## Question

What does the following paragraph mean?

*To improve the accuracy for measurement that is too small, measure a few together and then find the average. For example, if we are finding the density of a marble, we take the mass of eg 5 marbles and divide by volume of the 5 marbles – and not find the density of individual marbles. If we are finding the average diameter of a marble, we measure the length of eg 5 marbles in a straight line and then divide by 5 – and not find diameter of individual marble and take average.*

## Answer

When finding the measurement of a small quantity, it is more desirable to find the total quantity due to a few items and then divide to obtain the quantity of a single item.

For example, we have a stick which is 3.58854 cm long. When measuring the length using a ruler, we will obtain 3.6 cm because a ruler can only read to 0.1 cm.

However, if we take 10 identical sticks and join them end-to-end and measure, we can obtain a total length of 35.8 cm. By dividing, we can find the length of each stick.

We know that there are exactly 10 sticks, not 10.1 or 9.9, hence the value of 10 is infinitely accurate and precise. We can assume that it is correct to many significant figures. Hence, the limiting significant figures is due to the value 35.8.

Therefore, our final answer of 3.58 cm is to 3 significant figures.

**This method allows you to obtain an answer to more decimal places than the limit of the measuring instrument.**

We can proceed further and join 100 identical sticks together and get a value which is even closer to the true value of the length.

With 100 sticks, the total length measured would be 358.8 cm when using a measuring tape. Then, dividing to find the length of each stick,

However, once we join so many sticks together, there will be other factors contributing to an error in measurement. For example, we need to ensure that the sticks are identical, are joined in a straight line and there are no spaces between all the sticks.