Question

A and B can complete a task together in 12 days, while B and C can finish it in 16 days. After A works on it for 5 days and B for 7 days, C takes the remaining 13 days to finish the work. How many days would it take for C to complete the work alone?

• A. 24 days
• B. 16 days
• C. 20 days
• D. 18 days

Answer 1

Answer:

A. 24 days

Let the daily work rates of A, B, and C be (a), (b), and (c) respectively.

Step 1: Form equations from the given data

A and B together finish the work in 12 days:

$a+b=\frac{1}{12}$

B and C together finish the work in 16 days:

$b+c=\frac{1}{16}$

Also, A works for 5 days, B for 7 days, and C for 13 days to complete the whole work:

$5a+7b+13c=1$

Step 2: Express (a) and (c) in terms of (b)

From the first two equations:

$a=\frac{1}{12}-b$
$c=\frac{1}{16}-b$

Substitute into the third equation:

$5\left(\frac{1}{12}-b\right)$
$+7b$
$+13\left(\frac{1}{16}-b\right)=1$
$\frac{5}{12}-5b+7b+\frac{13}{16}-13b=1$
$\frac{20+39}{48}-11b=1$
$\frac{59}{48}-11b=1$
$11b=\frac{59}{48}-1=\frac{11}{48}$
$b=\frac{1}{48}$

Then

$c=\frac{1}{16}-\frac{1}{48}$
$=\frac{3-1}{48}$
$=\frac{2}{48}$
$=\frac{1}{24}$

Step 3: Find C's time alone

$\text{Time taken by C alone}=\frac{1}{c}=24$
$\boxed{24\text{ days}}$

So, C alone can complete the task in 24 days.