Efficiency of Amit is 40\% less than Bharat and 20\% more than Chandra. If Amit and Bharat work together, they will take $$ \frac{1}{4} $$ days less than the time taken by Bharat and Chandra working together. Find in how many days Bharat can complete the work alone.

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**Solution:**

1. Let Bharat's efficiency be $ B $.

2. Amit's efficiency is 40\% less than Bharat's, so:
   $$
   A = B - 0.4B = 0.6B
   $$

3. Amit's efficiency is also 20\% more than Chandra's, so:
   $$
   $A = C + 0.2C = 1.2C \implies C = \frac{A}{1.2} = \frac{0.6B}{1.2} = 0.5B$
   $$

4. Time taken is inversely proportional to efficiency. Let the total work be 1 unit.

5. Time taken by Amit and Bharat working together:
   $$
   $T_{AB} = \frac{1}{A + B} = \frac{1}{0.6B + B} = \frac{1}{1.6B}$
   $$

6. Time taken by Bharat and Chandra working together:
   $$
  $ T_{BC} = \frac{1}{B + C} = \frac{1}{B + 0.5B} = \frac{1}{1.5B}$
   $$

7. Given that:
   $$
  $ T_{BC} - T_{AB} = \frac{1}{4}$
   $$

Substitute values:
   $$
  $ \frac{1}{1.5B} - \frac{1}{1.6B} = \frac{1}{4}$
   $$

8. Simplify the left side:
   $$
  $ \frac{1}{1.5B} - \frac{1}{1.6B} = \frac{1.6 - 1.5}{1.5 \times 1.6 \times B} = \frac{0.1}{2.4B} = \frac{1}{24B}$
   $$

9. So:
   $$
  $ \frac{1}{24B} = \frac{1}{4} \implies 24B = 4 \implies B = \frac{4}{24} = \frac{1}{6} $
   $$

10. Since efficiency $ $B = \frac{1}{6} $$, Bharat can complete the work alone in 6 days.

**Answer:** 6 days

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