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Fractions and Decimals

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Introduction to Fractions and Decimals

Fractions and decimals are two fundamental ways to represent parts of a whole. In everyday life, we often use fractions and decimals - such as when measuring quantities, calculating prices in rupees and paise, or understanding ratios. For example, if a pizza is divided into 4 equal parts and you eat one part, you have eaten one-fourth or 0.25 of the pizza. Understanding these concepts is crucial, especially in competitive entrance exams, where you may encounter questions requiring quick and accurate conversions and calculations involving fractions and decimals.

Both fractions and decimals express a quantity smaller than one whole unit, but they do so differently. Fractions use two integers - a numerator and denominator - to show division, while decimals use place value notation in powers of ten. Mastering these will sharpen your number sense and calculation skills, providing a strong base for advanced topics like percentages, ratios, and interest calculations.

Definition of Fractions

A fraction is a number that expresses a part of a whole. It is written as \(\frac{a}{b}\), where:

  • Numerator (a) is the number on top, representing how many parts are considered.
  • Denominator (b) is the number at the bottom, representing the total equal parts into which the whole is divided.

The denominator cannot be zero because dividing by zero is undefined.

Fractions can be classified into three types:

  • Proper Fractions: Numerator is less than denominator, e.g., \(\frac{3}{5}\). These represent values less than 1.
  • Improper Fractions: Numerator is equal to or greater than denominator, e.g., \(\frac{7}{4}\). These represent values equal to or greater than 1.
  • Mixed Fractions: A whole number combined with a proper fraction, e.g., 1 \frac{3}{4}. Equivalent to an improper fraction but easier to understand visually.
3/5 1 7/4 = 1 3/4

The pie charts above visually show how the numerator represents the shaded parts and the denominator the total parts of the whole.

Decimal System Basics

Decimals are numbers expressed in the base-10 place value system, extending beyond whole numbers to represent parts of a whole. A decimal number has two parts separated by a decimal point: the integer part and the fractional part.

Each digit to the right of the decimal point represents a fraction with denominators as powers of ten:

  • The first place is tenths (\(\frac{1}{10}\)),
  • The second is hundredths (\(\frac{1}{100}\)),
  • The third is thousandths (\(\frac{1}{1000}\)), and so on.
0.1 (tenths) 0.01 (hundredths) 0.001 (thousandths) 0.0001 (ten-thousandths)

For example, the decimal 12.345 means:

  • 12 whole units,
  • 3 tenths (0.3 = \frac{3}{10}),
  • 4 hundredths (0.04 = \frac{4}{100}), and
  • 5 thousandths (0.005 = \frac{5}{1000}).

Hence, decimals and fractions are closely related - each decimal corresponds to a fraction whose denominator is a power of 10.

Conversion Between Fractions and Decimals

Being able to convert between fractions and decimals is crucial for solving many problems efficiently.

Fraction to Decimal Conversion

Convert a fraction to a decimal by dividing the numerator by the denominator.

Example: Convert \(\frac{3}{4}\) to decimal.

Divide 3 by 4: \(3 \div 4 = 0.75\).

So, \(\frac{3}{4} = 0.75\).

Decimal to Fraction Conversion

Write the decimal as a fraction using the place value of the last digit, then simplify the fraction.

Example: Convert 0.125 to a fraction.

0.125 has three decimal places, so it is \(\frac{125}{1000}\). Simplify by dividing numerator and denominator by 125:

\[\frac{125}{1000} = \frac{125 \div 125}{1000 \div 125} = \frac{1}{8}.\]

Thus, 0.125 = \(\frac{1}{8}\).

graph LR  Fraction -->|Divide numerator by denominator| Decimal  Decimal -->|Express decimal as fraction over 10^n| Fraction  Fraction -->|Simplify fraction if possible| Fraction  Decimal -->|Simplify fraction after conversion| Fraction

Operations on Fractions and Decimals

Performing operations on fractions and decimals requires understanding specific rules for each type and their similarities.

Operation Fractions Decimals
Addition Find LCM of denominators, convert to equivalent fractions, then add numerators. Align decimal points vertically and add digits column-wise.
Subtraction Find LCM, convert, subtract numerators, simplify if needed. Align decimal points and subtract similarly.
Multiplication Multiply numerators and denominators directly; simplify. Multiply numbers ignoring decimal, then place decimal in the product based on total decimal places.
Division Multiply by reciprocal of divisor. Convert divisor to whole number by shifting decimal, do same for dividend, then divide.

Worked Example 1: Adding Fractions with Different Denominators

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

Step 1: Find the least common multiple (LCM) of denominators 3 and 4.

LCM of 3 and 4 is 12.

Step 2: Convert the fractions to equivalent fractions 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.

\[ \frac{8}{12} + \frac{9}{12} = \frac{8 + 9}{12} = \frac{17}{12}. \]

Step 4: Since \(\frac{17}{12}\) is an improper fraction, convert it to a mixed fraction.

\[ 17 \div 12 = 1 \text{ remainder } 5, \quad \text{so } \frac{17}{12} = 1 \frac{5}{12}. \]

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

Worked Example 2: Converting a Repeating Decimal to a Fraction

Example 2: Converting Repeating Decimal 0.666... to Fraction Medium
Convert the repeating decimal \(0.\overline{6}\) (i.e., 0.666...) to a fraction.

