Welcome to the World of Decimals!
In this chapter, we are exploring Decimal Fractions. You see decimals every single day—especially when you are dealing with money (like £1.50) or measuring your height (like 1.65m). Understanding decimals is like having a "maths magnifying glass" because they allow us to work with parts of numbers that are smaller than a whole. Don't worry if you find them a bit "fiddly" at first; we will break them down step-by-step!
1. Understanding Place Value
Before we do any math, we need to know where we are! The decimal point is the anchor. It separates the whole numbers (on the left) from the parts (on the right).
The Place Value Table:
... Hundreds | Tens | Units . Tenths | Hundredths | Thousandths ...
... 100 | 10 | 1 . \( \frac{1}{10} \) | \( \frac{1}{100} \) | \( \frac{1}{1000} \) ...
Example: In the number 4.25, the '2' is in the tenths column and the '5' is in the hundredths column.
Quick Review: Every time you move one space to the right after the decimal point, the value gets 10 times smaller!
2. Converting Decimals and Fractions (2.02a)
Decimals and fractions are like different languages saying the same thing. We need to be able to "translate" between them.
A. Decimal to Fraction
This is all about Place Value. Say the number out loud in your head using the column names!
Step 1: Identify the place value of the last digit.
Step 2: Write the digits as the numerator (top) and the place value as the denominator (bottom).
Step 3: Simplify the fraction if you can.
Example: Convert 0.4 to a fraction.
The '4' is in the tenths column. So, it is \( \frac{4}{10} \).
Simplified, it becomes \( \frac{2}{5} \).
B. Fraction to Decimal (The "Bus Stop" Method)
To turn a fraction into a decimal, remember that the fraction bar just means "divide." To turn \( \frac{1}{4} \) into a decimal, you do \( 1 \div 4 \).
1. Terminating Decimals: These are decimals that end (like 0.5 or 0.25).
2. Recurring Decimals: These go on forever in a pattern (like \( 0.333... \)). We show this with a dot over the repeating digit, like \( 0.\dot{3} \).
Did you know? The fraction \( \frac{1}{7} \) creates a repeating pattern of six digits: \( 0.\dot{1}4285\dot{7} \)!
Higher Tier Only: Recurring Decimals to Fractions
If you see a recurring decimal like \( 0.\dot{4}\dot{1} \), you can convert it back to an exact fraction. A quick trick for simple cases is to put the repeating digits over the same number of 9s.
Example: \( 0.\dot{4}\dot{1} = \frac{41}{99} \)
Key Takeaway: All simple fractions can be written as either a terminating or recurring decimal.
3. Adding and Subtracting Decimals (2.02b)
The golden rule here is: Line up the dots!
If the decimal points are lined up in a vertical column, the rest is just normal addition or subtraction. If one number has fewer digits, just add placeholder zeros at the end.
Example: \( 12.5 - 3.24 \)
12.50 (Added a placeholder zero!)
- 03.24
----------
09.26
Common Mistake: Forgetting to line up the decimal points. Never line up the digits from the right-hand side like you do with whole numbers!
4. Multiplying Decimals (2.02b)
Multiplying decimals can feel scary, but here is a simple trick to make it easy:
Step 1: Ignore the decimal points and multiply the numbers as if they were whole numbers.
Step 2: Count how many digits were behind the decimal points in the original question.
Step 3: Put the decimal point back into your answer so it has that same number of digits behind it.
Example: \( 0.3 \times 0.07 \)
1. Treat it as \( 3 \times 7 = 21 \).
2. Count the decimal places: 0.3 (1 place) and 0.07 (2 places). Total = 3 places.
3. Move the point 3 places left in 21: 0.021.
Memory Aid: "Multiply first, count the jumps later!"
5. Division of Decimals (2.02c)
A. Dividing a Decimal by a Whole Number
This is just like normal division. Just keep the decimal point in the answer exactly above the decimal point in the question.
Example: \( 0.24 \div 6 \)
How many 6s go into 0? (0).
How many 6s go into 2? (0, carry the 2).
How many 6s go into 24? (4).
Answer: 0.04.
B. Dividing by a Decimal (Foundation/Higher)
It is very hard to divide by a decimal, so we change the question to make it easier! We use equivalent fractions to make the divisor (the number you are dividing by) a whole number.
Example: \( 0.3 \div 0.6 \)
1. Write it as a fraction: \( \frac{0.3}{0.6} \).
2. Multiply top and bottom by 10 to get rid of the decimals: \( \frac{3}{6} \).
3. Now divide: \( 3 \div 6 = 0.5 \).
Key Takeaway: Always "Shift the decimal" for both numbers until the number you are dividing by is a whole number.
6. Ordering Decimals (2.04a)
To see which decimal is bigger, it helps to make them all the same length by adding placeholder zeros.
Question: Order these from smallest to largest: 0.7, 0.72, 0.09
1. Add zeros: 0.70, 0.72, 0.09
2. Now it’s easy to see: 0.09 is the smallest, then 0.70, then 0.72.
Summary Checklist:
- Can I identify tenths, hundredths, and thousandths?
- Can I use the "Bus Stop" method for fractions?
- Do I remember to "Line up the dots" for adding/subtracting?
- Can I count decimal places for multiplication?
- Can I make the divisor a whole number for division?