# Problem of the Week

Problem A and Solution

Counting Collections

## Problem

The students in Riverside Public School love to play a game called
“Counting Collections”. In this game, the students collect objects, and
the student who has collected the most objects wins.

Santosh has collected \(262\) erasers, \(451\) buttons and \(173\) pencils. Alyssa has collected \(489\) straws and \(446\) rocks.

To estimate who collected the most objects, they round the total
number of each type of object, and then use the rounded numbers to
calculate the total for each player. Santosh rounds all the numbers to
the nearest \(100\). Alyssa rounds all
the numbers to the nearest \(10\).

Based on his rounded calculation, who does Santosh think won the
game?

Based on her rounded calculation, who does Alyssa think won the
game?

Who is correct? Justify your answer.

## Solution

Rounding to the nearest \(100\),
Santosh collected approximately \(300 + 500 +
200 = 1000\) objects and Alyssa collected approximately \(500 + 400 = 900\) objects. So Santosh
thinks that he is the winner.

Rounding to the nearest \(10\),
Santosh collected approximately \(260 + 450 +
170 = 880\) objects and Alyssa collected approximately \(490 + 450 = 940\) objects. So Alyssa thinks
that she is the winner.

The actual totals are \(262 + 451 + 173
= 886\) for Santosh and \(489 + 446 =
935\) for Alyssa. So Alyssa collected more objects and is
therefore the winner. The estimation when rounding to the nearest \(10\) is more accurate than when rounding to
the nearest \(100\).

**Teacher’s Notes**

When we use rounding to estimate the results of calculated values it
is important to remember that this is an approximation of the actual
result. Estimations can be valuable, especially in knowing when an
answer is reasonable or unreasonable. However, there is a margin of
error when we use rounding.

In this problem, when we see estimated totals that differ by \(100\) when the numbers were rounded to the
nearest \(100\), we should not assume
that we have enough information to make a conclusion about which actual
total is greater. Rounding to the nearest \(10\) gives us a better estimation, but it
is still an approximation. However, given that we only have five numbers
in our calculations, and the difference between our estimated totals is
more than \(5 \times 10\), we should
have more confidence that our conclusion is correct in this case.