“tom, peter and john shared a box of sweets. tom took 10 more than \frac{1}{3} of the total number of sweets. peter took 9 more than \frac{1}{2} of the remaining sweets. john took the last 5 sweets. how many sweets were there in the box?”

Let's denote the total number of sweets in the box as \( x \).

1. **Tom's Share**:
   Tom took 10 more than \( \frac{1}{3} \) of the total sweets. Therefore, Tom's share can be written as:
   \[
   T = \frac{1}{3}x + 10
   \]

2. **Remaining Sweets after Tom's Share**:
   After Tom took his share, the remaining sweets are:
   \[
   \text{Remaining sweets} = x - \left( \frac{1}{3}x + 10 \right) = \frac{2}{3}x - 10
   \]

3. **Peter's Share**:
   Peter took 9 more than \( \frac{1}{2} \) of the remaining sweets. So, Peter's share is:
   \[
   P = \frac{1}{2} \left( \frac{2}{3}x - 10 \right) + 9
   \]
   Simplifying Peter's share:
   \[
   P = \frac{1}{3}x - 5 + 9 = \frac{1}{3}x + 4
   \]

4. **Remaining Sweets after Peter's Share**:
   After Peter took his share, the remaining sweets are:
   \[
   \text{Remaining sweets} = \left( \frac{2}{3}x - 10 \right) - \left( \frac{1}{3}x + 4 \right)
   \]
   Simplifying:
   \[
   \text{Remaining sweets} = \frac{2}{3}x - 10 - \frac{1}{3}x - 4 = \frac{1}{3}x - 14
   \]

5. **John's Share**:
   We are told that John took the last 5 sweets, so:
   \[
   \frac{1}{3}x - 14 = 5
   \]

6. **Solving for \( x \)**:
   To find the total number of sweets \( x \), solve the equation:
   \[
   \frac{1}{3}x - 14 = 5
   \]
   Add 14 to both sides:
   \[
   \frac{1}{3}x = 19
   \]
   Multiply both sides by 3:
   \[
   x = 57
   \]

Thus, the total number of sweets in the box is \( 57 \).

Would you like a detailed breakdown of each step or further explanation?

Here are 5 related questions you can explore:

1. If Tom took 5 more sweets instead of 10, how many sweets would be in the box?
2. How would the solution change if John took 10 sweets instead of 5?
3. If the total number of sweets was a prime number, could the distribution still work out?
4. How would the solution be different if Tom took exactly \( \frac{1}{3} \) of the sweets without the extra 10?
5. What if Peter took \( \frac{1}{4} \) of the remaining sweets instead of \( \frac{1}{2} \)?

**Tip**: When working with fractions of totals, always break the problem into clear steps by keeping track of how much remains after each person's share.

tom, peter and john shared a box of sweets. tom took 10 more than \frac{1}{3} of the total number of sweets. peter took 9 more than \frac{1}{2} of the remaining sweets. john took the last 5 sweets. how many sweets were there in the box?

Discover how to solve a problem where Tom, Peter, and John share a box of sweets using algebraic techniques. This step-by-step solution covers concepts like fractions, linear equations, and logical reasoning, suitable for students in Grades 6-8.

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