Theory & Concepts

SSC CGL Fractions and Decimals Questions, Formulas & Tricks

Get comprehensive theory, expert shortcuts, and hand-picked practice questions for Fractions & Decimals specifically designed for the SSC CGL 2025-26 pattern.

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Practice Notes

Fractions and Decimals are the "building blocks" of calculation. In SSC CGL, speed is currency, and knowing how to quickly compare fractions or convert recurring decimals into fractions can save you minutes in the exam.

This module covers the essential shortcuts for fraction comparison, the logic of bars in decimals, and the simplification of complex ladder fractions.

Learning path

  • Comparison Shortcuts
  • Recurring Decimals (Bar)
  • Continued (Ladder) Fractions
  • LCM/HCF of Fractions
  • 10 Exam-Level Examples

1. Comparison of Fractions

Don't use the division method for every fraction. Use these 3 shortcuts instead:

Cross Multi

To compare a/ba/b and c/dc/d, calculate adad and bcbc. If ad>bcad > bc, then a/b>c/da/b > c/d.

Difference Method

If the difference between Num and Denom is the same, the fraction with the larger numerator is larger (for proper fractions).

Decimal Approximation

Use for standard values: 1/8=0.1251/8 = 0.125, 1/7≈0.1421/7 \approx 0.142, 1/9=0.1111/9 = 0.111.

2. Recurring Decimals (Bar Notation)

Converting recurring decimals to vulgar fractions is a frequent Tier-1 pattern.

Rule of 9s

Write the number without the decimal and bar. Subtract the non-recurring part. Put as many 9s as recurring digits and as many 0s as non-recurring digits in denominator.

Pure Recurring

0.aˉ=a90.\bar{a} = \frac{a}{9}
0.ab‾=ab990.\overline{ab} = \frac{ab}{99}

Mixed Recurring

0.abˉ=ab−a900.a\bar{b} = \frac{ab - a}{90}
0.abcd‾=abcd−ab99000.ab\overline{cd} = \frac{abcd - ab}{9900}

3. Continued (Ladder) Fractions

Ladder fractions look intimidating but can be solved from bottom to top using simple addition.

Shortcut for Ladder Fractions:

Start with the last fraction a/ba/b. Write a,ba, b. Perform addition/multiplication operations as you move up. The second last number becomes the numerator and the last number becomes the denominator.

4. 10 Solved examples

Question 01Exam Pattern

Which of the following fractions is the largest? \( 2/3, 3/5, 8/11, 11/17 \)

8/11
2/3
3/5
11/17
Correct answer: a) 8/11

Solution

Step 1: Calculate approximate decimal values for each fraction.
\( 2/3 \approx 0.666 \)
\( 3/5 = 0.600 \)
\( 8/11 \approx 0.727 \)
\( 11/17 \approx 0.647 \)
Final calculation: Comparing the values, \( 0.727 \) is the largest. Thus, \( 8/11 \) is the answer.
Question 02Exam Pattern

Convert \( 0.3\bar{7} \) into a vulgar fraction.

17/45
37/99
34/90
37/90
Correct answer: a) 17/45

Solution

Step 1: Use the formula for mixed recurring decimals: \( \frac{\text{Total} - \text{Non-recurring}}{9\dots0\dots} \).
Step 2 (Numerator): Write the number as 37 and subtract the non-recurring part 3. \( 37 - 3 = 34 \).
Step 3 (Denominator): Put one 9 (for recurring 7) and one 0 (for non-recurring 3) = 90.
Final calculation: Fraction = \( 34/90 \). Simplifying by 2, we get \( 17/45 \).
Question 03Exam Pattern

Find the value of \( 0.\bar{3} + 0.\bar{6} + 0.\bar{7} + 0.\bar{8} \).

2.66...
2.33...
2.55...
2.77...
Correct answer: a) 2.66...

Solution

Step 1: Convert each recurring decimal into its fraction form: \( 3/9, 6/9, 7/9, 8/9 \).
Step 2: Find the sum of the fractions. Sum = \( (3+6+7+8)/9 = 24/9 \).
Step 3: Simplify the fraction to \( 8/3 \).
Final calculation: Divide 8 by 3 to get the recurring decimal: \( 2.666... = 2.\bar{6} \).
Question 04Exam Pattern

What is the HCF of \( 2/3, 8/9, 64/81, 10/27 \)?

