Five divided by the difference of a number and 8 equals the quotient of 10 and the sum of a number and 7. Find the number.

If Jon can paint the fence in 3 hours and Karen can paint it in 5 hours, find how long takes them to paint it together.

Step-by-step explanation:

Let the number be x

Five divided by the difference of the number and 8 means

5 / (x-8) and this is

equals the quotient of 10 and the sum of a number and 7. This means

10 / (7+x)

Therefore,

5 / (x-8) = 10 / (7+x)

Cross multiplying,

5 (7+x) = 10 (x-8)

35 + 5x = 10x - 80

5x - 10x = - 80 - 35

-5x = - 115

x = - 115/5 = 23

2)

If Jon can paint the fence in 3 hours

This means Jon can paint the fence in 1/3 per hour

If Karen can paint it in 5 hours

This means Karen can paint the fence in 1/5 hours

Their combined rate is additive since they work simultaneously. Jon doesn't stop for Karen to work and Karen doesn't stop for Jon to work.

Their combined rate is

1/3 + 1/5 = 8/15

Let's assume the combined time is t hours.

The combined time per hour is 1/t per hour. Therefore 1/t = 8/15

t = 15/8 = 1.875 hours