Decimals — division, terminating, non-terminating, recurring

In science, measurements almost never come out as neat whole numbers. When you read a thermometer, measure a liquid in a graduated cylinder, or record the mass of a sample on a balance, you are working with decimals every day.

Knowing how to divide decimals lets you:

• Calculate the average of repeated measurements (add them up, then divide)
• Share a total quantity equally between samples or trials
• Convert between units (e.g. dividing millilitres by 1000 to get litres)
• Analyse results and spot patterns in data

Before you calculate, it is always a good habit to estimate your answer first. For $32.5÷5$, you might think: "$30÷5=6$, so my answer should be a little more than $6$." Estimation helps you catch errors — if your calculator shows $65$, you know straight away something has gone wrong!

In this subtopic, you will learn how to divide decimals confidently and how to recognise whether the result is a terminating or recurring decimal — both of which appear regularly in scientific data.

Dividing a decimal by a whole number works just like dividing whole numbers — you just need to keep track of the decimal point.

Here are the key steps:

1. Divide each place value in turn, starting with the largest place value (left to right).
2. If a digit is too small to be divided on its own, write the remainder and bring down the next digit.
3. Place the decimal point in the answer directly above the decimal point in the question.
4. If needed, write extra zeros at the end of the decimal number you are dividing into.

Here is how the long division layout looks:

    6 . 5
  -------
5 ) 3 2 . 5
    3 0
    ---
      2 5
      2 5
      ---
        0

Notice how the decimal point in the answer sits directly above the decimal point in $32.5$. Always write the decimal point in your answer line before you start dividing — this way you cannot forget it.