Decimals and Percentages

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🔹 1. Decimals

➤ What is a Decimal?

A decimal is a number that has a whole number part and a fraction part separated by a decimal point.

✏️ Example:

$$7.5 \text{ means 7 wholes and 5 tenths } \left(\frac{5}{10}\right)$$ $$3.25 = 3 + \frac{2}{10} + \frac{5}{100}$$

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🔹 2. Types of Decimals

a) Terminating Decimals

• Decimals that come to an end.

Example:

$$0.5, \quad 3.75, \quad 2.125$$

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b) Recurring (Repeating) Decimals

• Decimals that have one or more digits repeating forever.

Example:

$$0.333\ldots = 0.\overline{3}, \quad 1.666\ldots = 1.\overline{6}$$

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c) Non-Terminating, Non-Repeating Decimals

• Decimals that never end and don’t repeat.

• These are irrational numbers.

Example:

$$\pi = 3.14159265\ldots \quad \text{or} \quad \sqrt{2} = 1.4142135\ldots$$

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🔹 3. Operations on Decimals

➤ a) Addition

Line up the decimal points and add.

Example:

$$2.35 + 1.7 = 4.05$$

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➤ b) Subtraction

Align decimal points and subtract.

Example:

$$5.8 - 2.45 = 3.35$$

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➤ c) Multiplication

Multiply as whole numbers, then place the decimal point in the answer.

Example:

$$2.3 \times 1.2 = 2.76$$

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➤ d) Division

Move the decimal point to make the divisor a whole number, then divide normally.

Example:

$$4.8 \div 0.2 = 24$$

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🔹 4. Percentages

➤ What is a Percentage?

A percentage is a number expressed as a fraction of 100.

The symbol % means "per hundred".

✏️ Examples:

$$50\% = \frac{50}{100} = 0.5 \quad 25\% = \frac{25}{100} = 0.25$$

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➤ Converting Between Forms

Form	Example
Fraction to %	$\frac{3}{4} = 0.75 = 75\%$
Decimal to %	$0.6 = 60\%$
% to Decimal	$25\% = 0.25$

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➤ Finding a Percentage of a Number

Example:
What is 20% of 150?

$$= \frac{20}{100} \times 150 = 30$$

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➤ Increase or Decrease by a Percentage

Increase:

$$\text{New Value} = \text{Original} + (\text{Percentage} \times \text{Original})$$

Decrease:

$$\text{New Value} = \text{Original} - (\text{Percentage} \times \text{Original})$$

Example:
Increase 200 by 10%

$$= 200 + (0.10 \times 200) = 220$$

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✅ Summary Table:

Concept	Example
Decimal (Terminating)	2.75
Decimal (Recurring)	0.666... = 0.\overline{6}
Add Decimals	1.25 + 2.4 = 3.65
Multiply Decimals	0.5 × 0.2 = 0.10
Convert % to Decimal	40% = 0.4
Find the % of a Number	10% of 80 = 8

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