Question

The ratio of the number of apples to oranges to pears is 7:11:9.
Timmy ate 21 fruits.
As a result, the ratio of the number of apples to oranges to pears became 2:3:3. How many fruits were left ?

Answer 1

Answer :

Given : The ratio of the number of apples to oranges to pears is 7 : 11 : 9

Let ratio coefficient = x , So
Total apples = 7x ,
Total oranges = 11x
And
Total pears = 9x
And
Total fruits at starting = 7x + 11x + 9x = 27x

Now Timmy ate 21 fruits and then ratio of the number of apples to oranges to pears became is 2 : 3 : 3
Let ratio coefficient = y , So
Total fruits left = 2y + 3y + 3y = 8y
So , we get

27x - 21 = 8y

x = $\frac{8y+21}{27}$

We know

27 = 3 $\times$ 3 $\times$ 3, so 8y + 21 must be divisible by 3,
And
21 = 3 $\times$ 7 , So 21 is a multiple of 3, so 8y must also be a multiple of 3.

Therefore " y " must be a multiple of 3.

We know multiples of 3 = 3 , 6 , 9 , 12 , 15 , 18 , 21 , ...

Now we choose value of y as the whole ( 8y + 21 ) is divisible by 27

Let y = 3 , than
8 ( 3 ) + 21 = 24 + 21 = 45 , that is not divisible by 27
And
Let y = 12 , than
8 ( 12 ) + 21 = 96 + 21 = 117 , that is not divisible by 27

Trying various multiples of 3, we find that the lowest value of a that makes this come out in integers is 21.

y = 21,

8 ( 21 ) + 21 = 21 ( 8 + 1 ) = 21 $\times$ 9 = 189 that is divisible by 27 , after that we can also find some more multiple of 3 that can form ( 8y + 21 ) is divisible by 27 . But we take the lowest , So that is
y = 21,
So,
Total number of fruits left = 8 $\times$ 21 = 168 ( Ans )