Computation of Whole Numbers, Decimals, and Fractions, Relationships Between Numbers

Welcome! In this section, we're going to dive deep into the fundamental building blocks of quantitative aptitude: whole numbers, decimals, and fractions. Understanding how these numbers work and their relationships is crucial for solving a wide range of problems. We'll cover how to perform calculations with them and how to easily convert between different forms.

Whole Numbers: The Foundation

Whole numbers are the simplest form of numbers we use daily. They include zero and all positive integers. Think of them as the numbers you'd use to count objects: 0, 1, 2, 3, and so on, extending infinitely.

Properties of Whole Numbers

Whole numbers have some key properties that make calculations predictable:

• Closure Property: The sum or product of any two whole numbers is always a whole number. (e.g., 3 + 5 = 8, 4 × 6 = 24).
• Commutative Property: The order of numbers doesn't affect the sum or product. (e.g., a + b = b + a, a × b = b × a). For example, 7 + 2 is the same as 2 + 7 (both equal 9), and 3 × 5 is the same as 5 × 3 (both equal 15).
• Associative Property: When adding or multiplying three or more numbers, the grouping doesn't change the result. (e.g., (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)). For instance, (2 + 3) + 4 = 5 + 4 = 9, and 2 + (3 + 4) = 2 + 7 = 9. Similarly, (2 × 3) × 4 = 6 × 4 = 24, and 2 × (3 × 4) = 2 × 12 = 24.
• Distributive Property: Multiplication distributes over addition. (e.g., a × (b + c) = (a × b) + (a × c)). For example, 5 × (2 + 3) = 5 × 5 = 25, and (5 × 2) + (5 × 3) = 10 + 15 = 25.
• Identity Property: Adding 0 to any whole number doesn't change it (0 is the additive identity). Multiplying any whole number by 1 doesn't change it (1 is the multiplicative identity). (e.g., a + 0 = a, a × 1 = a).

Decimals: Parts of a Whole

Decimals are a way to represent numbers that are less than one or have fractional parts. They use a decimal point to separate the whole number part from the fractional part. For example, 0.5 means "five-tenths," and 3.14 means "three and fourteen-hundredths."

Understanding Place Value in Decimals

The digits to the right of the decimal point represent fractions with denominators that are powers of 10.

• The first digit to the right of the decimal point is the tenths place (1/10).
• The second digit is the hundredths place (1/100).
• The third digit is the thousandths place (1/1000), and so on.

For example, in the number 12.345:

• 1 is in the tens place.
• 2 is in the ones place.
• 3 is in the tenths place (3/10).
• 4 is in the hundredths place (4/100).
• 5 is in the thousandths place (5/1000).

Operations with Decimals

Performing arithmetic with decimals is similar to whole numbers, but you must align the decimal points correctly.

• Addition and Subtraction: Line up the decimal points vertically and add or subtract as usual. Add trailing zeros if needed to make the number of decimal places equal.

  Example: 12.5 + 3.078

  12.500
  + 3.078
  -------
  15.578

• Multiplication: Multiply the numbers as if they were whole numbers, ignoring the decimal points initially. Then, count the total number of decimal places in the original numbers and place the decimal point in the product so it has that many decimal places.

  Example: 2.5 × 1.2

  Multiply 25 × 12 = 300.

  2.5 has one decimal place. 1.2 has one decimal place. Total = 2 decimal places.

  So, the answer is 3.00 or simply 3.

• Division: To divide by a decimal, first convert the divisor into a whole number by moving its decimal point to the right. Move the decimal point in the dividend the same number of places to the right. Then, perform the division. Place the decimal point in the quotient directly above the decimal point in the new dividend.

  Example: 15.6 ÷ 0.3

  Move the decimal in 0.3 one place to the right to make it 3.

  Move the decimal in 15.6 one place to the right to make it 156.

  Now divide 156 ÷ 3 = 52.

Fractions: Representing Parts of a Whole

A fraction represents a part of a whole. It consists of two parts: the numerator (the top number) and the denominator (the bottom number). The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have.

• Proper Fraction: The numerator is smaller than the denominator (e.g., 1/2, 3/4).
• Improper Fraction: The numerator is greater than or equal to the denominator (e.g., 5/4, 7/7).
• Mixed Number: A whole number combined with a proper fraction (e.g., 1 1/4, 2 1/2).

Operations with Fractions

Working with fractions requires attention to their denominators.

• Addition and Subtraction: To add or subtract fractions, they must have a common denominator. If they don't, find the Least Common Multiple (LCM) of the denominators. Convert each fraction to an equivalent fraction with the LCM as the new denominator. Then, add or subtract the numerators.

  Example: 1/3 + 1/4

  LCM of 3 and 4 is 12.

  1/3 = 4/12 (multiply numerator and denominator by 4)

  1/4 = 3/12 (multiply numerator and denominator by 3)

  So, 4/12 + 3/12 = 7/12.

