#
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.

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.

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.

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

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 \(19\).

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 \(20\).

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.