(Q2).Kavitha thought of constructing 2 more rooms in her house. She enquired about the labour. She came to know that 6 men and 8 women could finish this work in 14 days. But she wishes to complete that work in only 10 days. When she enquired, she was told that 8 men and 12 women could finish the work in 10 days. Find out how much time would be taken to finish the work if one man or one woman worked alone.

Let the time taken by one man to finish the work = x days.

The portion of work done by one man in one day =
1

Let the time taken by one woman to finish the work = y days.

The portion of work done by one woman in one day =
1

Now, 8 men and 12 women can finish the work in 10 days.

So the portion of work done by 8 men and 12 women in one day =
1
..............(1)
Also, the portion of work done by 8 men in one day is ×
1
 =
Similarly, the portion of work done by 12 women in one day is ×
1
 =
Total portion of work done by 8 men and 12 women in one day =
x
 +
y
 ......................(2)
Equating equations (1) and (2) (
x
 +
y
 ) =
1
(
x
 +
y
 ) = 1
x
 +
y
 = 1 ...............(3)

Also, 6 men and 8 women can finish the work in 14 days.

The portion of work done by 6 men and 8 women in one day =
x
 +
y
 =
1
⇒ (
x
 +
y
 ) = 1
(
x
 +
y
 ) = 1 ...............(4)
Observe equations (3) and (4). Are they linear equations? How do we solve them then? We can convert them into linear equations by substituting
1
x
 = and
1
y
 = .
Equation (3) becomes u + v = 1 ................(5)
Equation (4) becomes u + v = 1 ................(6)

L.C.M. of and is . Using the elimination method,

Equation (3) × (21 × )u + ( × 120)v = 21
Equation (4) × (20 × )u + (20 × )v = 20
u+v = 21
u+v = 20 Same sign for u, so subtract
(-) (-) (-)
v =
v =
Substitute in equation (5) 80u + 120 ×
1
 = 1
80u = 1 -
=
-
=
u =
×
1
 =
1

So one man alone can finish the work in days and one woman alone can finish the work in days.