Simple and Decimal Fractions
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Important Formulas & Concepts
Study MaterialSimple and Decimal Fractions
Simple and Decimal Fractions are among the most important topics in Quantitative Aptitude. Questions based on fractions, decimal operations, recurring decimals, simplification, approximation, and percentage calculations are frequently asked in SSC, Banking, Railway, CAT, CDS, NDA, UPSC, and placement examinations.
Understanding fractions and decimals improves calculation speed and helps in solving arithmetic questions accurately and efficiently.
Why Simple and Decimal Fractions are Important?
- Frequently asked in aptitude and arithmetic sections.
- Useful in percentage, ratio, profit-loss, and interest problems.
- Improves simplification and approximation skills.
- Forms the foundation for higher arithmetic concepts.
- Helps in fast calculations during competitive exams.
What are Fractions?
A fraction represents a part of a whole quantity and is written in the form:
Numerator / Denominator
| Term | Description |
|---|---|
| Numerator | The upper part of a fraction |
| Denominator | The lower part of a fraction |
Example: In 5/8, 5 is the numerator and 8 is the denominator.
Types of Fractions
1. Proper Fraction
A fraction in which numerator is smaller than denominator.
Example: 3/5, 7/9
2. Improper Fraction
A fraction in which numerator is greater than or equal to denominator.
Example: 9/5, 11/7
3. Mixed Fraction
A combination of a whole number and a proper fraction.
Example: 2 â…“
4. Decimal Fraction
Fractions whose denominators are powers of 10 are called decimal fractions.
| Fraction | Decimal Form |
|---|---|
| 1/10 | 0.1 |
| 1/100 | 0.01 |
| 99/100 | 0.99 |
| 7/1000 | 0.007 |
Conversion of Decimal into Fraction
To convert a decimal into a vulgar fraction:
- Remove the decimal point.
- Write the number over 1 followed by zeros.
- Simplify the fraction.
Example 1:
0.25 = 25/100 = 1/4
Example 2:
2.008 = 2008/1000 = 251/125
Annexing Zeros in Decimal Fractions
Adding zeros to the extreme right side of a decimal number does not change its value.
| Original Number | Equivalent Forms |
|---|---|
| 0.8 | 0.80 = 0.800 |
| 5.2 | 5.20 = 5.200 |
✔ Annexing zeros is useful while comparing decimal numbers.
Removing Decimal Signs
If numerator and denominator contain equal decimal places, remove the decimal signs directly.
Example:
1.84 / 2.99
= 184 / 299
Operations on Decimal Fractions
1. Addition and Subtraction
Arrange the numbers such that decimal points lie in the same column and perform normal addition or subtraction.
Example:
12.56 + 8.004 + 0.9
= 21.464
2. Multiplication by Power of 10
Move the decimal point to the right according to the number of zeros.
| Expression | Result |
|---|---|
| 5.9632 × 100 | 596.32 |
| 0.073 × 10000 | 730 |
3. Multiplication of Decimal Fractions
Multiply the numbers normally without decimal points. Then place the decimal point according to total decimal places.
Example:
0.2 × 0.02 × 0.002
2 × 2 × 2 = 8
Total decimal places = 1 + 2 + 3 = 6
Result = 0.000008
4. Division of Decimal Fractions
For division:
- Convert divisor into whole number.
- Multiply dividend and divisor by suitable powers of 10.
- Perform normal division.
Example:
0.00066 ÷ 0.11
= (0.00066 × 100) / (0.11 × 100)
= 0.066 / 11
= 0.006
Comparison of Fractions
To compare fractions:
- Convert them into decimals.
- Compare decimal values directly.
Example: Arrange 3/5, 6/7, and 7/9 in descending order.
3/5 = 0.6
6/7 = 0.857
7/9 = 0.777...
Therefore:
6/7 > 7/9 > 3/5
Recurring Decimals
A decimal number in which one digit or a set of digits repeats continuously is called a recurring decimal.
| Fraction | Recurring Decimal |
|---|---|
| 1/3 | 0.333... |
| 22/7 | 3.142857142857... |
Pure Recurring Decimal
A decimal in which all digits after the decimal point repeat continuously.
Examples:
- 0.333...
- 0.545454...
- 0.678678678...
Conversion Formula
Write repeating digits once in numerator and corresponding 9s in denominator.
| Recurring Decimal | Fraction |
|---|---|
| 0.5Ì… | 5/9 |
| 0.53Ì… | 53/99 |
| 0.067Ì… | 67/999 |
Mixed Recurring Decimal
A decimal in which some digits do not repeat and some digits repeat continuously.
Example: 0.173333...
Conversion Formula
Numerator:
(All digits after decimal including repeated part once) − (Non-repeating part)
Denominator:
As many 9s as repeating digits followed by as many 0s as non-repeating digits.
Example:
0.16Ì…
= (16 − 1) / 90
= 15/90
= 1/6
Important Algebraic Formulae
| Formula Name | Formula |
|---|---|
| Difference of Squares | (a + b)(a − b) = a² − b² |
| Square of Sum | (a + b)² = a² + b² + 2ab |
| Square of Difference | (a − b)² = a² + b² − 2ab |
| Cube Sum | a³ + b³ = (a + b)(a² − ab + b²) |
| Cube Difference | a³ − b³ = (a − b)(a² + ab + b²) |
Important Exam Tips
- Practice decimal multiplication and division regularly.
- Memorize recurring decimal conversion methods.
- Always align decimal points during addition and subtraction.
- Use annexing zeros while comparing decimal values.
- Learn fraction-to-decimal and decimal-to-fraction conversions thoroughly.
- Practice simplification questions daily for better speed.
Simple and Decimal Fractions form the base of arithmetic aptitude. Strong understanding of decimal operations, recurring decimals, fraction comparison, and conversions helps candidates solve quantitative aptitude questions faster and more accurately.