• Multiplication: To multiply fractions, multiply the numerators together and multiply the denominators together. Simplify the resulting fraction if possible.

  Example: 2/3 × 3/4

  (2 × 3) / (3 × 4) = 6/12.

  Simplify 6/12 to 1/2.

• Division: To divide fractions, invert the second fraction (find its reciprocal) and multiply.

  Example: 1/2 ÷ 3/4

  Invert 3/4 to get 4/3.

  Now multiply: 1/2 × 4/3 = (1 × 4) / (2 × 3) = 4/6.

  Simplify 4/6 to 2/3.

Relationships Between Numbers: Conversion and Comparison

Understanding how whole numbers, decimals, and fractions relate to each other is key to solving problems efficiently.

Converting Between Forms

You'll often need to convert between these forms.

• Fraction to Decimal: Divide the numerator by the denominator.

  Example: 3/4 = 3 ÷ 4 = 0.75

  Example: 1/8 = 1 ÷ 8 = 0.125

• Decimal to Fraction: Write the decimal as a fraction using its place value. Then, simplify the fraction.

  Example: 0.6 = 6/10 = 3/5

  Example: 0.125 = 125/1000. Divide both by 125 to get 1/8.

• Mixed Number to Improper Fraction: Multiply the whole number by the denominator, add the numerator, and keep the same denominator.

  Example: 2 1/4 = (2 × 4 + 1) / 4 = 9/4

• Improper Fraction to Mixed Number: Divide the numerator by the denominator. The quotient is the whole number part, the remainder is the new numerator, and the denominator stays the same.

  Example: 7/3 = 7 ÷ 3. Quotient is 2, remainder is 1. So, 7/3 = 2 1/3.

Comparing Numbers

To compare numbers in different forms, it's easiest to convert them to the same form, usually decimals or fractions with a common denominator.

• Comparing Decimals: Start from the leftmost digit. The number with the larger digit in the first place where they differ is the larger number.

  Example: Compare 0.56 and 0.61. The digits in the tenths place are 5 and 6. Since 6 is greater than 5, 0.61 is larger than 0.56.

• Comparing Fractions: If denominators are the same, compare numerators. If denominators are different, find a common denominator and then compare numerators. Alternatively, convert to decimals.

  Example: Compare 2/3 and 3/4.

  Common denominator is 12.

  2/3 = 8/12

  3/4 = 9/12

  Since 9/12 > 8/12, then 3/4 > 2/3.

  Using decimals: 2/3 ≈ 0.667, 3/4 = 0.75. Since 0.75 > 0.667, 3/4 is larger.

Important Concepts and Formulas

Let's summarize some key computational aspects.

• LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more numbers. Essential for adding/subtracting fractions.
• GCD (Greatest Common Divisor) / HCF (Highest Common Factor): The largest positive integer that divides two or more numbers without leaving a remainder. Useful for simplifying fractions.
• Formula for LCM and GCD: For two numbers 'a' and 'b', $a \times b = LCM(a, b) \times GCD(a, b)$. This is a very handy shortcut.

Practice Problems and Strategies

When faced with a quantitative aptitude problem involving these numbers, follow these steps:

1. Identify the numbers: Are they whole numbers, decimals, or fractions?
2. Understand the operation: Are you adding, subtracting, multiplying, or dividing?
3. Convert if necessary: If numbers are in different forms, convert them to a consistent form (e.g., all fractions or all decimals) for easier calculation.
4. Apply the correct rules: Remember the rules for operating with fractions (common denominators) and decimals (aligning decimal points).
5. Simplify: Always simplify your answer, especially if it's a fraction.
6. Check your work: If possible, estimate your answer or check it using a different method.

Example Problem:

Calculate: $3 \frac{1}{2} + 0.75 \times \frac{4}{3}$

Step 1: Convert all to fractions.

• $3 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2}$
• $0.75 = \frac{75}{100} = \frac{3}{4}$
• $\frac{4}{3}$ is already a fraction.

The expression becomes: $\frac{7}{2} + \frac{3}{4} \times \frac{4}{3}$

Step 2: Perform multiplication first (Order of Operations - BODMAS/PEMDAS).

$\frac{3}{4} \times \frac{4}{3} = \frac{3 \times 4}{4 \times 3} = \frac{12}{12} = 1$

The expression is now: $\frac{7}{2} + 1$

Step 3: Perform addition.

$1$ can be written as $\frac{2}{2}$.

$\frac{7}{2} + \frac{2}{2} = \frac{7+2}{2} = \frac{9}{2}$

Step 4: Convert to mixed number or decimal if required.

$\frac{9}{2} = 4 \frac{1}{2}$ or $4.5$

The final answer is $4.5$ or $4 \frac{1}{2}$.