5 * (2A + 120)).
Simplifying:
5 * (A/4) + 5 * (2/13) * (A + 120) = 2A + 120.
Multiplying both sides by 4 to eliminate the fraction:
5A + 20 * (2/13) * (A + 120) = 8A + 480.
Simplifying:
5A + 40/13 * (A + 120) = 8A + 480.
Expanding:
5A + (40/13)A + (40/13) * 120 = 8A + 480.
Simplifying:
(65/13)A + (40/13) * 120 = 8A + 480.
Simplifying further:
(65/13)A + (40/13) * 120 - 8A = 480.
Multiplying both sides by 13 to eliminate the fractions:
65A + 40 * 120 - 104A = 480 * 13.
Simplifying:
65A + 4800 - 104A = 6240.
Combining like terms:
-39A + 4800 = 6240.
Subtracting 4800 from both sides:
-39A = 1440.
Dividing both sides by -39:
A = -1440/-39.
A = 37.97 (rounded to 2 decimal places).
Since the number of apples must be a whole number, we can assume it is 38.
The number of pears would be 38 + 120 = 158.
The total number of fruits in the orchard would be 38 + 158 = 196.
Therefore, there are 196 fruits in the orchard.