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

Step 2: Multiply both sides by 10 to move the decimal one place right:

\[ 10x = 6.666... \]

Step 3: Subtract the original equation from this:

\[ 10x - x = 6.666... - 0.666... \implies 9x = 6. \]

Step 4: Solve for \(x\):

\[ x = \frac{6}{9} = \frac{2}{3}. \]

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

Worked Example 3: Multiplying Decimals in Currency Problems

Example 3: Multiplying Decimal Amounts in Currency Context Easy
Find the total cost of buying 4.2 kg of mangoes priced at Rs.12.5 per kg.

Step 1: Multiply 4.2 by 12.5.

Ignore decimals for now: 42 x 125 = 5250.

Step 2: Count total decimal places: 4.2 has 1 decimal place; 12.5 has 1 decimal place. Total = 2.

Step 3: Place decimal 2 places from the right in 5250 -> 52.50.

Answer: Total cost = Rs.52.50

Worked Example 4: Subtracting Mixed Fractions with Borrowing

Example 4: Subtracting Mixed Fractions (3 \(\frac{1}{2}\) - 1 \(\frac{3}{4}\)) Medium
Calculate \(3 \frac{1}{2} - 1 \frac{3}{4}\).

Step 1: Convert mixed fractions to improper fractions.

\[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}, \quad 1 \frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}. \]

Step 2: Find LCM of denominators 2 and 4, which is 4.

<[ \frac{7}{2} = \frac{7 \times 2}{2 \times 2} = \frac{14}{4}.

Step 3: Subtract numerators:

\[ \frac{14}{4} - \frac{7}{4} = \frac{14 - 7}{4} = \frac{7}{4}. \]

Step 4: Convert \(\frac{7}{4}\) back to mixed fraction:

\[ 7 \div 4 = 1 \text{ remainder } 3, \quad \Rightarrow 1 \frac{3}{4}. \]

Answer: \(1 \frac{3}{4}\)

Worked Example 5: Decimal Division in Metric Conversion

Example 5: Dividing Decimals in Metric Conversion Hard
Convert 250 cm to meters by dividing by 100.

Step 1: Understand that 1 meter = 100 centimeters.

Step 2: Divide 250 by 100:

\[ 250 \div 100 = ? \]

Ignoring decimals initially, 250 / 100 can be thought of as \( \frac{250}{100} \).

Step 3: Simplify fraction by dividing numerator and denominator by 50:

\[ \frac{250 \div 50}{100 \div 50} = \frac{5}{2} = 2.5. \]

Step 4: So, 250 cm = 2.5 meters.

Answer: 2.5 meters

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
\[\text{Decimal Number} = \frac{\text{Decimal Number without Point}}{10^{n}}\]
where: \(n\) = number of decimal places
Fraction to Decimal Conversion
\[\frac{a}{b} = a \div b\]
where: \(a, b\) are integers; \(b eq 0\)

Tips & Tricks

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

When to use: To quickly determine which fraction is larger during exams for speed and accuracy.

Tip: Memorize common fraction-decimal equivalents such as \(\frac{1}{2} = 0.5\), \(\frac{1}{4} = 0.25\), and \(\frac{1}{3} = 0.333...\).

When to use: To speed up conversions and estimations during tests.

Tip: Always simplify fractions after performing operations to avoid mistakes in future steps.

When to use: After addition, subtraction, multiplication, or division of fractions.

Tip: When adding or subtracting decimals, align decimal points vertically to avoid place value errors.

When to use: During all decimal addition and subtraction problems.

Tip: Use algebraic multiplication strategies to convert repeating decimals into exact fractions efficiently.

When to use: For converting recurring decimals quickly in exams.

Common Mistakes to Avoid

❌ Adding fractions by adding numerators and denominators directly (e.g., \(\frac{1}{2} + \frac{1}{3} = \frac{2}{5}\))
✓ Find a common denominator and add equivalent fractions: \(\frac{1}{2} = \frac{3}{6}\), \(\frac{1}{3} = \frac{2}{6}\), so sum = \(\frac{5}{6}\).
Why: Fractions are not like whole numbers; denominators must be the same before adding.
❌ Confusing decimal place values when converting decimals to fractions, e.g., writing 0.03 as \(\frac{3}{10}\) instead of \(\frac{3}{100}\).
✓ Count the number of decimal places carefully; 0.03 has two decimal places, so it equals \(\frac{3}{100}\).
Why: Misunderstanding place value leads to incorrect denominator choice.
❌ Forgetting to flip the second fraction (taking reciprocal) during division of fractions.
✓ Multiply the first fraction by the reciprocal of the second fraction, e.g., \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\).
Why: Division of fractions is defined as multiplication by reciprocal.
❌ Misaligning decimal points while adding or subtracting decimals.
✓ Always write decimals with points aligned vertically before performing the operation.
Why: Misalignment causes place value shifts and incorrect sums.
❌ Incorrectly converting repeating decimals to fractions by ignoring the length of the repeating pattern.
✓ Use the algebraic method considering repeat length to accurately convert recurring decimals to fractions.
Why: Oversimplifies conversion, resulting in wrong fractions.
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