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Fractions and Decimals

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Introduction to Fractions and Decimals

Fractions and decimals are two ways to represent parts of a whole. They are essential in everyday life, from measuring ingredients in cooking to handling money and understanding distances. For example, when you buy 1.5 kilograms of rice or pay Rs.75.50 for groceries, decimals are used. Fractions help when you divide a pizza into equal slices or share sweets among friends.

Understanding fractions and decimals is crucial for competitive exams because many problems test your ability to work with parts of numbers accurately and quickly. This section will guide you through the basics, operations, conversions, and applications of fractions and decimals, building your skills step-by-step.

Types of Fractions

A fraction represents a part of a whole and is written as numerator/denominator, where the numerator is the number of parts taken, and the denominator is the total number of equal parts.

There are three main types of fractions:

  • Proper Fraction: The numerator is less than the denominator. Example: \(\frac{3}{4}\) means 3 parts out of 4.
  • Improper Fraction: The numerator is equal to or greater than the denominator. Example: \(\frac{7}{4}\) means 7 parts out of 4, which is more than one whole.
  • Mixed Fraction: A whole number combined with a proper fraction. Example: \(1 \frac{3}{4}\) means 1 whole and 3 parts out of 4.
Proper: 3/4 Improper: 7/4 Mixed: 1 3/4

Operations on Fractions

Fractions can be added, subtracted, multiplied, and divided. Each operation has specific steps to follow.

graph TD    A[Start] --> B{Operation?}    B --> C[Addition/Subtraction]    B --> D[Multiplication]    B --> E[Division]    C --> F[Find LCM of denominators]    F --> G[Convert fractions to equivalent with LCM denominator]    G --> H[Add or subtract numerators]    H --> I[Simplify the result]    D --> J[Multiply numerators]    J --> K[Multiply denominators]    K --> I    E --> L[Take reciprocal of divisor fraction]    L --> M[Multiply fractions]    M --> I

Let's understand each operation:

  • Addition and Subtraction: To add or subtract fractions, first find a common denominator (usually the Least Common Multiple, LCM, of the denominators). Then convert each fraction to an equivalent fraction with this common denominator. Add or subtract the numerators and keep the denominator the same. Finally, simplify the fraction.
  • Multiplication: Multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. Simplify if possible.
  • Division: To divide by a fraction, multiply by its reciprocal (flip numerator and denominator of the divisor). Then multiply as usual.

Decimal Place Value and Operations

Decimals represent parts of a whole using a decimal point. Each digit after the decimal point has a place value:

Place Value Example Digit
Tenths 1/10 or 0.1 0.3 (3 in tenths place)
Hundredths 1/100 or 0.01 0.07 (7 in hundredths place)
Thousandths 1/1000 or 0.001 0.005 (5 in thousandths place)

When adding or subtracting decimals, always align the decimal points to keep place values correct. Multiplying decimals involves multiplying as whole numbers and then placing the decimal point in the product so that the total number of decimal places equals the sum of decimal places in the factors. Division of decimals involves shifting the decimal point to make the divisor a whole number, then dividing as usual.

Conversion between Fractions and Decimals

You can convert a fraction to a decimal by dividing the numerator by the denominator. For example, \(\frac{3}{4} = 3 \div 4 = 0.75\).

To convert a decimal to a fraction, write the decimal number without the decimal point as the numerator and use \(10^n\) as the denominator, where \(n\) is the number of digits after the decimal point. For example, 0.625 has 3 digits after the decimal, so:

\[0.625 = \frac{625}{10^3} = \frac{625}{1000}\]

Then simplify the fraction to lowest terms.

Recurring decimals (decimals with repeating digits) can also be converted to fractions using algebraic methods, which we will explore in the worked examples.

Formula Bank

Formula Bank

Addition of Fractions
\[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \]
where: \(a,b,c,d\) are integers; \(b,d eq 0\)
Subtraction of Fractions
\[ \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \]
where: \(a,b,c,d\) are integers; \(b,d eq 0\)
Multiplication of Fractions
\[ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \]
where: \(a,b,c,d\) are integers; \(b,d eq 0\)
Division of Fractions
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \]
where: \(a,b,c,d\) are integers; \(b,c,d eq 0\)
Decimal to Fraction Conversion
\[ \text{Decimal} = \frac{\text{Decimal number without point}}{10^{n}} \]
where: \(n\) = number of digits after decimal point
Fraction to Decimal Conversion
\[ \text{Decimal} = \frac{\text{Numerator}}{\text{Denominator}} \]
Numerator and Denominator are integers; Denominator \( eq 0\)

Worked Examples

Example 1: Adding Fractions with Different Denominators Easy
Add \(\frac{2}{3}\) and \(\frac{3}{4}\).

Step 1: Find the LCM of denominators 3 and 4. LCM(3,4) = 12.

Step 2: Convert each fraction to an equivalent fraction with denominator 12.

\(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)

\(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)

Step 3: Add the numerators: \(8 + 9 = 17\).

Step 4: Write the sum with denominator 12: \(\frac{17}{12}\).

