Total wages for a work is ? 1280. A alone can do a piece of work in 8 days, while B alone can do it in 12 days. If A and B work on alternate days, then find the share of A.

Let man be represented by m and woman be represented by w.
? 2m + 1w = 1/14
? 14 x (2m + 1w ) = 1 ...(i)

and 4w + 2m = 1/8
8 x (4w + 2m) = 1 ...(ii)

On equating Eqs. (i) and (ii), we get
14 (2m + 1w) = 8 (4w + 2m)
? 28m + 14w = 32w + 16m
? 28m - 16m = 32w - 14w
? 12m = 18w
? m/w = 18/12 = 3/2
So, efficiency of 1 man and 1 woman is 3 : 2.
So, their wages must be in the same ratio
90/x = 3/2
[here, x = wages of a woman ]
? x = 90 x 2 / 3 = ? 60

Let the numbers of men, women and
children are 3k, 2k and k, respectively.
Given, 3k = 90
? k = 30
Number of women = 60
and number of children = 30

Let the men's, women's and children's
wages be ? 5p, ? 3p and ? 2p, respectively.
According to the question,
Total daily wages = ? 10350
? 90 x 5p + 60 x 3p + 30 x 2p = 10350
? p x (450 + 180 + 60) = 10350
? p = 10350/690 = 15
? Daily wages of a man = 15 x 5 = ? 75

Let the wages of 1 man, 1 woman and 1 boy are ? P, ? Q and ? R, respectively. According to the question,
3P + 2Q + 4R = 26 ....(i)
3P = 4Q ....(ii)
and 2Q = 3R ....(iii)

From Eqs. (i) and (ii), we get
4Q + 2Q + 4R = 26
? 6Q + 4R = 26.........................(iv)

From Eqs. (iii) and (iv), we get
9R + 4R = 26
? 13R = 26
? R = 2
From Eqs. (ii) and (iii), we get
Q = 3 and P = 4

? Wages of 4 men, 3 women and 2 boys
= (4 x 4) + (3 x 3) + (2 x 2)
= 16 + 9 + 4 = ? 29

Amount received by ( A + B + C) per day
= 5400/36 = ? 150
? A + B + C = ? 150.................(i)
Similarly, amount received by (A + C) per day
= 1880/20 = ? 94
? A + C = ? 94 ........................(ii)
Amount received by (B + C) per day
= 3040/40 = ? 76
? B + C = ? 76 ........................(iii)
From Eqs. (i) and (iii), we get
A + 76 = ? 150
A = 150 - 76 = ? 74
By putting the value of A in Eq. (ii), we get
74 + C = ? 94
? C = 94 - 74 = ? 20
Hence, amount received by C per day
= ? 20

Let the number of days to complete the work be N.
According to the question,
k/20 + (k - 2)/24 + (k - 5)/ 30 = 1
? [6k + 5(k - 2) + 4(k - 5)] / 120 = 1
? 6k + 5k + 4k = 120 + 10 + 20
? 15k = 150
? k = 10
? Work done by A = 10/20 = 1/2
? Share of A from the assured money
= (1/2) x 5400
= ? 2700

? In 5 days , (4 men + 6 women ) get ? 1600.
? In 1 day, (4 men + 6 women) get ? 1600/5 = ? 320 ....(i)

In 1 day number of person to get ? 1 = 320 / 4 men + 6 women ....(ii)

Similarly, in second condition,
In 1 day, number of person to get ? 1 = (1740 / 6) x (3 men + 7 women)
= 290 / (3 men + 7 women) ....(iii)

From Eqs. (ii) and (iii), we get
320 / (4 men + 6 women) = 290 / (3 men + 7 women)
96 men + 224 women = 116 men + 174 women
? 20 men = 50 women
? Man / Women = 5/2
? 1 women = 2/5 man
From Eq. (i), 1 day,
(4 men + 6 women) = (4 men + 6 x 2/5 men) = 32/5 men get ? 320
? In 1 day, 1 man get = 320 x 5 /32 = ? 50
? In 1 day, 1 woman get = 50 x 2/5 = ? 20
? In 1 day, (7 men + 6 women) get
7 x 50 + 6 x 20 = ? 470
? Required number of days = 3760 / 470 = 8 days