Question

A man sold one sort of oranges at 5 for $2, and another sort at 16 for $12. If he had sold them all at $0.5 each, he would have received $2 less. If he had sold all at 3 for $2, he would have received $18 more. How many of each sort did he sell?

Answer 1

Let's call the amounts `xandy` and the money he got `z`

Explanation:

What he did can be translated into:
Equation (1): `z=2/5x+12/16y=2/5x+3/4y`

The second option can be translated into:
Equation (2): `z-2=0.5*(x+y)=1/2(x+y)`

The last option would be:
Equation (3): `z+18=2/3(x+y)`

If we look at the last two equations, we see that the difference between selling 2 for $1 or 3 for $2 is $20, or:

`2/3(x+y)-1/2(x+y)=20->`

`(2/3-1/2)(x+y)=(4/6-3/6)(x+y)=1/6(x+y)=20`

`->x+y=120->z-2=1/2*120=60->z=62`

We now have the total number of oranges and the price he got according to the original scheme. We can write `y=120-x`

Putting these into the first equation, we get:

`62=2/5x+3/4(120-x)=2/5x+90-3/4x->`

`62=8/20x+90-15/20x=90-7/20x->7/20x=28->`

`x=560/7=80->y=120-80=40`

Answer : 80 of the first sort, 40 of the other.
Note : Not quite satisfactory, as he could not have sold the second sort by the 16's, but 4 for $3 would do.