Step 5: Since \(\frac{17}{12}\) is an improper fraction, convert to mixed fraction:

\(17 \div 12 = 1\) remainder \(5\), so \(\frac{17}{12} = 1 \frac{5}{12}\).

Answer: \(1 \frac{5}{12}\)

Example 2: Multiplying a Decimal by a Fraction Medium
Multiply 0.6 by \(\frac{3}{5}\).

Step 1: Convert the fraction \(\frac{3}{5}\) to decimal.

\(\frac{3}{5} = 3 \div 5 = 0.6\)

Step 2: Multiply decimals: \(0.6 \times 0.6 = 0.36\).

Answer: \(0.36\)

Example 3: Converting a Recurring Decimal to a Fraction Hard
Convert the recurring decimal \(0.\overline{72}\) (where 72 repeats) to a fraction.

Step 1: Let \(x = 0.727272...\)

Step 2: Multiply both sides by 100 (since 2 digits repeat):

\(100x = 72.727272...\)

Step 3: Subtract the original \(x\) from this:

\(100x - x = 72.727272... - 0.727272...\)

\(99x = 72\)

Step 4: Solve for \(x\):

\(x = \frac{72}{99}\)

Step 5: Simplify the fraction by dividing numerator and denominator by 9:

\(\frac{72 \div 9}{99 \div 9} = \frac{8}{11}\)

Answer: \(0.\overline{72} = \frac{8}{11}\)

Example 4: Solving Word Problem Involving Fractions and INR Medium
Ramesh bought \(\frac{3}{4}\) kg of apples at Rs.120 per kg and \(\frac{2}{3}\) kg of oranges at Rs.90 per kg. Find the total amount he paid.

Step 1: Calculate cost of apples:

\(\frac{3}{4} \times 120 = \frac{3 \times 120}{4} = \frac{360}{4} = 90\) INR

Step 2: Calculate cost of oranges:

\(\frac{2}{3} \times 90 = \frac{2 \times 90}{3} = \frac{180}{3} = 60\) INR

Step 3: Add the costs:

\(90 + 60 = 150\) INR

Answer: Ramesh paid Rs.150 in total.

Example 5: Comparing Fractions and Decimals Easy
Which is greater: \(\frac{5}{8}\) or 0.62?

Method 1: Convert fraction to decimal

\(\frac{5}{8} = 5 \div 8 = 0.625\)

Compare 0.625 and 0.62

Since \(0.625 > 0.62\), \(\frac{5}{8}\) is greater.

Method 2: Cross multiplication

Write 0.62 as \(\frac{62}{100}\).

Cross multiply:

\(5 \times 100 = 500\)

\(8 \times 62 = 496\)

Since 500 > 496, \(\frac{5}{8} > \frac{62}{100}\).

Answer: \(\frac{5}{8}\) is greater than 0.62.

Summary of Key Formulas

  • Addition: \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\)
  • Subtraction: \(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\)
  • Multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
  • Division: \(\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}\)
  • Decimal to Fraction: Decimal = \(\frac{\text{number without decimal}}{10^n}\)
  • Fraction to Decimal: Decimal = Numerator / Denominator

Tips & Tricks

Tip: Always simplify fractions before performing operations to reduce calculation complexity.

When to use: Before adding, subtracting, multiplying, or dividing fractions.

Tip: Convert fractions to decimals when dealing with metric measurements for easier calculations.

When to use: When measurements are given in decimals or require precise calculations.

Tip: Use cross multiplication to compare two fractions quickly without converting to decimals.

When to use: When asked to find which fraction is greater or to order fractions.

Tip: For recurring decimals, use algebraic methods to convert to fractions instead of long division.

When to use: When decimals repeat infinitely and need exact fraction form.

Tip: Memorize common fraction-decimal equivalents (e.g., \(\frac{1}{2} = 0.5\), \(\frac{1}{4} = 0.25\)) for quick recall.

When to use: During quick calculations or multiple-choice questions.

Common Mistakes to Avoid

❌ Adding fractions by adding numerators and denominators directly (e.g., \(\frac{1}{2} + \frac{1}{3} = \frac{2}{5}\))
✓ Find common denominator first, then add numerators (e.g., \(\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\))
Why: Students confuse fraction addition with whole number addition.
❌ Ignoring place value while adding or subtracting decimals (e.g., adding 0.5 and 0.25 as 0.75 without aligning decimals)
✓ Always align decimal points before performing operations.
Why: Misalignment leads to incorrect sums or differences.
❌ Not simplifying fractions after operations.
✓ Always simplify fractions to their lowest terms for clarity.
Why: Simplification makes answers clearer and easier to compare.
❌ Confusing recurring decimals with terminating decimals.
✓ Identify repeating patterns and use algebraic methods for conversion.
Why: Misinterpretation leads to incorrect fraction equivalents.
❌ Incorrectly converting fractions to decimals by dividing numerator by denominator incorrectly.
✓ Use long division carefully or a calculator for precise decimal values.
Why: Calculation errors cause wrong decimal results.
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