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Fractions Decimals

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Introduction to Fractions and Decimals

Fractions and decimals are two fundamental ways to represent parts of a whole. They appear frequently in daily life - whether measuring ingredients in a recipe, calculating money, or understanding distances. In competitive exams, a strong grasp of these concepts is essential for solving a variety of quantitative problems quickly and accurately.

Understanding the relationship between fractions and decimals helps you switch between these forms effortlessly, making calculations easier. For example, when buying 1.5 kg of rice or calculating discounts of 25%, decimals and fractions come into play.

In this section, we will start from the basics, explore how to convert between fractions and decimals, perform operations on them, and apply these concepts to real-world problems, especially involving metric units and Indian currency (INR).

Definition and 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 considered, and the denominator is the total number of equal parts the whole is divided into.

For example, if a chocolate bar is divided into 4 equal pieces and you eat 3, the fraction representing the eaten part is 3/4.

Types of Fractions

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

Visualizing fractions on a number line helps understand their size and relation.

0 3/4 Proper 7/4 Improper 1 3/4 Mixed 2

Decimal Numbers

A decimal number is another way to represent fractions, especially those parts of a whole divided into powers of ten. Decimals use a decimal point to separate the whole number part from the fractional part.

For example, 0.75 means 75 parts out of 100, which is the decimal form of the fraction 75/100.

Place Value in Decimals

Each digit after the decimal point represents a fraction with denominator as a power of 10:

  • First digit after decimal = tenths (\( \frac{1}{10} \))
  • Second digit = hundredths (\( \frac{1}{100} \))
  • Third digit = thousandths (\( \frac{1}{1000} \)), and so on.
3 . 7 5 Tenths (7 x 1/10) Hundredths (5 x 1/100)

Terminating and Recurring Decimals

Terminating decimals have a finite number of digits after the decimal point (e.g., 0.25, 3.5).

Recurring decimals have digits that repeat infinitely (e.g., 0.666..., 2.121212...). Recognizing recurring decimals is important for exact fraction conversion.

Conversion Between Fractions and Decimals

Understanding how to convert fractions to decimals and vice versa is crucial for solving problems efficiently.

graph TD    A[Start with Fraction] --> B[Divide Numerator by Denominator]    B --> C[Get Decimal Equivalent]    D[Start with Decimal] --> E[Count Decimal Places (n)]    E --> F[Multiply Decimal by 10^n]    F --> G[Write as Fraction over 10^n]    G --> H[Simplify Fraction]

Fraction to Decimal: Divide numerator by denominator using long division.

Decimal to Fraction: Multiply decimal by \(10^n\) where \(n\) is number of decimal places, write over \(10^n\), then simplify.

Operations on Fractions and Decimals

Performing addition, subtraction, multiplication, and division with fractions and decimals requires careful attention to rules and place values.

Addition and Subtraction

  • Fractions: Find a common denominator (usually the LCM of denominators), convert fractions, then add or subtract numerators.
  • Decimals: Align decimal points and add or subtract digit by digit.

Multiplication and Division

  • Fractions: Multiply numerators and denominators directly. For division, multiply by the reciprocal.
  • Decimals: Multiply as whole numbers, then place decimal point in the product by counting total decimal places. For division, move decimal points to make divisor a whole number, then divide.

Worked Examples

Example 1: Convert 3/8 to Decimal Easy
Convert the fraction \( \frac{3}{8} \) into its decimal equivalent.

Step 1: Divide numerator 3 by denominator 8 using long division.

3 / 8 = 0.375

Answer: \( \frac{3}{8} = 0.375 \)

Example 2: Add 1/4 and 0.75 Medium
Find the sum of \( \frac{1}{4} \) and 0.75.

Step 1: Convert \( \frac{1}{4} \) to decimal.

\( \frac{1}{4} = 0.25 \)

Step 2: Add decimals: 0.25 + 0.75 = 1.00

Answer: 1.00 or simply 1

Example 3: Convert Recurring Decimal 0.666... to Fraction Hard
Express the recurring decimal \(0.\overline{6}\) as a fraction.

Step 1: Let \(x = 0.666...\)

Step 2: Multiply both sides by 10 (since one digit repeats):

\(10x = 6.666...\)

Step 3: Subtract original equation from this:

\(10x - x = 6.666... - 0.666...\)

\(9x = 6\)

Step 4: Solve for \(x\):

\(x = \frac{6}{9} = \frac{2}{3}\)

Answer: \(0.\overline{6} = \frac{2}{3}\)

Example 4: Calculate Cost of 2.5 kg Apples at Rs.120.75/kg Medium
Find the total cost of buying 2.5 kg of apples priced at Rs.120.75 per kg.

