Working Together Gets The Job Done Faster

Let Alex's 1 day work be $\frac{1}{A}$ of the total work.

and Bella's 1 day work be $\frac{1}{B}$ of the total work.

and Charlie's 1 day work be $\frac{1}{C}$ of the total work.

A/Q, $\frac{1}{A}+\frac{1}{B}=\frac{1}{18}---(1)$

and $\frac{1}{B}+\frac{1}{C}=\frac{1}{24}---(2)$

and $\frac{1}{A}+\frac{1}{C}=\frac{1}{36}---(3)$

We need to find $X$ in: $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{X}$

Adding $(1)$ , $(2)$ and $(3)$ we get:

$\implies 2 \times (\frac{1}{A}+\frac{1}{B}+\frac{1}{C})=\frac{1}{18}+\frac{1}{24}+\frac{1}{36}$

$\implies \frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{16}=\frac{1}{X}$

$\implies X=\boxed {16}.$

Let's say Alex takes $A$ days to complete a work alone, Bella takes $B$ days to complete the same work and Charlie takes $C$ to complete the same work alone.

Hence, Alex completes $\frac{1}{A}$ th of the work in a day, Bella completes $\frac{1}{B}$ th of the work in a day and Charlie completes $\frac{1}{C}$ th of the work in a day.

Since Alex and Bella can paint the fence in 18 days, we can write

$\frac{1}{A} \times 18 + \frac{1}{B} \times 18 = 1$

$\frac{18}{A} + \frac{18}{B} = 1$

$18(A + B) = AB$ .......... (i)

Since Bella and Charlie can paint the same fence in 24 days, we can write

$\frac{1}{B} \times 24 + \frac{1}{C} \times 24 = 1$

$24(B + C) = BC$ .......... (ii)

Since Alex and Charlie can paint the same fence in 36 days, we can write

$\frac{1}{A} \times 36 + \frac{1}{C} \times 36 = 1$

$36(A + C) = AC$ .......... (iii)

Now, we need to calculate $A$ , $B$ and $C$ from equations (i), (ii) and (iii).

From equation (i), we can write

$18A + 18B = AB$

$A = \frac{18B}{B - 18}$ .......... (iv)

From equation (ii), we can write

$24B + 24C = BC$

$C = \frac{24B}{B - 24}$ .......... (v)

Substituting equations (iv) and (v) in equation (iii), we have

$36 \times (\frac{18B}{B - 18} + \frac{24B}{B - 24}) = \frac{18B \times 24B}{(B - 18)(B - 24)}$

$\frac{3}{B - 18} + \frac{4}{B - 24} = \frac{2B}{(B - 18)(B - 24)}$

$\frac{3(B - 24) + 4(B - 18)}{(B - 18)(B - 24)} = \frac{2B}{(B - 18)(B - 24)}$

$3B - 72 + 4B - 72 = 2B$

$B = \frac{144}{5}$ .......... (vi)

Substituting equation (vi) in (iv) and (v) to get $A$ and $C$ ,

$A = 48$

$C = 144$

Hence, if Alex, Bella and Charlie work together and if they take $x$ days to complete it, we can write

$\frac{1}{A} \times x + \frac{1}{B} \times x + \frac{1}{C} \times x = 1$

$x \times (\frac{1}{A} + \frac{1}{B} + \frac{1}{C}) = 1$

Substitute $A$ , $B$ and $C$ ,

$x \times (\frac{1}{48} + \frac{5}{144} + \frac{1}{144}) = 1$

$x \times \frac{3 + 5 + 1}{144} = 1$

$x \times \frac{9}{144} = 1$

$x \times \frac{1}{16} = 1$

$x = 16$

That's the answer!

First, let's see the statement this way: Since Alex and Bella can paint a fence in 18 days, we can assume that on average, they can paint 1/18 part of the fence each day. By this logic, we can also say that Bella and Charlie can paint 1/24 part and Alex and Charlie can paint 1/36 part each day.

Let

a = part of the fence Alex can paint ALONE each day

b = part of the fence Bella can paint ALONE each day

c = part of the fence Charlie can paint ALONE each day

a + b = 1/18

b + c = 1/24

a + c = 1/36

Solve for a, b, c using Substitution and Elimination Method we will get:

a = 3/144

b = 5/144

c = 1/144

So, when Alex, Bella and Charlie work together, they can finish:

a + b + c = 9/144 = 1/16 part of the fence each day

Hence, it will take them 16 days to complete painting the whole fence

Since Alex and Bella can do $\frac {1}{18}$ of the work in a day, Bella and Charlie can do $\frac {1}{24}$ of the work in a day, and Charlie and Alex can do $\frac {1}{36}$ of the work in a day, if you add these equations you get $2(A + B + C)=\frac {1}{18} + \frac {1}{24} + \frac {1}{36}$ , Alex, Bella and Charlie can do it together in $\frac {1}{18} * \frac {1}{2}$ in a day which means that together the can do all the work in 16 days.

