Question

Bernard only had durians. Timothy only had apples. They gave each other, half of their fruits. Bernard sold 8 durians and Timothy sold 40 apples. In the end, the ratio of durians to apples for Bernard and Timothy is 1:7 and 1:6 respectively. How many durians did Bernard have at first?

Answer 1

Ifthe amount of durians is x, then the amount of apples will be y.
If Bernard and Timothy give each other half of their fruits, it
will be x/2 and y/2. If Bernard sold 8 durians and Timothy sold 40 apples, it
will be x/2 - 8 and y/2 - 40. If the ratio of durians to apples for Bernard and
Timothy is 1:7 and 1:6 respectively, we can construct the following system of equations.
(x/2 - 8)/(y/2) = 1 / 7 ;
(x/2) / (y/2 - 40) = 1 / 6.
Solution:
7*(x / 2 - 8) = 1 * (y/2);
6 * (x/2)= 1 * (y/2 - 40).

7x / 2 - 56 = y / 2;
3 x = (y/2 - 40).

if y/2 = 7x / 2 - 56
Now substitute this expression for y/2 into the other equation

3 x = 7x / 2 - 56 - 40
This results in a single equation involving only variable x.
3 x = 7x / 2 - 96;
3 x = 3.5x - 96;
3 x - 3.5x = - 96;
-0.5x = -96;
x = (-96)/(-0.5)
x = 192;
So the amount of durians was 192.
And the amount of apples was
y/2 = 7 * 192 / 2 - 56;
y/2 = 1344/2 - 56;
y/2 = 672 - 56;
y/2 = 616;
y = 1232.