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Fractions and decimals are two common ways to represent parts of a whole number. Understanding both forms and how to convert between them is essential for solving many numerical problems, especially in competitive exams like the Delhi Police Constable (DPC) Examination. Whether you are calculating distances in kilometers, prices in INR, or quantities in liters, fractions and decimals help express values that are not whole numbers.
In this section, you will learn how to convert fractions to decimals and vice versa, and how to perform the four basic arithmetic operations - addition, subtraction, multiplication, and division - on both fractions and decimals. Mastering these concepts will improve your speed and accuracy in solving exam questions.
A fraction is a number expressed as the ratio of two integers, written as numerator over denominator, for example, \(\frac{3}{4}\). A decimal is a number expressed in base 10 with a decimal point, such as 0.75.
To convert between these two forms, we use division and place value concepts.
graph TD A[Start] --> B{Fraction to Decimal?} B -- Yes --> C[Divide numerator by denominator] C --> D[Write quotient as decimal] B -- No --> E{Decimal to Fraction?} E -- Yes --> F[Count decimal places (n)] F --> G[Multiply decimal by \(10^n\)] G --> H[Write as fraction over \(10^n\)] H --> I[Simplify fraction] E -- No --> J[End]To convert a fraction to a decimal, divide the numerator by the denominator. For example, to convert \(\frac{3}{8}\) to a decimal, divide 3 by 8:
\(3 \div 8 = 0.375\)
This decimal is called a terminating decimal because it ends after a few digits.
To convert a decimal to a fraction, count the number of digits after the decimal point. Suppose the decimal is 0.625, which has 3 digits after the decimal. Multiply the decimal by \(10^3 = 1000\) to get 625, then write it as a fraction over 1000:
\[0.625 = \frac{625}{1000}\]
Now simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD), which is 125:
\[\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}\]
Some decimals repeat a pattern infinitely, such as \(0.333...\) or \(0.142857142857...\). These are called recurring decimals. They can also be converted to fractions using algebraic methods, which will be discussed in advanced topics.
When adding or subtracting fractions, the denominators (bottom numbers) must be the same. If they are different, find the Least Common Multiple (LCM) of the denominators to get a common denominator.
Steps to add or subtract fractions:
| Step | Addition Example: \(\frac{2}{5} + \frac{3}{7}\) | Subtraction Example: \(\frac{5}{6} - \frac{1}{4}\) |
|---|---|---|
| Find LCM of denominators | LCM of 5 and 7 = 35 | LCM of 6 and 4 = 12 |
| Convert fractions | \(\frac{2}{5} = \frac{14}{35}, \quad \frac{3}{7} = \frac{15}{35}\) | \(\frac{5}{6} = \frac{10}{12}, \quad \frac{1}{4} = \frac{3}{12}\) |
| Add/Subtract numerators | 14 + 15 = 29 | 10 - 3 = 7 |
| Result | \(\frac{29}{35}\) | \(\frac{7}{12}\) |
Multiplying and dividing fractions is simpler than addition and subtraction because you do not need a common denominator.
Multiplication: Multiply the numerators together and the denominators together.
Division: Multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
Decimals are handled differently from fractions but follow logical rules based on place value.
| Operation | Rule | Example |
|---|---|---|
| Addition & Subtraction | Align decimal points vertically, then add or subtract digits column-wise. | 3.6 - 1.25 = 2.35 |
| Multiplication | Ignore decimals and multiply as integers. Count total decimal places in factors and place decimal in product accordingly. | 0.75 x 0.4 = 0.30 |
| Division | Multiply divisor and dividend by \(10^n\) to make divisor a whole number, then divide as integers. | Divide 1.2 by 0.4 -> Multiply both by 10 -> 12 / 4 = 3 |
Step 1: Divide numerator by denominator: \(3 \div 8\).
Step 2: Perform the division: \(3 \div 8 = 0.375\).
Answer: \(\frac{3}{8} = 0.375\).
Step 1: Find LCM of denominators 5 and 7: LCM = 35.
Step 2: Convert fractions to have denominator 35:
\(\frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35}\)
\(\frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35}\)
Step 3: Add numerators: \(14 + 15 = 29\).
Step 4: Write the result over common denominator: \(\frac{29}{35}\).
Answer: \(\frac{2}{5} + \frac{3}{7} = \frac{29}{35}\).
Step 1: Ignore decimals and multiply as integers: \(75 \times 4 = 300\).
Step 2: Count total decimal places: 0.75 has 2, 0.4 has 1, total = 3.
Step 3: Place decimal point in product so it has 3 decimal places: \(0.300\).
Step 4: Simplify decimal: \(0.300 = 0.3\).
Answer: \(0.75 \times 0.4 = 0.3\).
Step 1: Find reciprocal of divisor \(\frac{2}{3}\) -> \(\frac{3}{2}\).
Step 2: Multiply dividend by reciprocal:
\(\frac{7}{9} \times \frac{3}{2} = \frac{7 \times 3}{9 \times 2} = \frac{21}{18}\).
Step 3: Simplify fraction by dividing numerator and denominator by 3:
\(\frac{21 \div 3}{18 \div 3} = \frac{7}{6}\).
Answer: \(\frac{7}{9} \div \frac{2}{3} = \frac{7}{6}\).
Step 1: Align decimal points:
3.60
-1.25
Step 2: Subtract digits column-wise:
0 - 5 cannot be done, borrow 1 -> 10 - 5 = 5
5 (borrowed) - 2 = 3
6 - 1 = 5
Step 3: Write the result with decimal point aligned:
2.35
Answer: \(3.6 - 1.25 = 2.35\).
When to use: When adding or subtracting fractions with unlike denominators.
When to use: When multiplying decimals to avoid confusion with decimal placement.
When to use: When dividing decimals to simplify calculation.
When to use: When dealing with repeating decimal conversions in exams.
When to use: During practice to enhance conceptual clarity and retention.
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