A alone can finish the work in 15 hours and B alone can finish the work in 20 hours while C alone can finish the work in N hours. All three worked for 5 hours, after that A left the work and remaining work gets completed in 3 hours. Find the value of N.

The correct answer is Option 1: 30.

Step-by-step Explanation:

Let's find the rate of work done per hour by each individual.

Time taken by A alone to finish the work = 15 hours.
Work done by A in 1 hour = $\frac{1}{15}$

Time taken by B alone to finish the work = 20 hours.
Work done by B in 1 hour = $\frac{1}{20}$

Time taken by C alone to finish the work = N hours.
Work done by C in 1 hour = $\frac{1}{\mathrm{N}}$

Work done in the first 5 hours:
All three (A, B, and C) worked together for 5 hours.

Combined work done by A, B, and C in 1 hour = $\frac{1}{15}+\frac{1}{20}+\frac{1}{\mathrm{N}}$

Work done by all three in 5 hours = $5\times \left(\frac{1}{15}+\frac{1}{20}+\frac{1}{\mathrm{N}}\right)$

Simplifying the fraction inside the parentheses:
$\frac{1}{15}+\frac{1}{20}=\frac{4+3}{60}=\frac{7}{60}$

So, work done in 5 hours by A, B, and C = $5\times \left(\frac{7}{60}+\frac{1}{\mathrm{N}}\right)=\frac{35}{60}+\frac{5}{\mathrm{N}}=\frac{7}{12}+\frac{5}{\mathrm{N}}$

Work done in the remaining 3 hours:
After 5 hours, A left the work. The remaining work was completed by B and C working together for 3 hours.

Combined work done by B and C in 1 hour = $\frac{1}{20}+\frac{1}{\mathrm{N}}$

Work done by B and C in 3 hours = $3\times \left(\frac{1}{20}+\frac{1}{\mathrm{N}}\right)=\frac{3}{20}+\frac{3}{\mathrm{N}}$

Setting up the total work equation:
Total Work = (Work done in first 5 hours) + (Work done in next 3 hours) = 1

$\left(\frac{7}{12}+\frac{5}{\mathrm{N}}\right)+\left(\frac{3}{20}+\frac{3}{\mathrm{N}}\right)=1$

Grouping the terms together:
$\left(\frac{7}{12}+\frac{3}{20}\right)+\left(\frac{5}{\mathrm{N}}+\frac{3}{\mathrm{N}}\right)=1$

Find a common denominator for $\frac{7}{12}$ and $\frac{3}{20}$ (which is 60):
$\frac{35}{60}+\frac{9}{60}=\frac{44}{60}=\frac{11}{15}$

Substituting this back into the equation:
$\frac{11}{15}+\frac{8}{\mathrm{N}}=1$

$\frac{8}{\mathrm{N}}=1-\frac{11}{15}$

$\frac{8}{\mathrm{N}}=\frac{4}{15}$

Cross-multiplying to solve for N:
$4\times \mathrm{N}=8\times 15$

$4\mathrm{N}=120$

$\mathrm{N}=\frac{120}{4}=30$

Thus, the value of N is 30.