Decimal Fractions

Operations with decimals and fractions

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Decimal fractions represent parts of a whole using base-10 notation, where each digit after the decimal point represents tenths, hundredths, thousandths, etc. Understanding decimal operations, conversions between fractions and decimals, and recurring decimals is essential for solving quantitative aptitude problems efficiently.

Key Formulas

Decimal to Fraction: Write the decimal as numerator without decimal point, denominator as 1 followed by zeros equal to decimal places. Example: 0.75 = 75/100 = 3/4\text{Decimal to Fraction: Write the decimal as numerator without decimal point, denominator as 1 followed by zeros equal to decimal places. Example: 0.75 = 75/100 = 3/4}
Fraction to Decimal: Divide numerator by denominator. Example: 3/4 = 0.75\text{Fraction to Decimal: Divide numerator by denominator. Example: 3/4 = 0.75}
Terminating Decimal: A fraction p/q in lowest terms has terminating decimal expansion iff prime factors of q are only 2 and/or 5\text{Terminating Decimal: A fraction p/q in lowest terms has terminating decimal expansion iff prime factors of q are only 2 and/or 5}
Recurring Decimal: 0.a b = (ab)/99, 0.a bc = (abc)/999, 0.0a b = (ab)/990\text{Recurring Decimal: 0.a b = (ab)/99, 0.a bc = (abc)/999, 0.0a b = (ab)/990}
Pure Recurring: All digits repeat (e.g., 0.333... = 1/3). Mixed Recurring: Some digits don’t repeat (e.g., 0.1666... = 1/6)\text{Pure Recurring: All digits repeat (e.g., 0.333... = 1/3). Mixed Recurring: Some digits don't repeat (e.g., 0.1666... = 1/6)}
Comparing Decimals: Compare digit by digit from left to right after equalizing decimal places\text{Comparing Decimals: Compare digit by digit from left to right after equalizing decimal places}
Decimal Addition/Subtraction: Align decimal points vertically, then perform operation\text{Decimal Addition/Subtraction: Align decimal points vertically, then perform operation}
Decimal Multiplication: Multiply as whole numbers, then place decimal point counting total decimal places in factors\text{Decimal Multiplication: Multiply as whole numbers, then place decimal point counting total decimal places in factors}
Decimal Division: Shift decimal points to make divisor whole number, then divide\text{Decimal Division: Shift decimal points to make divisor whole number, then divide}

Key Concepts

Types of Decimals

Terminating decimals end after finite digits (e.g., 0.5, 0.25). Non-terminating decimals continue infinitely: pure recurring where all digits repeat (0.333...), and mixed recurring where some digits don't repeat (0.1666...). A fraction gives terminating decimal only when denominator (in lowest terms) has prime factors 2 and/or 5 only.

Converting Recurring Decimals to Fractions

For pure recurring: 0.a = a/9, 0.ab = ab/99, 0.abc = abc/999. For mixed recurring like 0.1a b : subtract non-repeating part from number with bar, divide by 9s for repeating digits followed by 0s for non-repeating. Example: 0.16 = (16-1)/90 = 15/90 = 1/6. The general formula is (entire number - non-repeating part) / (9s for repeating digits, 0s for non-repeating digits).

Decimal Operations

Addition/Subtraction: Line up decimal points. You may add trailing zeros to equalize places. Multiplication: Ignore decimals, multiply as integers, then count total decimal places in factors for the result. Division: Convert divisor to whole number by shifting decimal points equally in both numbers, then divide normally.

Comparing and Ordering Decimals

To compare decimals, first write them with equal decimal places by adding trailing zeros. Then compare digit by digit from left. For example, comparing 0.3, 0.29, 0.305: write as 0.300, 0.290, 0.305. Comparing: 0.305 > 0.300 > 0.290. When arranging in ascending/descending order, this method ensures accuracy.

Rounding and Approximation

To round to n decimal places: look at digit at (n+1)th place. If it's 5 or more, round up the nth digit; otherwise keep it. For significant figures, count from first non-zero digit. Rounding is crucial for quick estimations in multiple-choice questions where exact calculation isn't necessary.

Solved Examples

Problem 1:

Convert 0.125 to a fraction in lowest terms.

Solution:

Step 1: Write 0.125 as 125/1000 (3 decimal places means denominator is 1000).
Step 2: Find GCD of 125 and 1000. GCD(125, 1000) = 125.
Step 3: Divide both by 125: 125/125 / 1000/125 = 1/8.
Answer: 1/8

Problem 2:

Express 0.45 (0.4555...) as a fraction.

Solution:

Step 1: Identify this as a mixed recurring decimal (4 doesn't repeat, 5 repeats).
Step 2: Formula: (entire number without decimal - non-repeating part) / (9 for each repeating digit, 0 for each non-repeating digit).
Step 3: Entire number without decimal point and bar: 45. Non-repeating part: 4.
Step 4: Numerator = 45 - 4 = 41.
Step 5: Denominator = 90 (one 9 for repeating 5, one 0 for non-repeating 4).
Step 6: Fraction = 41/90.
Answer: 41/90

Problem 3:

Calculate: 0.2 + 0.3 + 0.4

Solution:

Step 1: Convert each to fraction. 0.2 = 2/9, 0.3 = 3/9 = 1/3, 0.4 = 4/9.
Step 2: Sum = 2/9 + 3/9 + 4/9 = 9/9 = 1.
Alternative method: 0.222... + 0.333... + 0.444... = 0.999... = 1.
Answer: 1

Problem 4:

If 1/7 = 0.142857 (repeating), what is the 100th digit after the decimal point?

Solution:

Step 1: The repeating block '142857' has 6 digits.
Step 2: Find 100 mod 6 = 4 (since 100 = 16x6 + 4).
Step 3: The 4th digit in '142857' is 8 (1st=1, 2nd=4, 3rd=2, 4th=8).
Answer: 8

Tips & Tricks

  • Memorize common fraction-decimal conversions: 1/2=0.5, 1/4=0.25, 3/4=0.75, 1/5=0.2, 1/8=0.125, 1/3~=0.333, 1/6~=0.167, 1/7~=0.143, 1/9~=0.111
  • For recurring decimals, count the length of the repeating cycle - this determines the denominator pattern (9, 99, 999, etc.)
  • When adding/subtracting decimals, always align decimal points vertically to avoid place value errors
  • A fraction has terminating decimal expansion only if denominator (in lowest terms) = 2^m x 5^n for non-negative integers m, n
  • For quick comparison, convert all decimals to same number of places by adding trailing zeros
  • In mixed recurring decimals like 0.ab c , the numerator is (abc - a) and denominator is 990 (two 9s for repeating digits, one 0 for non-repeating)
  • Use estimation to check answers: 0.4 x 0.5 should be around 0.2, not 2 or 0.02

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