2/81
2/3
10/81
8/27
Correct answer: a) 2/81

Solution

Step 1: Formula for HCF of Fractions = \( \frac{HCF \text{ of Numerators}}{LCM \text{ of Denominators}} \).
Step 2: Find HCF of numerators (2, 8, 64, 10). The HCF is 2.
Step 3: Find LCM of denominators (3, 9, 81, 27). The LCM is 81.
Final calculation: Result = \( 2/81 \).
Question 05Exam Pattern

Find the value of \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{3}}} \).

7/4
11/7
7/11
4/7
Correct answer: b) 11/7

Solution

Step 1 (Bottom layer): \( 1 + 1/3 = 4/3 \).
Step 2 (Middle layer): \( 1 + 1/(4/3) = 1 + 3/4 = 7/4 \).
Step 3 (Top layer): \( 1 + 1/(7/4) = 1 + 4/7 = 11/7 \).
Final calculation: The final simplified value is \( 11/7 \).
Question 06Exam Pattern

The product of two decimals is 0.768. If one of them is 1.6, find the other.

0.48
0.42
0.52
0.38
Correct answer: a) 0.48

Solution

Step 1: Set up the equation: \( 1.6 \times x = 0.768 \).
Step 2: Divide both sides by 1.6. \( x = 0.768 / 1.6 \).
Step 3: To simplify, multiply both by 10 to remove one decimal: \( 7.68 / 16 \).
Final calculation: \( 16 \times 4 = 64 \), remainder 12. \( 16 \times 8 = 128 \). Thus, \( x = 0.48 \).
Question 07Exam Pattern

Which of the following is in descending order (Largest to Smallest)?

5/8, 9/13, 11/17
11/17, 9/13, 5/8
9/13, 11/17, 5/8
9/13, 5/8, 11/17
Correct answer: c) 9/13, 11/17, 5/8

Solution

Step 1: Calculate approximate decimal values: \( 9/13 \approx 0.692 \), \( 5/8 = 0.625 \), \( 11/17 \approx 0.647 \).
Step 2: Rank them from largest to smallest: \( 0.692 > 0.647 > 0.625 \).
Final calculation: The correct descending order is \( 9/13, 11/17, 5/8 \). Option d matches this. (Correction: Option c label used in code, ensuring match).
Question 08Exam Pattern

Value of \( 3.\bar{8} - 2.\bar{7} + 1.\bar{2} \).

2.\bar{3}
2.\bar{1}
2.\bar{2}
2.\bar{4}
Correct answer: a) 2.\bar{3}

Solution

Step 1: Separate the whole numbers and the recurring parts.
Step 2 (Whole numbers): \( 3 - 2 + 1 = 2 \).
Step 3 (Recurring parts): \( 8/9 - 7/9 + 2/9 = 3/9 = 1/3 \).
Final calculation: Total = \( 2 + 1/3 = 2.333... = 2.\bar{3} \).
Question 09Exam Pattern

Simplify: \( \frac{0.1 \times 0.1 \times 0.1 + 0.02 \times 0.02 \times 0.02}{0.2 \times 0.2 \times 0.2 + 0.04 \times 0.04 \times 0.04} \).

0.125
0.25
0.5
0.0625
Correct answer: a) 0.125

Solution

Step 1: Let \( a = 0.1, b = 0.02 \).
Step 2: Observe that the denominator is exactly \( (2a)^3 + (2b)^3 = 8(a^3 + b^3) \).
Step 3: The expression simplifies to \( \frac{a^3 + b^3}{8(a^3 + b^3)} = 1/8 \).
Final calculation: \( 1/8 = 0.125 \).
Question 010Exam Pattern

If \( \frac{1}{a + \frac{1}{b + \frac{1}{c}}} = \frac{9}{26} \), find \( a + b + c \).

11
10
12
13
Correct answer: a) 11

Solution

Step 1: Inverse both sides: \( a + \frac{1}{b + \frac{1}{c}} = 26/9 \).
Step 2: Express as mixed fraction: \( 26/9 = 2 + 8/9 \). So, \( a = 2 \).
Step 3: Now, \( b + 1/c = 9/8 = 1 + 1/8 \). So, \( b = 1 \) and \( c = 8 \).
Final calculation: \( a + b + c = 2 + 1 + 8 = 11 \).

5. Strategy errors to avoid

Error 01Reciprocal slip: Confusing HCF of numerators/LCM of denominators when calculating LCM of fractions.
Error 02Ladder lag: Starting ladder fractions from the top instead of the bottom.
Error 03Decimal trap: Dividing by 9 instead of 90 for mixed recurring decimals.
Error 04Approximation risk: Rounding off too early in comparison questions.
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