1. Two numbers differ by 8. If one-half the larger is added to one-fourth the smaller, the resulting sum is 22.Find the numbers.

Question

2. Ten years ago a man was three times as old as his son. In 12 more years the son is three-fifths as old
as his father. How old are they now?
3. A and B working togrther can complete a piece of work in 30 days. After they have worked together for 18 days, A has to leave, and B finishes to work alone for 20 more days. Find the time it takes for each to finish the work alone.

Answer 1

1.
smaller number --- x
larger number --- x+8

(1/2)(x+8) + (1/4)x = 22

etc

2.
son's age 10 years ago --- x
man's age 10 years ago ---- 3x

in 12 years, son's age = x+22
in 12 years, man's age = 3x+22
x=22 = (3/5)(3x+22)
etc

3.
Let job to be done be 1 unit
let A's rate be 1/x, where x is the number of days
let B's rate be 1/y, where y is the number of days
combined rate = 1/x + 1/y = (x+y)/(xy)
1 /( (x+y)/(xy) ) = 30
xy/(x+y) = 30
xy = 30(x + y) **
They work together for 18 days, so when A leaves 12 days of work is left
or 12(x+y)/(xy) of the job is to be done by B

12(x+y)/(xy) ÷ (1/y) = 20
12(x+y)(y)/(xy) = 20
3(x+y)/x = 5
5x = 3(x+y) ***

*** ÷ **

5x/xy = 3(x+y)/(30(x+y))
5/y = 1/10
y = 50

sub in ***
5x = 3(x+50)
5x = 3x + 150
2x = 150
x = 75

Working alone, A would take 75 days, and B would work 50 days.
Check my arithmetic and algebra.