Civil Service Exam (Subprofessional) Numerical Ability — Fractions — Operations, Conversion & ComparisonStudy Notes

Understanding the three main types of fractions is crucial for solving problems effectively. Each type has distinct characteristics and applications in mathematical operations. Proper Fractions are fractions where the numerator (top number) is less than the denominator (bottom number). These fractions represent values less than one whole. Examples include 1/2, 3/8, 5/7, and 10/15. When you see a proper fraction, you immediately know its value is between 0 and 1. Improper Fractions have numerators greater than or equal to their denominators. These represent values equal to or greater than one whole. Examples include 5/2, 4/3, 6/4, 12/8, 9/7, and 50/15. These fractions can be converted to mixed numbers for easier understanding. Mixed Fractions combine a whole number with a proper fraction. They provide a clear visual representation of quantities greater than one. Examples include 2 1/2, 1 1/3, 1 2/4, 1 4/8, 1 2/7, and 3 5/15. Mixed fractions are commonly used in everyday situations like cooking measurements or construction.

Equivalent fractions are different fractions that represent the same value. Understanding this concept is essential for fraction operations and comparisons. The key principle is that multiplying or dividing both the numerator and denominator by the same non-zero number creates equivalent fractions. To find equivalent fractions, multiply both the numerator and denominator by the same number. For example, 1/3 = 2/6 = 3/9 = 4/12. Each fraction represents the same portion of a whole, just divided into different numbers of equal parts. Simplifying fractions involves reducing them to their lowest terms by finding the Greatest Common Factor (GCF) of the numerator and denominator, then dividing both by this factor. For instance, 12/18 can be simplified by finding the GCF of 12 and 18, which is 6. Dividing both by 6 gives us 2/3. Another method for simplification is successive division by prime numbers (2, 3, 5, 7, 11, 13...) until no further reduction is possible. This method is particularly useful when the GCF is not immediately obvious.

Converting between mixed and improper fractions is a fundamental skill required for fraction operations. This conversion process follows specific steps that ensure accuracy in calculations. To convert a mixed fraction to an improper fraction: (1) Multiply the whole number by the denominator, (2) Add this product to the numerator to get the new numerator, (3) Keep the same denominator. For example, 2 5/8 becomes (2×8+5)/8 = 21/8. To convert an improper fraction to a mixed number: (1) Divide the numerator by the denominator, (2) The quotient becomes the whole number, (3) The remainder becomes the new numerator, (4) Keep the same denominator. For example, 17/5 = 3 2/5 because 17÷5 = 3 remainder 2. These conversions are essential for addition, subtraction, multiplication, and division of fractions, especially when dealing with mixed numbers in word problems.

Adding and subtracting fractions requires different approaches depending on whether the denominators are the same or different. Mastering these operations is crucial for solving complex mathematical problems. For fractions with the same denominator (similar fractions), simply add or subtract the numerators and keep the denominator unchanged. For example, 2/7 + 3/7 = 5/7 and 5/8 - 2/8 = 3/8. For fractions with different denominators (dissimilar fractions), you must first find the Least Common Denominator (LCD) or Least Common Multiple (LCM). Then convert each fraction to an equivalent fraction with the LCD as the denominator before performing the operation. When adding or subtracting mixed numbers, you can either convert them to improper fractions first, or handle the whole numbers and fractions separately. Always simplify your final answer to its lowest terms. Special cases include subtracting a fraction from a whole number (convert the whole number to a mixed number) and borrowing when the fraction being subtracted is larger than the fraction it's being subtracted from.

Multiplication and division of fractions follow straightforward rules that are often easier than addition and subtraction because you don't need to find common denominators. To multiply fractions, simply multiply the numerators together and multiply the denominators together: (a/b) × (c/d) = (a×c)/(b×d). Then simplify if possible. When multiplying by whole numbers, convert the whole number to a fraction with denominator 1. To divide fractions, multiply by the reciprocal (flip) of the divisor: (a/b) ÷ (c/d) = (a/b) × (d/c). This is often remembered as 'keep, change, flip' - keep the first fraction, change division to multiplication, flip the second fraction. For mixed numbers, convert to improper fractions first, then perform the operation. Cross-cancellation can simplify calculations by reducing numbers before multiplying. Division of fractions often appears in word problems involving rates, proportions, and 'how many times' questions.

Comparing fractions allows us to determine which fraction is larger, smaller, or if they are equal. This skill is essential for solving problems involving rankings, measurements, and proportions. When fractions have the same numerator, the fraction with the smaller denominator is larger because it represents larger pieces. For example, 1/3 > 1/4 > 1/6. When fractions have the same denominator, the fraction with the larger numerator is larger. For example, 5/8 > 3/8 > 1/8. For fractions with different numerators and denominators, convert them to equivalent fractions with the same denominator (using LCD), then compare numerators. Alternatively, cross-multiply to compare: for a/b and c/d, if a×d > b×c, then a/b > c/d. When comparing mixed numbers, first compare the whole number parts. If they're equal, then compare the fractional parts using the methods above.

Converting fractions to decimals is a valuable skill that helps in calculations and comparisons, especially when using calculators or working with mixed number formats. The most direct method is to divide the numerator by the denominator. This can be done using long division or by recognizing decimal equivalents of common fractions. For fractions with denominators of 10, 100, 1000, etc., the conversion is straightforward: place the decimal point in the numerator counting from right to left. For 3/10, place the decimal point one place from the right to get 0.3. For 25/100, place it two places to get 0.25. For other fractions, you may need to create equivalent fractions with denominators of powers of 10, or simply perform the division. Some fractions result in repeating decimals, while others terminate. Common fraction-decimal equivalents that are useful to memorize include: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125, and 1/3 = 0.333...