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compulsory · CSS study notes

6. Fractions

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 6 (Fractions)

Past Paper Weightage

  • Appears both directly and embedded in word problems (ratios, average, percentage) — high indirect frequency.

Key Concepts

  • Proper fraction: numerator < denominator (e.g., 3/4); Improper: numerator ≥ denominator (7/4); Mixed number: whole + proper fraction (1¾)
  • Equivalent fractions: same value, different form — found by multiplying/dividing numerator and denominator by the same number
  • LCD (Least Common Denominator): needed before adding/subtracting fractions with different denominators
  • Operations:
    • Add/Subtract: convert to common denominator first, then add/subtract numerators
    • Multiply: multiply numerators together and denominators together (cross-cancel first to simplify)
    • Divide: multiply by the reciprocal of the second fraction ("keep, change, flip")
  • Comparing fractions: cross-multiply, or convert to a common denominator

Worked Example

  • 2/3 + 3/4 = 8/12 + 9/12 = 17/12 = 1 5/12
  • 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4

Past Paper Style Numericals (Solved)

  1. A tank is 3/5 full. If 40 liters more are added, it becomes completely full. Find the tank's capacity.
    • Step 1: If the tank is 3/5 full, the empty portion is 1 − 3/5 = 2/5 of the total capacity.
    • Step 2: That empty 2/5 portion equals the 40 liters added, so 2/5 × (capacity) = 40.
    • Step 3: Solve for capacity: capacity = 40 ÷ (2/5) = 40 × (5/2) = 100 liters.
  2. Simplify: (2½ + 1⅓) ÷ (1¾)
    • Step 1: Convert all mixed numbers to improper fractions: 2½ = 5/2, 1⅓ = 4/3, 1¾ = 7/4.
    • Step 2: Add the first two: 5/2 + 4/3 — common denominator 6 → 15/6 + 8/6 = 23/6.
    • Step 3: Divide by 7/4 (multiply by its reciprocal): 23/6 × 4/7 = 92/42, which simplifies (÷2) to 46/21 = 2 4/21.
  3. Rehan spends 1/4 of his salary on rent and 1/3 on food. What fraction of his salary remains?
    • Step 1: Find a common denominator for 1/4 and 1/3, which is 12: 1/4 = 3/12, 1/3 = 4/12.
    • Step 2: Add the spent fractions: 3/12 + 4/12 = 7/12.
    • Step 3: Subtract from the whole: 1 − 7/12 = 12/12 − 7/12 = 5/12 remains.
  4. Evaluate: 3/4 of 2/5 of 60
    • Step 1: Work from left to right (or innermost first) — first find 2/5 of 60: (2/5)×60 = 120/5 = 24.
    • Step 2: Now find 3/4 of that result (24): (3/4)×24 = 72/4.
    • Step 3: Simplify: 72/4 = 18.

MCQ Traps

  • You do NOT need a common denominator to multiply or divide fractions — only for adding/subtracting
  • Dividing by a fraction less than 1 makes the result larger, not smaller (common intuition error)

Self-Test Questions

  1. Simplify: 3/5 − 1/4
  2. Evaluate: 7/8 × 4/21
  3. Which is larger, 5/9 or 7/12?

Past-Paper MCQs (FPSC)

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The FPSC MCQ papers you shared (2024 and 2025) had no direct fraction question.

Past-Paper Questions with Answers (FPSC)

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Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me.

  • CSS 2025 Q7(d): Arrange the following fractions in descending order. (5)

    SetDecimal valuesDescending order
    (a) 7/12, 4/7, 9/14, 25/420.583, 0.571, 0.643, 0.5959/14 > 25/42 > 7/12 > 4/7
    (b) 5/8, 7/12, 5/9, 13/240.625, 0.583, 0.556, 0.5425/8 > 7/12 > 5/9 > 13/24
    (c) 6/35, 5/21, 2/7, 4/150.171, 0.238, 0.286, 0.2672/7 > 4/15 > 5/21 > 6/35
    (d) 2/5, 6/11, 7/15, 9/20, 13/250.400, 0.545, 0.467, 0.450, 0.5206/11 > 13/25 > 7/15 > 9/20 > 2/5
    (e) 1/5, 2/7, 3/8, 4/13, 5/170.200, 0.286, 0.375, 0.308, 0.2943/8 > 4/13 > 5/17 > 2/7 > 1/5
    • Methods: convert each to a decimal, or use cross-multiplication for pairs, or compare with a benchmark such as 1/2

CSS Exam Perspective

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  • Most repeated area: ordering fractions (2025, with five sets in one question)
  • Tip: dividing numerator by denominator to 3 decimals is the fastest and safest method with no calculator