Welcome to the World of Fractions, Decimals, and Percentages!

In this chapter, we are going to explore three different ways to talk about parts of a whole: Fractions, Decimals, and Percentages. You might see a sign in a shop saying "50% off," or follow a recipe that asks for \( \frac{1}{2} \) a cup of milk, or look at a price tag that says £2.50. These are all related!

By the end of these notes, you will see that these three are like "mathematical triplets"—they might look different, but they belong to the same family and represent the same values. Let's dive in!

Part 1: Fantastic Fractions

A fraction tells us how many parts of a whole we have. The top number is the Numerator (how many parts we have), and the bottom number is the Denominator (how many parts make the whole).

Simplifying Fractions

Sometimes fractions use big numbers, but we can make them simpler. This is called simplifying or cancelling down. To do this, we find a number that both the numerator and denominator can be divided by. This is called a common factor.

Example: Simplify \( \frac{8}{12} \)
1. Think of a number that divides into both 8 and 12. Let's use 4.
2. \( 8 \div 4 = 2 \)
3. \( 12 \div 4 = 3 \)
4. So, \( \frac{8}{12} \) simplified is \( \frac{2}{3} \).
Top Tip: Always try to find the Highest Common Factor to simplify in just one step!

Adding and Subtracting Fractions

Don't worry if this seems tricky at first! The secret rule is: The denominators MUST be the same.

If they aren't the same, we find a Common Denominator by finding a common multiple.
Example: \( \frac{1}{3} + \frac{1}{6} \)
1. We can turn the 3 into a 6 by multiplying by 2.
2. Remember: Whatever you do to the bottom, you must do to the top!
3. \( \frac{1 \times 2}{3 \times 2} = \frac{2}{6} \)
4. Now we can add: \( \frac{2}{6} + \frac{1}{6} = \frac{3}{6} \).
5. (Bonus point: \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \)!)

Multiplying Fractions

This is actually easier than adding! You don't need a common denominator.
Step 1: Multiply the numerators (top numbers).
Step 2: Multiply the denominators (bottom numbers).
Example: \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \)

Dividing Fractions by Whole Numbers

To divide a fraction by a whole number, we just multiply the denominator by that number.
Example: Imagine you have \( \frac{1}{2} \) of a pizza and you share it between 2 people.
\( \frac{1}{2} \div 2 = \frac{1}{4} \)
Memory Aid: If you divide your slice of cake, the pieces get smaller, so the denominator gets bigger!

Key Takeaway:

Always simplify your final answer if you can, and never add the denominators together when adding fractions!

Part 2: Dazzling Decimals

Decimals are another way of showing parts of a whole using a decimal point. In Year 6, we look at numbers up to three decimal places: tenths, hundredths, and thousandths.

Place Value Breakdown

In the number 0.125:
• 1 is in the tenths column (\( \frac{1}{10} \))
• 2 is in the hundredths column (\( \frac{2}{100} \))
• 5 is in the thousandths column (\( \frac{5}{1000} \))

Multiplying and Dividing by 10, 100, and 1000

When we multiply or divide by these numbers, the digits shift places.
• Multiply: Digits move to the left (the number gets bigger).
• Divide: Digits move to the right (the number gets smaller).
Example: \( 0.45 \times 10 = 4.5 \)
Example: \( 45 \div 100 = 0.45 \)

Multiplying Decimals

When multiplying a decimal by a whole number, you can pretend the decimal point isn't there, do the multiplication, and then put it back.
Example: \( 0.03 \times 7 \)
1. Think: \( 3 \times 7 = 21 \)
2. Since 0.03 has two decimal places, our answer must have two decimal places.
3. Answer: 0.21

Key Takeaway:

The decimal point never moves! It's the digits that shift positions around it.

Part 3: Powerful Percentages

The word Percent comes from "per centum," which means "for every hundred." So, 50% means 50 out of every 100.

Finding Percentages of Amounts

Here are some "cheat codes" for your mental maths:
• To find 50%: Divide by 2 (it's half!).
• To find 25%: Divide by 4 (half of a half).
• To find 10%: Divide by 10.
• To find 5%: Find 10% and then halve it.
• To find 1%: Divide by 100.

Example: Find 15% of 200.
1. 10% of 200 = 20
2. 5% of 200 = 10 (half of 10%)
3. 15% = 20 + 10 = 30.

Quick Review:

Did you know? Percentages are just fractions with a denominator of 100! 75% is the same as \( \frac{75}{100} \).

Part 4: Connecting the Triplets (Conversions)

Because fractions, decimals, and percentages are related, we can swap between them easily. Here are the most important ones to remember:

• \( \frac{1}{2} = 0.5 = 50\% \)
• \( \frac{1}{4} = 0.25 = 25\% \)
• \( \frac{3}{4} = 0.75 = 75\% \)
• \( \frac{1}{5} = 0.2 = 20\% \)
• \( \frac{1}{10} = 0.1 = 10\% \)

How to convert:

Fraction to Percentage: Try to make the denominator 100. For example, \( \frac{2}{5} \) is the same as \( \frac{40}{100} \), which is 40%.
Decimal to Percentage: Multiply the decimal by 100. For example, \( 0.34 \times 100 = 34\% \).
Percentage to Decimal: Divide the percentage by 100. For example, \( 8\% \div 100 = 0.08 \).

Common Mistake to Avoid:

Be careful with single digits! 5% is 0.05, not 0.5 (0.5 is 50%!). Always think: "Is this out of 10 or out of 100?"

Final Summary Checklist

• Fractions: Keep denominators the same for adding/subtracting. Multiply across for multiplying.
• Decimals: Know your tenths, hundredths, and thousandths columns.
• Percentages: Always remember they are "out of 100."
• Relationship: Use your "cheat code" conversions for common values like \( \frac{1}{2} \) and \( \frac{1}{4} \).

Great job! You've covered the essentials of Year 6 Fractions, Decimals, and Percentages. Keep practicing, and these "triplets" will become second nature to you!