Step 1: Multiply weight by price per kg:

2.5 x 120.75

Step 2: Calculate:

2.5 x 120.75 = (2 x 120.75) + (0.5 x 120.75) = 241.5 + 60.375 = 301.875

Answer: Total cost = Rs.301.875

Example 5: Simplify \( \left(\frac{3}{4} + \frac{5}{6}\right) \div \left(\frac{7}{8} - \frac{1}{3}\right) \) Hard
Simplify the expression \( \left(\frac{3}{4} + \frac{5}{6}\right) \div \left(\frac{7}{8} - \frac{1}{3}\right) \).

Step 1: Find common denominator for \( \frac{3}{4} + \frac{5}{6} \).

LCM of 4 and 6 is 12.

\( \frac{3}{4} = \frac{9}{12}, \quad \frac{5}{6} = \frac{10}{12} \)

Sum = \( \frac{9}{12} + \frac{10}{12} = \frac{19}{12} \)

Step 2: Find common denominator for \( \frac{7}{8} - \frac{1}{3} \).

LCM of 8 and 3 is 24.

\( \frac{7}{8} = \frac{21}{24}, \quad \frac{1}{3} = \frac{8}{24} \)

Difference = \( \frac{21}{24} - \frac{8}{24} = \frac{13}{24} \)

Step 3: Divide the two results:

\( \frac{19}{12} \div \frac{13}{24} = \frac{19}{12} \times \frac{24}{13} \)

Step 4: Simplify before multiplying:

\( \frac{24}{12} = 2 \), so

\( \frac{19}{12} \times \frac{24}{13} = 19 \times 2 \div 13 = \frac{38}{13} \)

Answer: \( \frac{38}{13} \) or approximately 2.923

Fraction to Decimal Conversion

\[\text{Decimal} = \frac{\text{Numerator}}{\text{Denominator}}\]

Divide numerator by denominator to get decimal equivalent

Numerator = Top part of fraction
Denominator = Bottom part of fraction

Decimal to Fraction Conversion

\[\text{Fraction} = \frac{\text{Decimal} \times 10^n}{10^n}\]

Multiply decimal by 10^n (where n = number of decimal places) and write over 10^n, then simplify

Decimal = Given decimal number
n = Number of digits after decimal point

Pro Tips

  • Memorize decimal equivalents of common fractions with denominators 2, 4, 5, 8, 10 (e.g., 1/2 = 0.5, 3/4 = 0.75) to save time.
  • Use the algebraic method to convert recurring decimals to fractions quickly.
  • Always find the LCM of denominators before adding or subtracting fractions to avoid errors.
  • Convert fractions to decimals when adding or subtracting with decimals for easier calculation.
  • Use metric units in word problems to relate decimals to real-life measurements effectively.

Tips & Tricks

Tip: Memorize decimal equivalents of fractions with denominators 2, 4, 5, 8, 10.

When to use: During time-bound exams to save calculation time.

Tip: Use the algebraic method for converting recurring decimals to fractions.

When to use: When exact fraction form of recurring decimals is required.

Tip: Always find the LCM of denominators before adding or subtracting fractions.

When to use: To avoid mistakes in fraction operations.

Tip: Convert fractions to decimals when adding or subtracting with decimals.

When to use: When fractions and decimals appear together in calculations.

Tip: Use metric units in word problems to relate decimals to real-life measurements.

When to use: In application-based problems involving weights, lengths, or currency.

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 and add adjusted numerators: \( \frac{1}{2} + \frac{1}{3} = \frac{3+2}{6} = \frac{5}{6} \)
Why: Students confuse fraction addition with whole number addition.
❌ Converting decimals to fractions without simplifying (e.g., 0.50 as \( \frac{50}{100} \) instead of \( \frac{1}{2} \))
✓ Always simplify fraction to lowest terms after conversion.
Why: Lack of practice in fraction simplification.
❌ Ignoring recurring nature of decimals and treating them as terminating decimals.
✓ Recognize recurring decimals and use algebraic method for exact fraction conversion.
Why: Misunderstanding of decimal types.
❌ Incorrect placement of decimal point after multiplication or division of decimals.
✓ Count total decimal places in factors and place decimal accordingly in product or quotient.
Why: Lack of attention to decimal place value.
❌ Not converting mixed fractions to improper fractions before operations.
✓ Convert mixed fractions to improper fractions to perform multiplication or division correctly.
Why: Misconception about mixed fraction operations.
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