Shortcut Techniques
Problem on Numbers · Study material
Shortcut Techniques – Problem on Numbers
Shortcut techniques in Problems on Numbers help candidates solve arithmetic and logical number-based questions quickly and accurately in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement examinations.
Instead of lengthy calculations, these methods focus on:
- Fast equation formation
- Consecutive number tricks
- Digit-based shortcuts
- Divisibility techniques
- Algebraic identity methods
- Mental calculation tricks
- Logical elimination techniques
Why Learn Shortcut Techniques?
- Improves equation-solving speed.
- Reduces lengthy arithmetic calculations.
- Helps solve digit-based puzzles quickly.
- Useful in algebraic word problems.
- Improves logical reasoning ability.
Shortcut #1: Always Assume the Unknown as x
In most Problems on Numbers, begin by assuming the unknown number as:
Let the number be x
Then convert the given condition into an equation.
Example:
A number increased by 15 becomes 40.
x + 15 = 40
x = 25
Shortcut #2: Consecutive Number Representation
Use standard representations directly instead of creating separate variables.
| Type | Representation |
|---|---|
| Two consecutive numbers | x, x + 1 |
| Three consecutive numbers | x, x + 1, x + 2 |
| Consecutive even numbers | 2x, 2x + 2 |
| Consecutive odd numbers | 2x + 1, 2x + 3 |
✔ This reduces unnecessary variables and simplifies equations.
Shortcut #3: Two-Digit Number Trick
If:
- x = tens digit
- y = units digit
Then:
Two-digit number = 10x + y
Reversed number:
10y + x
Example:
If digits are 4 and 7:
Number = 10(4) + 7
= 47
Reversed number:
= 74
Shortcut #4: Divisibility Rules Shortcut
Memorizing divisibility rules saves significant time.
| Number | Shortcut Rule |
|---|---|
| 2 | Last digit even |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 5 | Last digit 0 or 5 |
| 6 | Divisible by 2 and 3 |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 10 | Last digit 0 |
Shortcut #5: Algebraic Identity Trick
Use identities for fast simplification.
| Identity | Formula |
|---|---|
| Difference of Squares | (a + b)(a − b) = a² − b² |
| Square of Sum | (a + b)² = a² + b² + 2ab |
| Square of Difference | (a − b)² = a² + b² − 2ab |
| Sum of Cubes | a³ + b³ = (a + b)(a² − ab + b²) |
| Difference of Cubes | a³ − b³ = (a − b)(a² + ab + b²) |
Example:
99 × 101
= (100 − 1)(100 + 1)
= 100² − 1²
= 9999
Shortcut #6: Sum of Consecutive Numbers
For consecutive numbers:
Average = (First Number + Last Number) / 2
Then:
Sum = Average × Number of Terms
Example:
Find sum of numbers from 11 to 21.
Average:
= (11 + 21)/2
= 16
Total terms = 11
Sum:
= 16 × 11
= 176
Shortcut #7: Difference Between Number and Reversed Number
For a two-digit number:
Original number = 10x + y
Reversed number = 10y + x
Difference:
9(x − y)
✔ Difference between a two-digit number and its reverse is always divisible by 9.
Shortcut #8: Sum of Number and Reversed Number
For a two-digit number:
(10x + y) + (10y + x)
= 11(x + y)
✔ Sum of a two-digit number and its reverse is always divisible by 11.
Shortcut #9: Even-Odd Logic
| Operation | Result |
|---|---|
| Even + Even | Even |
| Odd + Odd | Even |
| Even + Odd | Odd |
| Even × Any Number | Even |
| Odd × Odd | Odd |
Example:
(17 × 9) + 4
Odd × Odd = Odd
Odd + Even = Odd
Result is Odd
Shortcut #10: Digital Sum Technique
Digital sum helps verify calculations quickly.
Add digits repeatedly until a single digit remains.
Example:
Find digital sum of 5876.
5 + 8 + 7 + 6 = 26
2 + 6 = 8
Shortcut #11: Fast Sum Formulae
| Concept | Formula |
|---|---|
| Sum of First n Natural Numbers | n(n + 1)/2 |
| Sum of Squares | n(n + 1)(2n + 1)/6 |
| Sum of Cubes | [n(n + 1)/2]² |
Example:
Sum of first 20 natural numbers:
= 20 × 21 / 2
= 210
Shortcut #12: Elimination Technique in MCQs
Use divisibility, parity, and digit properties to eliminate wrong options quickly.
- Check even/odd nature.
- Check divisibility conditions.
- Check digit sum properties.
- Verify unit digits.
✔ Very useful in competitive exam objective questions.
Shortcut #13: Number Ending Trick
A square number never ends with:
- 2
- 3
- 7
- 8
A cube number may end with any digit.
Shortcut #14: Ratio-Based Number Trick
If two numbers are in ratio a:b:
Assume numbers as:
ax and bx
Example:
Ratio = 3:5
Numbers = 3x and 5x
Shortcut #15: Equation Balancing Trick
Always simplify equations before solving.
- Remove brackets first.
- Combine like terms.
- Shift variables to one side.
- Shift constants to another side.
Example:
3x + 12 = 36
3x = 24
x = 8
Important Exam Tips
- Always assume unknown numbers properly.
- Translate statements carefully into equations.
- Memorize algebraic identities and divisibility rules.
- Practice digit-based number formation regularly.
- Use elimination techniques in MCQs.
- Check parity and unit digit whenever possible.
- Practice mental calculations daily.
Shortcut techniques in Problems on Numbers help candidates improve logical reasoning, reduce calculation time, and solve aptitude questions efficiently in competitive examinations.