Let:

The days needed for Alex to paint a fence alone is A

The days needed for Bella to paint a fence alone is B

The days needed for Charlie to paint a fence alone is C

Therefore we can set up the following simultaneous questions:

$\frac{1}{A}$ + $\frac{1}{B}$ = $\frac{1}{18}$ ... (1)

$\frac{1}{B}$ + $\frac{1}{C}$ = $\frac{1}{24}$ ... (2)

$\frac{1}{A}$ + $\frac{1}{C}$ = $\frac{1}{36}$ ... (3)

(1)+(2)+(3):

$\frac{1}{A}$ + $\frac{1}{B}$ + $\frac{1}{B}$ + $\frac{1}{C}$ + $\frac{1}{A}$ + $\frac{1}{C}$ = $\frac{1}{18}$ + $\frac{1}{24}$ + $\frac{1}{36}$

2( $\frac{1}{A}$ + $\frac{1}{B}$ + $\frac{1}{C}$ )= $\frac{1}{8}$

$\frac{1}{A}$ + $\frac{1}{B}$ + $\frac{1}{C}$ = $\frac{1}{16}$

Therefore it takes 16 days for all three of them to paint a fence together.

let, Alex, Bella and Charlie can complete the work singly in A, B and C days respectively. $$ $A^{-1}+B^{-1}=18^{-1}$ $B^{-1}+C^{-1}=24^{-1}$ $C^{-1}+A^{-1}=36^{-1}$ $----------------$ $=>2(A^{-1}+B^{-1}+C^{-1})=8^{-1}$ $=>(A^{-1}+B^{-1}+C^{-1})=16^{-1}$ so, 16 days

One day's work of AB & BC is added and from the sum one days work of AC is subtracted this gives one days work of 2B. This is halved to give one days work of B. It is 5/144. This is added to one days work of AC ie 1/36. This gives 1/16. This is one day's work of A,B and C thus they will complete the work in 16 days.

I did it all in my head, quite simply. If you add up all the people and their jobs done in 36 days, you find that Alex and Bella paint 2 fences in this time, Bella and Charlie paint 1.5, and Charlie and Alex paint 1. So theoretically all 6 of them would paint 4.5 fences in 36 days. But the question is asking for only one of each person so multiply the time it takes by 2 to compensate: 4.5 fences by the trio in 72 days. For the time taken for one fence, simply divide 72 by 4.5 and get $16$ .

A and B paint 1/18 of fence in 1 day. B and C paint 1/24 of fence in 1 day. C and A paint 1/36 of fence in 1 day. Adding the above, we get A, B and C will paint 9/144 of fence in 1 day. Hence they will take 144/9 = 16 days to paint the fence.

A + B will complete 1/18 the work in 1 day
B + C will complete 1/24 the work in 1 day
C + A will complete 1/36 the work in 1 day
2A+2B+2C will complete (1/18 +1/24+1/36) the work in 1 day
i.e 1/8 the work in 1 day
A+B+C will complete 1/16 the work in 1 day
i.e A+B+C will complete the work in 16 days

add all of them up 2*(1/a+a/b+1/c)= 1/8 (1/a+a/b+1/c)= 1/16 So answer is 16 days when all work together

Since the work of any two of them out of three are given at a time we need to multiply their work by $2$ to add them.

Total work done:

$2 (A + B + C) = \frac{1}{18}+\frac{1}{24}+\frac{1}{36}= \frac{9}{72} = \frac{1}{8}$

Then: $A + B + C = \frac{\frac{1}{8}}{2}=\frac{1}{8}\times\frac{1}{2}=\frac{1}{16}$

Reciprocate it to get the no. of days in which the job would be done $= \boxed{16}$ days

(1/A+1/B)=1/18 (1/B+1/C)=1/24 (1/C+1/A)=1/36 ... We are asked to find How many days it would take to complete a work when all three works together i.e

(1/A+1/B+1/C)N=1 Where N is no.of days took for 3 members to complete the task. On solving first 3 equations we will get (1/A+1/B+1/C) =9/144

Hence 9N/144=1 N=16.

a and b one day work =>1/18,similarly b and c on day work=>1/24,c&a=>1/36 hence, a,b and c's one day work=>1/a+1/b+1/b+1/c+1/c+1/a=1/18+1/24+1/36 =>2(1/a+1/b+1/c)=1/8 =>1/a+1/b+1/c=1/16 hence,a,b&c can paint the fence in 16 days

'

First we get that: $\frac{f}{A+B}=18$ $\frac{f}{B+C}=24$ $\frac{f}{A+C}=36$ This converts to: $\frac{f}{18}=A+B$ $\frac{f}{24}=B+C$ $\frac{f}{36}=A+C$ We now add all the equations up and get: $\frac{f}{8}=2(A+B+C)$ $\frac{f}{16}=A+B+C$ Thus the time it will take them to paint the whole fence is $\frac{f}{\frac{f}{16}}=\boxed{16}$ .

