# 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 $$10$$.

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

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

## Solution

1. 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.

2. 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.

3. 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.