## Work and Wages

When a person receives some money for a certain work, the received money is called wages of the person for that particular work.

Total wages = Wages of 1 day work × Total number of days

Example: If Rohan's monthly wages ₹ 5400 and he work for all 30 days, then what will be his daily wages?
Solution: Let the daily wages be x.
Total wages = Daily wages × Total number of days
⇒ 5400 = x × 30

 ⇒ x = 5400 = 180 30
∴ Daily wages = ₹ 180

### Important Points

1. Wages is directly proportional to the work done.
2. More money will be received for more work and less money will be received for less work.
3. Wages is indirectly proportional to the time taken by the individual.

Example: If A can do a piece of work in 15 days and B can do the same piece of work in 20 days. Then, ratio of A and B's daily wage will be
20 : 15 = 4 : 3.

### Some Important Rule

Rule 1: If P can do a piece of work in x days and Q can do the same work in y days, the ratio of their wages will be y : x. Then the wages earned by P and Q will be

 P's wages = Total wages × y (x + y)
 Q's wages = Total wages × x (x + y)

Example:
A can do a piece of work in 5 days, while B can do the same work in 7 days. If the total amount to be given for this work is ₹ 7200, then what will be the share of A, if both work together?

Solution:- Time taken by A = 5 days

 A's one day work = 1 5
Time taken by B = 7 days
 B's one day work = 1 7
Total amount earned = ₹ 7200
 Ratio of their incomes = 1 : 1 = 7 : 5 5 7
 ∴ A's share = 7200 × 5 = 7200 × 5 ( 7 + 5 ) 12
= 600 × 5 = ₹ 3000

Example: A can do a piece of work in 5 days while B can do the same work in 6 days. If they work together for a total wages of ₹ 5500, how much does A get?
Solution: Time taken by A = 5 days
 A's one day work = 1 5
Time taken by B = 6 days
 A's one day work = 1 6
 Ratio of their incomes = 1 : 1 = 6 : 5 5 6
Total incomes = ₹ 5500
 ∴ A's share = 5500 × 5 = 5500 × 6 ( 6 + 5 ) 11
= 500 × 6 = ₹ 3000

Rule 2: If P, Q and R can do a piece of work in x, y and z days respectively, the ratio of their wages will be yz : xz : xy. Then, wages earned by P, Q and R respectively will be

 P's wages = Total wages × yz ( xy + yz + zx )
 Q's wages = Total wages × xz ( xy + yz + zx )
 R's wages = Total wages × xy ( xy + yz + zx )

Example: A, B and C take ₹ 470 for doing a piece of work together. If working alone, each takes 3 days, 4 days and 5 days respectively, then find the share of each.
Solution:- Total amount earned = ₹ 470
 Ratio of their incomes = 1 : 1 : 1 3 4 5
LCM of 3, 4 and 5 = 60
 = 1 × 60 : 1 × 60 : 1 × 60 = 20 : 15 : 12 3 4 5
 ∴ A's share = 470 × 20 = ₹ 200 47
 B's share = 470 × 15 = ₹ 150 47
 C's share = 470 × 12 = ₹ 120 47

Rule 3: P can do a piece of work in x days. With the help of Q, P can do the same work in y days. If they get ₹ a for that work, then

 Share of P = ₹ ay x
 Share of Q = ₹ a( x - y ) x

Example: A Can do a work in 30 days. A and B together do the same work in 25 days. If they got ₹ 3000 for that work, then find the share of A and B.
Solution:
 A's one day work = 1 30
 ( A + B )'s one day work = 1 25
 B's one day work = 1 - 1 = 6 - 5 25 30 150
 B's one day work = 1 150
 Ratio of their income = 1 : 1 = 5 : 1 30 150
Total income = ₹ 3000
 ∴ A's share = 3000 × 5 = 3000 × 5 = ₹ 2500 ( 5 + 1 ) 6
 B's share = 3000 × 1 = 3000 × 1 = ₹ 500 ( 5 + 1 ) 6

By formula,
Here, x = 30, y = 25 and a = 3000

 Share of A = ₹ ay = 3000 x 25 = ₹ 2500 x 30
 Share of B = ₹ a( x - y ) = 3000( 30 - 25 ) = ₹ 500 x 30

Rule 4: P, Q and R undertake to do a work for ₹ a. If together they do only x/y of the work and rest is done by R alone, then

 Share of R = a 1 - x y

Example: A, B and C undertake to do a work for ₹ 600. A and B together do 1/3 of the work and rest is done by the C alone. How much should C get?
Solution:-

 Work done by ( A + B ) = 1 3
 Work done by C = 1 - 1 = 2 3 3
Total income = ₹ 600 Ratio of incomes = ( A + B ) : c
 Ratio of incomes = 1 : 2 = 1 : 2 3 3
 ∴ Share of C = 600 × 2 = 600 × 2 = ₹ 400 ( 1 + 2 ) 3
By formula, Here, a = ₹ 600, x = 1 and y = 3
 ∴ Share of C = 600 1 - 1 = 600 × 2 = ₹ 400 3 3
Example: A and B undertaken to do a piece of work for ₹ 1200. A alone can do it in 8 days, while B can do it in 6 days. with the help of C they complete it in 3 days. Find C's share.
Solution:-
 Work done by ( A + B ) = 1 3
 Work done by C = 1 - 1 = 2 3 3
Total income = ₹ 600
Ratio of incomes = ( A + B ) : c
 Ratio of incomes = 1 : 2 = 1 : 2 3 3
 ∴ Share of C = 600 × 2 = 600 × 2 = ₹ 400 ( 1 + 2 ) 3
By formula, Here, a = ₹ 600, x = 1 and y = 3
 ∴ Share of C = 600 1 - 1 = 600 × 2 = ₹ 400 3 3
 A's one day work = 1 8
 B's one day work = 1 6
 C's one day work = 1 - 1 + 1 3 8 6
 C's one day work = 8 - 3 - 4 24 24 24
 C's one day work = 8 - 7 = 1 24 24 24
 Ratio of their incomes = 1 : 1 : 1 = 3 : 4 : 1 8 6 24
Total income = ₹ 1200
 ∴ Share of C = 1200 × 1 = 1200 × 1 = ₹ 150 ( 3 + 4 + 1 ) 8