04 — The Fraction Engine
SSC CGL TIER-I & II / CALCULATION FOUNDATION
A fraction is not a strange form of a number.
IT IS SIMPLY A PRECISE WAY OF DESCRIBING A PART OF A WHOLE, A RATIO, OR A DIVISION.
IT IS SIMPLY A PRECISE WAY OF DESCRIBING A PART OF A WHOLE, A RATIO, OR A DIVISION.
Fractions will become the language connecting Percentage, Ratio, Profit & Loss, Interest, Mixture, Time & Work and DI. We are going to make them comfortable, not merely memorized.
04.1 What is a Fraction?
A fraction has two parts:
ab
Numerator (Top) Tells us how many parts we are considering.
Denominator (Bottom) Tells us how many equal parts the whole has been divided into.
Example:
35
means 3 parts out of 5 equal parts.
04.2 The Absolute Rule
A fraction may have numerator 0, but its denominator can NEVER be 0. Division by zero is undefined.
The Three Forms
04.3 Proper Fraction
NUMERATOR < DENOMINATOR
Value is less than 1.
12
, 35
04.4 Improper Fraction
NUMERATOR ≥ DENOMINATOR
Value is at least 1.
72
, 95
04.5 Mixed Number
WHOLE NUMBER + PROPER FRACTION
Means Addition. 2
13
= 2 + 13
Conversions
04.6 Mixed → Improper
Convert 2 ⅓
1. Multiply whole number by denominator (2 × 3 = 6)
2. Add numerator (6 + 1 = 7)
3. Put over original denominator
2
13
= 73
04.7 Improper → Mixed
Convert 17/5
1. Divide numerator by denominator (17 ÷ 5 = 3)
2. Find remainder (17 - 15 = 2)
3. Whole = quotient, Numerator = remainder
175
= 3 25
The Core Fraction Law
04.8 Equivalent Fractions
If you multiply or divide both the numerator and denominator by the same non-zero number, the value of the fraction does not change. This is the foundation of all fraction manipulation.
12
=
24
=
510
=
50100
They look different, but represent the exact same value.
Simplification
04.10 Simplifying
Divide top and bottom by a common factor.
1824
→ divide by 6 → 34
04.11 Lowest Form
A fraction is in lowest terms when numerator and denominator have no common factor greater than 1.
04.12 The Fastest SSC Habit
Before doing any serious fraction calculation, always ask: Can I cancel something first?
12 × 2515
→ cancel 12/15 →
4 × 255
→
20
Addition & Subtraction
04.15 & 04.16 Same Denominator
Add numerators:
27
+ 37
= 57
Subtract numerators:
67
- 27
= 47
04.17 & 04.19 Different Denominators
You must find a common denominator (LCM) first.
Addition:
12
+ 13
(LCM=6) → 36
+ 26
= 56
04.18 General Rule:
ab
+ cd
= ad + bcbd
Multiplication & Division
04.20 Multiplying
Numerator × Numerator
Denominator × Denominator
23
× 45
= 815
04.21 Cancel First!
Do not multiply blindly. Cancel cross-wise before multiplying to keep numbers small.
04.22 & 04.23 Division
Division by a fraction is multiplication by its reciprocal.
KEEPCHANGE (÷ to ×)FLIP
23
÷ 45
→
23
× 54
→
1012
=
56
Comparing Fractions
04.25 Between 0 & 1
A proper positive fraction always lies between 0 and 1.
04.26 Same Denominator
Larger Numerator = Larger Fraction.
37
< 57
04.26 Same Numerator
Smaller Denominator = Larger Pieces = Larger Fraction.
35
> 38
04.28 The Cross-Multiplication Rule
To compare different fractions rapidly.
35
? 47
3×7 vs 4×5
→
21 > 20, therefore
35
> 47
The Fundamental Landmarks
THESE SHOULD BECOME AUTOMATIC. THE BRIDGE TO PERCENTAGE.
Halves & Quarters
12
50%14
25%34
75%Thirds & Sixths
13
33 ⅓%23
66 ⅔%16
16 ⅔%Eighths
18
12.5%38
37.5%58
62.5%78
87.5%Other Key Roots
15
20%19
11 ⅑%110
10%111
9.09%112
8 ⅓%The Three-Form Triangle
You should gradually see these as the same quantity in three different languages.
34
↔
0.75
↔
75%
Fraction is Division: 3/4 means 3 ÷ 4.
To Decimal: Just divide it.
To Percentage: Multiply by 100%.
Percentage to Fraction: Put over 100 and simplify (e.g., 25% = 25/100 = 1/4).
Fraction of a Number
"Of" means multiplication. The denominator should make you think: "Divide first."
Method
78
of 64
First: 64 ÷ 8 = 8
Then: 8 × 7 = 56
Then: 8 × 7 = 56
Speed Application
720
of 800
800 ÷ 20 = 40
40 × 7 = 280
(Much faster than decimals)
40 × 7 = 280
(Much faster than decimals)
04.46 Sanity Checks
Is the fraction proper? Its value should be less than 1.
Did I multiply by a fraction < 1? The result should decrease. (100 × 3/4 = 75)
Did I divide by a fraction < 1? The result should increase. (20 ÷ 1/2 = 40)
The Practice Labs
Self-Test: Basics & Operations
Press and hold [?] to reveal.
Simplify:
1218
23
25
+ 15
35
34
- 14
12
23
× 910
35
35
÷ 67
710
58
+ 14
78
Self-Test: Numbers & Conversion
Apply "Of" and the Conversion Triangle.
34
of 8060
58
of 6440
710
of 300210
12
to %50%
34
to %75%
0.125 to frac
18
04.52 Master Test
Enter your calculated answers. Use basic formats like '5/8' or '175'.
Simp:
4872
=23
+ 16
=78
- 14
=35
× 109
=45
÷ 25
=710
of 250 =38
of 160 =62.5% as frac =
If 3/8 is completed, what fraction remains?
If 4/5 of a number is 64, what is the number?
23
of 34
of 90 =04.54 The Master Principles
A fraction is division.
Equivalent fractions have the same value.
Simplify before calculating.
Cancel before multiplying.
“Of” means multiplication.
Dividing by a fraction means multiplying by its reciprocal.
Fractions, decimals and percentages are three languages for the same quantity.
Always check whether the answer's size makes sense.
04.55 Mastery Standard
You have mastered the Fraction Engine when you can convert mixed to improper instantly, simplify and cancel rapidly before multiplying, move fluidly between fraction/decimal/percentage, recognise landmarks like 1/8 and 1/6, and use fractions naturally inside an SSC arithmetic problem.