Class 5 Notes - Fractions and Decimals

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Introduction

Fractions and decimals are essential concepts in mathematics that help us represent and solve problems involving parts of a whole or numbers between integers. This chapter explores the fundamental ideas of fractions and decimals, their operations, and applications.

Fractions

Definition

A fraction represents a part of a whole. It is written in the form a/b, where:

  • a (numerator): Represents the part.
  • b (denominator): Represents the total number of equal parts.

Types of Fractions

  • Proper Fraction: The numerator is less than the denominator (e.g., 3/4).
  • Improper Fraction: The numerator is greater than or equal to the denominator (e.g., 7/4).
  • Mixed Fraction: A combination of a whole number and a proper fraction (e.g., 2 1/3).

Operations on Fractions

1. Addition and Subtraction

To add or subtract fractions:

  • Make the denominators the same (find the LCM if necessary).
  • Add or subtract the numerators while keeping the denominator the same.
Example:
Add 1/4 and 2/4:
Numerators: 1 + 2 = 3
Denominator: 4
Result: 3/4

2. Multiplication

To multiply fractions:

  • Multiply the numerators.
  • Multiply the denominators.
Example:
Multiply 2/3 by 3/5:
Numerators: 2 × 3 = 6
Denominators: 3 × 5 = 15
Result: 6/15 = 2/5 (simplified)

3. Division

To divide fractions:

  • Invert (reciprocal) the second fraction.
  • Multiply the first fraction by the reciprocal of the second.
Example:
Divide 4/5 by 2/3:
Reciprocal of 2/3: 3/2
Multiply: 4/5 × 3/2 = 12/10 = 6/5 (simplified)

Decimals

Definition

Decimals are another way to represent numbers that are not whole. They are based on the powers of ten and use a decimal point to separate the whole number part from the fractional part.

Place Value in Decimals

Each digit in a decimal number has a place value based on its position relative to the decimal point:

  • On the left side of the decimal: Units, tens, hundreds, etc.
  • On the right side of the decimal: Tenths, hundredths, thousandths, etc.
Example:
In the number 23.456:
2 = Tens
3 = Units
4 = Tenths
5 = Hundredths
6 = Thousandths

Operations on Decimals

1. Addition and Subtraction

Align the decimal points and perform the operation column-wise.

Example:
Add 12.3 and 4.56:
Align: 12.30
+ 4.56
Result: 16.86

2. Multiplication

Ignore the decimal points and multiply as whole numbers. Place the decimal point in the result based on the total number of decimal places in the factors.

Example:
Multiply 1.2 by 3.4:
12 × 34 = 408
Decimal places: 1 (1.2) + 1 (3.4) = 2
Result: 4.08

3. Division

Move the decimal point in the divisor to make it a whole number. Do the same for the dividend and divide as usual.

Example:
Divide 4.2 by 0.7:
Adjust: 42 ÷ 7 = 6
Result: 6.0

Relationship Between Fractions and Decimals

Fractions can be converted to decimals by dividing the numerator by the denominator. Similarly, decimals can be converted to fractions by writing them as the ratio of the decimal number to a power of ten and simplifying.

Example:
Convert 3/4 to decimal:
3 ÷ 4 = 0.75
Convert 0.25 to fraction:
0.25 = 25/100 = 1/4

Applications

Fractions and decimals are used in real-life scenarios such as:

  • Measurements (e.g., length, weight, volume).
  • Financial calculations (e.g., currency, interest rates).
  • Data representation (e.g., statistics, percentages).

Practice Problems

  1. Simplify: 5/8 + 3/8
  2. Convert 0.6 to a fraction and simplify.
  3. Multiply 2.5 by 1.4.
  4. Divide 7/9 by 2/3.
  5. Convert 7/10 to a decimal.

Conclusion

Understanding fractions and decimals is crucial for solving mathematical problems and their applications in daily life. Practice regularly to strengthen your understanding and skills.