A can complete the work in 7 days, so A’s work rate is $\frac{1}{7}\backslash frac\{1\}\{7\}$71​ of the work per day.
B can complete it in 14 days, so B’s work rate is $\frac{1}{14}\backslash frac\{1\}\{14\}$141​ of the work per day.

When A and B work together for 2 days:

$Combined work rate=\frac{1}{7}+\frac{1}{14}=\frac{2}{14}+\frac{1}{14}=\frac{3}{14}\backslash text\{Combined\; work\; rate\}\; =\; \backslash frac\{1\}\{7\}\; +\; \backslash frac\{1\}\{14\}\; =\; \backslash frac\{2\}\{14\}\; +\; \backslash frac\{1\}\{14\}\; =\; \backslash frac\{3\}\{14\}$Combined work rate=71​+141​=142​+141​=143​In 2 days, they complete:

$Work done=2\times \frac{3}{14}=\frac{6}{14}=\frac{3}{7}\backslash text\{Work\; done\}\; =\; 2\; \backslash times\; \backslash frac\{3\}\{14\}\; =\; \backslash frac\{6\}\{14\}\; =\; \backslash frac\{3\}\{7\}$Work done=2×143​=146​=73​Remaining work:

$1-\frac{3}{7}=\frac{4}{7}1\; –\; \backslash frac\{3\}\{7\}\; =\; \backslash frac\{4\}\{7\}$1−73​=74​Let C’s work rate be $\frac{1}{x}\backslash frac\{1\}\{x\}$x1​. Together, A, B, and C complete the remaining work in 1 day:

$\frac{3}{14}+\frac{1}{x}=1\backslash frac\{3\}\{14\}\; +\; \backslash frac\{1\}\{x\}\; =\; 1$143​+x1​=1This simplifies to:

$\frac{1}{x}=1-\frac{3}{14}=\frac{11}{14}\backslash frac\{1\}\{x\}\; =\; 1\; –\; \backslash frac\{3\}\{14\}\; =\; \backslash frac\{11\}\{14\}$x1​=1−143​=1411​Thus,

$x=\frac{14}{11}x\; =\; \backslash frac\{14\}\{11\}$x=1114​Therefore, C can complete the work alone in $\frac{14}{11}\backslash frac\{14\}\{11\}$1114​ days, or approximately 1.27 days