Final answer:
It takes 1 man alone 11/9 days to finish the work and 1 woman alone 1/24 days.
Step-by-step explanation:
Let's assume that 1 man can complete the work in x days and 1 woman can complete the work in y days.
From the given information, we can set up two equations:
2x + 7y = 1/4 (Equation 1)
4x + 4y = 1/3 (Equation 2)
Multiplying Equation 1 by 2 and Equation 2 by 7, we get:
4x + 14y = 1/2 (Equation 3)
28x + 28y = 7/3 (Equation 4)
Subtracting Equation 3 from Equation 4, we get:
28x + 28y - (4x + 14y) = 7/3 - 1/2
24x + 14y = 17/6
Multiplying Equation 2 by 6, we get:
24x + 24y = 2/3 (Equation 5)
Substituting Equation 5 in the previous equation:
(24x + 24y) + 14y = 17/6
24x + 38y = 17/6
Multiplying Equation 1 by 38, we get:
76x + 266y = 19/6 (Equation 6)
Subtracting Equation 6 from Equation 5, we get:
(24x + 38y) - (76x + 266y) = 17/6 - 19/6
-52x - 228y = -2/6
Dividing by -4:
13x + 57y = 1/6 (Equation 7)
Multiplying Equation 1 by 13, we get:
26x + 91y = 13/4 (Equation 8)
Subtracting Equation 8 from Equation 7, we get:
(13x + 57y) - (26x + 91y) = 1/6 - 13/4
-13x - 34y = -79/12
Dividing by -1:
13x + 34y = 79/12 (Equation 9)
Multiplying Equation 7 by 34, we get:
442x + 1938y = 17/2 (Equation 10)
Subtracting Equation 10 from Equation 9, we get:
(13x + 34y) - (442x + 1938y) = 79/12 - 17/2
-429x - 1904y = -133/6
Dividing by -17:
429x + 112y = 133/36 (Equation 11)
Multiplying Equation 7 by 429, we get:
30069x + 13053y = 143/4 (Equation 12)
Subtracting Equation 12 from Equation 11, we get:
(429x + 112y) - (30069x + 13053y) = 133/36 - 143/4
-29640x - 12941y = -2879/36
Dividing by -37:
800x + 349y = 2879/36 (Equation 13)
Multiplying Equation 1 by 349, we get:
698x + 2443y = 349/4 (Equation 14)
Subtracting Equation 14 from Equation 13, we get:
(800x + 349y) - (698x + 2443y) = 2879/36 - 349/4
102x - 2094y = -721/36
Dividing by -6:
-17x + 349y = 1202/36 (Equation 15)
Adding Equation 15 and Equation 11, we get:
(-17x + 349y) + (429x + 112y) = 1202/36 + 133/36
412x + 461y = 1335/36
Multiplying Equation 8 by 461, we get:
4786x + 41951y = 6113/4 (Equation 16)
Subtracting Equation 16 from the previous equation:
(412x + 461y) - (4786x + 41951y) = 1335/36 - 6113/4
-4374x - 41528y = -8087/36
Dividing by -2:
2187x + 20764y = 4043/36 (Equation 17)
Multiplying Equation 7 by 20764, we get:
14968x + 515948y = 59297/2 (Equation 18)
Subtracting Equation 18 from Equation 17, we get:
(2187x + 20764y) - (14968x + 515948y) = 4043/36 - 59297/2
-12781x - 495184y = -34379/4
Dividing by -439:
29x + 1126y = 2933/36 (Equation 19)
Adding Equation 19 and Equation 15, we get:
(29x + 1126y) + (-17x + 349y) = 2933/36 + 1202/36
12x + 1475y = 4135/36
Multiplying Equation 1 by 1475, we get:
2950x + 10325y = 1475/4 (Equation 20)
Subtracting Equation 20 from the previous equation:
(12x + 1475y) - (2950x + 10325y) = 4135/36 - 1475/4
-2938x - 8849y = -3905/36
Dividing by -2938:
x + 3y = 35/36 (Equation 21)
Multiplying Equation 21 by 3, we get:
3x + 9y = 35/12 (Equation 22)
Subtracting Equation 22 from Equation 7, we get:
(13x + 57y) - (3x + 9y) = 1/6 - 35/12
10x + 48y = -17/12
Multiplying Equation 22 by 10, we get:
10x + 30y = 35/6 (Equation 23)
Subtracting Equation 23 from the previous equation:
(10x + 48y) - (10x + 30y) = -17/12 - 35/6
18y = -87/12
Dividing by 18:
y = -87/216
Simplifying, we get:
y = -1/24
Substituting the value of y in Equation 21, we get:
x + 3(-1/24) = 35/36
x - 1/8 = 35/36
x = 35/36 + 1/8
x = (35 + 9)/36
x = 44/36
Simplifying, we get:
x = 11/9
Therefore, it takes 1 man alone 11/9 days to finish the work and 1 woman alone 1/24 days to finish the work.