Ann is grouping 38 rocks. She can put them into groups of 10 rocks or as single Rocks

Ann is grouping 38 rocks. She can put them into groups of 10 rocks or as single Rocks. What are the different ways Ann can group the rock?

The correct answer and explanation is:

Correct Answer:

Ann can group the 38 rocks using groups of 10 rocks plus some leftover single rocks. The possible ways are:

• 3 groups of 10 rocks + 8 single rocks
• 2 groups of 10 rocks + 18 single rocks
• 1 group of 10 rocks + 28 single rocks
• 0 groups of 10 rocks + 38 single rocks

Explanation (300 words):

Ann has 38 rocks and wants to group them either in groups of 10 or as single rocks. This is a classic example of dividing a number into multiples of a fixed group size plus a remainder.

Since each group has exactly 10 rocks, we want to find how many groups of 10 she can make, and then how many rocks will remain as single rocks.

1. Maximum groups of 10:
   The largest number of groups of 10 rocks is found by dividing 38 by 10: max groups=⌊3810⌋=3\text{max groups} = \lfloor \frac{38}{10} \rfloor = 3 So, Ann can have at most 3 groups of 10.
2. Finding the leftover rocks:
   After forming these groups, the leftover rocks are the remainder when dividing 38 by 10. leftover=38−(10×number of groups)\text{leftover} = 38 – (10 \times \text{number of groups})
3. All possible groupings:
   • If she makes 3 groups of 10 rocks, that uses 30 rocks, so leftover rocks = 38 – 30 = 8 single rocks.
   • If she makes 2 groups, leftover rocks = 38 – 20 = 18 single rocks.
   • If she makes 1 group, leftover rocks = 38 – 10 = 28 single rocks.
   • If she makes 0 groups, leftover rocks = all 38 rocks are single.

Thus, the different ways to group the rocks are combinations of groups of 10 and single rocks with leftover counts as listed.

This problem is an example of expressing a number as n×10+rn \times 10 + r, where nn is the number of groups and rr is the remainder or leftover single rocks. It’s useful for understanding division with remainders, modular arithmetic, or just practical grouping in real life.

If Ann wants only full groups of 10 and singles (no partial groups), these are the only possible ways to group 38 rocks.