KS4. Number. Decimals

In order to use decimals, we need to understand the concept of face value and place value. When we look at the number 234.912, what is the place value and the face value of the digit 1? How would we write this digit as a fraction?

Remember that a digit that is one place further to the left in a number is 10 times as large. This applies regardless of whether it is to the left or to the right of the decimal point.

Examples

(1.) Put the following numbers in order: 0.09, 0.101, 0.029

(2.) Increase each of the numbers above by $\frac{3}{100}$

Exercise

Let’s check our knowledge of this by completing exercise 4 on pages 78 and 79 of the textbook:

The answers are below:

Challenging Question:

Which of the following recurring decimals is closest to 1?

Answer

Scales

We need to be able to identify the value of a number on a scale, where only the start and end-points are marked with a number, based on the number of steps the scale is broken up into.

Example

Exercise

Let’s complete exercise 5 on page 79 of the textbook:

The answers are below:

Multiplying and Dividing Decimals by 10, 100, 1000, etc.

Multiplying and decimals by 10, 100, 1000, etc. (powers of 10) is straightforward, as it simply moves all of the digits to the left or the right by a certain number of places. The decimal point is fixed and never moves. Let’s try these first with the teacher for a few examples.

Exercise

Let’s complete exercise 6 from page 80 of the textbook:

The answers are below:

Adding & Subtracting Decimals

Adding and subtracting decimals is done in the same was as adding and subtracting integers. We simply line up the numbers so that the corresponding places are in columns (i.e. the 10s, the 1s, the 1/10s, the 1/100s), and then add or subtract using our standard methods. Let’s try a few first with the teacher.

Exercise

Let’s complete exercise 7 on page 81 of the textbook:

Extra question (without answer given)

2 numbers added together give 1 and multiplied together give 0.09. What are the 2 numbers?

The answers are below:

Multiplying and Dividing Decimals

To multiply two decimals together, we can first multiply each of them by whichever of 10, 100, 1000, etc. is necessary to make them into integers. Then we multiply them together using our normal ways. At then end we must divide them back by whatever we multiplied them in the beginning.

For instance, if we had 8.3 x 9.21, we could first multiply the 8.3 by 10 to give 83 and the 9.21 by 100 to give 921. Once we had multiplied 83 by 921 we would then have to divide by 1000 to get our answer. Why by 1000?

Examples: (a) 0.8 x 0.2, (b) 0.4 x 0.0

To divide two decimals is essentially easier. We can multiply each of them by the same number from 10, 100, 1000, etc. Then when we have finished we don’t need to divide that number back out, because the answer will already be correct. This effectively relies upon the fact that 100a/100b = a/b. (Why?)

Examples: (a) 9.36 ÷ 0.4, (b) 0.0378 ÷ 0.07

Exercises

Let’s complete exercises 8 and 9 from pages 81 and 82 of the textbook: