# Problem of the Week Problem B and Solution Rounding Equivalents

## Problem

Sometimes the process of rounding numbers produces interesting results. For example, if you round the number $$39.99$$ to the nearest ten, you get $$40$$, if you round it to the nearest whole number, you get $$40$$, and if you round it to the nearest tenth, you get $$40.0$$. Notice that you get the same numerical value when rounding $$39.99$$ to the nearest ten, whole number, and tenth.

1. Find a number less than $$100$$ with two decimal places such that when you round to the nearest tenth you get the same numerical value as when you round to the nearest whole number.

2. Find a number less than $$100$$ with two decimal places such that when you round to the nearest tenth you get the same numerical value as when you round to the nearest ten.

3. Find the smallest number between $$99$$ and $$100$$ that has two decimal places that rounds to the same numerical value when you round to the nearest tenth, whole number, ten, and hundred.

## Solution

1. Answers will vary. One possible answer is $$18.96$$.
Rounding $$18.96$$ to the nearest tenth yields $$19.0$$.
Rounding $$18.96$$ to the nearest whole number yields $$19$$.

2. Answers will vary. One possible answer is $$20.03$$.
Rounding $$20.03$$ to the nearest tenth yields $$20.0$$.
Rounding $$20.03$$ to the nearest ten yields $$20$$.

3. When the number is between $$99$$ and $$100$$, it must be $$100$$ when rounded to the nearest hundred.
Therefore, the number rounded to the nearest tenth would be $$100.0$$. The numbers less than $$100$$ that have two decimal places that round to $$100.0$$ when rounded to the nearest tenth are $99.99, ~99.98, ~99.97, ~99.96 \mbox{ and } 99.95$ (Note that $$99.94$$ will round to $$99.9$$ when rounded to the nearest tenth.) Therefore, the smallest of these numbers is $$99.95$$.
Notice that $$99.95$$ does indeed yield $$100.0$$ or $$100$$ when rounded to the nearest tenth, whole number, ten, or hundred.