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# ALGEBRA WORK PROBLEM

+1
36
6
+28

A father and his son can dig a well if the father works 6 hours and his son works 12 hours, or they can do it if the father works 9 hours, and the son works 8 hours. How long will it take for the son to dig the well alone?

the answer is aparently 20, but i still dont know how to come up with it. pls help

May 14, 2023

#1
+1

Let's say that the father can dig a well in f hours, and the son can dig a well in s hours. We know that the father and son can dig a well if the father works 6 hours and the son works 12 hours, or they can do it if the father works 9 hours and the son works 8 hours. This means that:

61​+121​=f1​+s1​

and

91​+81​=f1​+s1​

We can solve this system of equations to find that f=18 and s=36. Therefore, it will take the son 36 hours to dig the well alone.

May 14, 2023
#2
+28
+1

thank you very much for answering but this is from practice exam where we get to see the correct answer, and the answer is 20(unless they're wrong). I just want to know how did they come up with 20.

Gravy  May 14, 2023
#3
+2

F=Father,  S=Son

F/ 6  + S / 12 =F/9 + S/8, solve for S in terms of F

S = (4F) / 3

Since his father worked a total of: 6 + 9 = 15 hours, the son could do the same job in:

S =( 4*15) / 3 = 20 hours.

It is also obvious that when the father works: 6+9=15 hours, the son works: 12 + 8=20 hours.

May 14, 2023
#5
+28
+1

i see now, thank you very much!

Gravy  May 15, 2023
#4
+387
+1

Think in terms of shovels.

If the father can dig, on average, f shovels per hour and it takes W shovels to dig the well, it would take the father, working alone, W/f hours to dig the well.

Similarly if the son can dig, on average, s shovels per hour, it would take him, working alone W/s hours to dig the well, (and notice in passing that this is the number to be calculated).

Now, the well can be dug if the father works for 6hrs and the son for 12 hrs, so

6f + 12s = W

and similarly

9f + 8s = W.

Multiply the first equation by 3 and the second by 2,

18f + 36s = 3W

18f + 16s + 2W,

and subtract,

20s = W,

so W/s = 20.

May 14, 2023
#6
+28
+1

i see now, thank you very much!

Gravy  May 15, 2023