# Answer to Question #32478 in Other Math for Yi Tong

Question #32478

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?

Expert's answer

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.

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.

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