Reciprocals everywhere. A+B=18. B+C=24. C+A=36. Adding all, 2(A+B+C)=8. A+B+C=16.

step 1- take assign Alex,Bella and charlie a variable each let Alex be A,Bella be B and Charlie be C step2- we know that alex and bella can paint the fence in 18 days so in 1 day the can paint 1/18 of the fence i.e A+B = 1/18 step3 - similiar for Bella and charlie ..... B+C = 1/ 24 PART OF FENCE step4- similiar for Alex and Charlie ....... A+C = 1/36 part of fence step 5- add the three together then we will get 2(A+B+C) = 1/24 + 1/18 + 1/36 step 6 solve the equation for A+B+C and that is the answer

If A and B can do a work in x days, B and C can do a work in y days and C and A can do that same work in z days, then all of them working together will complete the work in 2xyz/(xy+yz+zx)..

Alex=A,Bella=B,Charlie=C. 1/A+1/B=1/18, 1/B+1/C=1/24, 1/C+1/A= 1/36. Therefore , 1/A+1/B+1/C=9/144. A+B+C = 16.

A = ALEX, B= BELLA, C= CHARLIE

1/A+1/B=1/18

1/B+1/C=1/24

1/C+1/A=1/36

2(1/A+1/B+1/C)=1/18+1/24+1/36

1/A+1/B+1/C=1/16

Alex = A Bella = B Charlie = C

We want all of them at the same time but we only have two of each working together at the same time. So we just add them all together then divide by 2.

A + B does $\frac{1}{18}$ per day. B + C does $\frac{1}{24}$ per day. C + A does $\frac{1}{36}$ per day.

2(A + B + C) = $\frac{1}{18}$ + $\frac{1}{24}$ + $\frac{1}{36}$ = $\frac{1}{8}$ A + B + C = $\frac{1}{8}$ / 2 = $\boxed{ \frac{1}{16}}$

Let us take Alex, Bella and Charlie as A,B, C resp.

A and B can do the work in 18 days, so in 1 day, they can complete $\frac{1}{18}$ part of work.

B and C can do the work in 24 days, so in 1 day, they can complete $\frac{1}{24}$ part of work.

C and A can do the work in 36 days, so in 1 day, they can complete $\frac{1}{36}$ part of work.

So, (A and B)+(B and C)+(C and A) can together do in 1 day $= \frac{1}{18}+\frac{1}{24}+\frac{1}{36}=\frac{4+3+2}{72}=\frac{9}{72}=\frac{1}{8}$ part of work.

$\implies 2(A+B+C)$ do in 1 day together = $\frac{1}{8}$ part of work

$\implies$ A, B and C do in 1 day together $=\frac{1}{8\times 2} =\frac{1}{16}$ part of work.

So, they complete the total work together in $=\frac{1}{1/16} = \boxed{16}$ days.

A + B = 18

B + C = 24

A + C = 36

mean that if charlie join that paint so it must be fast 2 days

so A + B + C = 18 - 2 = $\boxed{16}$

assume that the job is x, Alex has rate a to finish the job, Bella has rate b to finish the job, and Charlie has rate c to finish the job,

so, $\frac{x}{a + b}$ = 18 ; $\frac{x}{b + c}$ = 24 ; $\frac{x}{a + c}$ = 36

then, a + b = $\frac{x}{18}$ ; b + c = $\frac{x}{24}$ ; a + c = $\frac{x}{36}$

(a+b) + (b+c) + (a+c) = 2.(a+b+c) = $\frac{x}{18}$ + $\frac{x}{24}$ + $\frac{x}{36}$

2.(a+b+c) = $\frac{9x}{72}$

$\frac{2 * 72}{9}$ = $\frac{x}{a + b + c}$

$\frac{x}{a + b + c}$ = 16

so, Alex, Bella, and Charlie can paint a fence together in 16 days

1/Alex+1/Bella=1/18 1/Charlie+1/Bella=1/24 1/Charlie+1/Alex=1/36 1/Alex+1/Bella+1/Charlie=1/16