To find out how long it will take A, B, and C to complete the work together, we start by determining their individual work rates:

• A can complete the work in 24 days, so A’s work rate = $\frac{1}{24}\backslash frac\{1\}\{24\}$241​ of the work per day.
• B can complete the work in 36 days, so B’s work rate = $\frac{1}{36}\backslash frac\{1\}\{36\}$361​ of the work per day.
• C can complete the work in 48 days, so C’s work rate = $\frac{1}{48}\backslash frac\{1\}\{48\}$481​ of the work per day.

Now, we find their combined work rate when all three work together:

$Combined work rate=\frac{1}{24}+\frac{1}{36}+\frac{1}{48}\backslash text\{Combined\; work\; rate\}\; =\; \backslash frac\{1\}\{24\}\; +\; \backslash frac\{1\}\{36\}\; +\; \backslash frac\{1\}\{48\}$Combined work rate=241​+361​+481​

Finding a common denominator (LCM of 24, 36, and 48 is 144):

• A’s work in 144 days = $\frac{6}{144}\backslash frac\{6\}\{144\}$1446​
• B’s work in 144 days = $\frac{4}{144}\backslash frac\{4\}\{144\}$1444​
• C’s work in 144 days = $\frac{3}{144}\backslash frac\{3\}\{144\}$1443​

Thus, the combined work rate is:

$\frac{6+4+3}{144}=\frac{13}{144} of the work per day\backslash frac\{6\; +\; 4\; +\; 3\}\{144\}\; =\; \backslash frac\{13\}\{144\}\; \backslash text\{\; of\; the\; work\; per\; day\}$1446+4+3​=14413​ of the work per day

In the first 4 days, all three work together:

$Work done in 4 days=4\times \frac{13}{144}=\frac{52}{144}=\frac{13}{36}\backslash text\{Work\; done\; in\; 4\; days\}\; =\; 4\; \backslash times\; \backslash frac\{13\}\{144\}\; =\; \backslash frac\{52\}\{144\}\; =\; \backslash frac\{13\}\{36\}$Work done in 4 days=4×14413​=14452​=3613​

This leaves:

$Remaining work=1-\frac{13}{36}=\frac{23}{36}\backslash text\{Remaining\; work\}\; =\; 1\; –\; \backslash frac\{13\}\{36\}\; =\; \backslash frac\{23\}\{36\}$Remaining work=1−3613​=3623​

Now, A and B continue working. Let $xx$x be the total days taken to complete the work. A leaves 3 days before the work is done, so they work for $x-3x\; –\; 3$x−3 days. Their combined work rate is:

$\frac{1}{24}+\frac{1}{36}=\frac{3+2}{72}=\frac{5}{72}\backslash frac\{1\}\{24\}\; +\; \backslash frac\{1\}\{36\}\; =\; \backslash frac\{3\; +\; 2\}\{72\}\; =\; \backslash frac\{5\}\{72\}$241​+361​=723+2​=725​

The equation for the remaining work is:

$(x-3)\times \frac{5}{72}=\frac{23}{36}(x\; –\; 3)\; \backslash times\; \backslash frac\{5\}\{72\}\; =\; \backslash frac\{23\}\{36\}$(x−3)×725​=3623​

Multiplying both sides by 72 to clear the fraction:

$5(x-3)=465(x\; –\; 3)\; =\; 46$5(x−3)=46

Solving for $xx$x:

$5x-15=46\hspace{0.25em} ⟹\hspace{0.25em} 5x=61\hspace{0.25em} ⟹\hspace{0.25em} x=\frac{61}{5}=12.2 days5x\; –\; 15\; =\; 46\; \backslash implies\; 5x\; =\; 61\; \backslash implies\; x\; =\; \backslash frac\{61\}\{5\}\; =\; 12.2\; \backslash text\{\; days\}$5x−15=46⟹5x=61⟹x=561​=12.2 days

Thus, the total time taken to complete the work is:

$4+12.2=16.2 days4\; +\; 12.2\; =\; 16.2\; \backslash text\{\; days\}$4+12.2=16.2 days

So, the work will be completed in approximately 16.2 days.