Of a collection of mango fruits, the king took 1/6; the queen took 1/5 of the remainder, and three chief princes took 1/4, 1/3, 1/2 of what remained at each step; and the youngest child took the remaining three mangoes. O you, who are clever in working miscellaneous problems on fractions, give out the measure of that collection of mangoes.

We are tasked with determining the total number of mangoes in a collection based on a series of fractional deductions taken by different people. The strategy is to work backward from the information that the youngest child took the remaining 3 mangoes.

### Steps:

1. **Let \( x \) be the total number of mangoes** in the collection.

2. **The king takes \( \frac{1}{6} \) of the total:**
   - After the king takes \( \frac{1}{6} \), the remaining mangoes are \( \frac{5}{6} \) of the total.
   - The remaining mangoes = \( \frac{5}{6}x \).

3. **The queen takes \( \frac{1}{5} \) of the remainder:**
   - After the queen takes \( \frac{1}{5} \) of \( \frac{5}{6}x \), she leaves \( \frac{4}{5} \) of the remaining.
   - The remaining after the queen = \( \frac{4}{5} \times \frac{5}{6}x = \frac{4}{6}x = \frac{2}{3}x \).

4. **The first chief prince takes \( \frac{1}{4} \) of what’s left:**
   - After the first prince takes \( \frac{1}{4} \) of \( \frac{2}{3}x \), the remaining is \( \frac{3}{4} \) of that.
   - The remaining after the first prince = \( \frac{3}{4} \times \frac{2}{3}x = \frac{1}{2}x \).

5. **The second chief prince takes \( \frac{1}{3} \) of the remaining:**
   - After the second prince takes \( \frac{1}{3} \) of \( \frac{1}{2}x \), the remaining is \( \frac{2}{3} \) of that.
   - The remaining after the second prince = \( \frac{2}{3} \times \frac{1}{2}x = \frac{1}{3}x \).

6. **The third chief prince takes \( \frac{1}{2} \) of the remaining:**
   - After the third prince takes \( \frac{1}{2} \) of \( \frac{1}{3}x \), the remaining is \( \frac{1}{2} \) of that.
   - The remaining after the third prince = \( \frac{1}{2} \times \frac{1}{3}x = \frac{1}{6}x \).

7. **The youngest child takes the remaining 3 mangoes:**
   - The remaining after all the deductions is \( \frac{1}{6}x \), and we know this equals 3.
   - So, \( \frac{1}{6}x = 3 \).

8. **Solve for \( x \):**
   - Multiply both sides of the equation by 6 to get the total number of mangoes.
   \[
   x = 6 \times 3 = 18.
   \]

Thus, the total number of mangoes in the collection is \( \boxed{18} \).

Would you like more details on any step or have any questions?

Here are some related problems to explore:
1. What if the king had taken \( \frac{1}{5} \) of the total instead?
2. How would the problem change if the queen took \( \frac{1}{6} \) of the remainder instead?
3. What happens if the third chief prince takes \( \frac{2}{3} \) of the remaining instead of \( \frac{1}{2} \)?
4. How would the solution change if the youngest child took 5 mangoes instead of 3?
5. What if the order of taking mangoes were reversed?

**Tip:** When solving problems involving fractions of a whole, it's often easier to work backward from the final known quantity.

This detailed guide explains how to solve a mango distribution problem using fractions and algebra. Learn how to calculate the total number of mangoes after successive fractional deductions by the king, queen, princes, and a final remainder taken by the youngest child. Suitable for Grades 6-8 students.

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