How do I represent consecutive numbers algebraically?

Represent three consecutive numbers as (n-1), n, (n+1) — this makes the algebra symmetric and easier to solve. Alternatively use n, n+1, n+2. For consecutive even or odd numbers use n, n+2, n+4.

How do I represent a two-digit number algebraically?

A two-digit number with tens digit 'a' and units digit 'b' is written as 10a + b. When digits are reversed, the new number is 10b + a. The sum of digits is a + b.

What does 'the number exceeds its reciprocal by X' mean?

It means: number − 1/number = X. Let the number be n. Then n − 1/n = X, which gives n² − Xn − 1 = 0. Solve using the quadratic formula.

What is the sum of the first 100 natural numbers?

Sum = n(n+1)/2 = 100 × 101 / 2 = 5050. This is a very common exam fact.

How do I solve 'a number is 5 more than twice another number' type problems?

Assign variables: let the smaller number = x. Then the other number = 2x + 5. Use the second condition (sum, product, or difference) to form a second equation and solve simultaneously.

Problems on Numbers — Formulas, Shortcuts, and Worked Examples | TNPSC, SSC, Bank

Quick Concept Summary

Problems on Numbers are word problems where you translate English statements about numbers into algebraic equations and solve them. The key skill is converting words to equations accurately.

Common phrasings and their equations:

English phrase	Algebraic meaning
"A number exceeds B by k"	A = B + k
"A is k times B"	A = kB
"Sum of two numbers is S"	x + y = S
"Product of two numbers is P"	xy = P
"Difference of two numbers is D"	x − y = D (x > y)
"A number added to its square is S"	n + n² = S
"A number and its reciprocal sum to S"	n + 1/n = S
"Digits are reversed"	10a+b → 10b+a
"Three consecutive numbers"	n-1, n, n+1 or n, n+1, n+2
"Three consecutive even numbers"	n, n+2, n+4

Key Formulas

What to find	Formula
Sum of first n natural numbers	n(n+1)/2
Sum of squares of first n naturals	n(n+1)(2n+1)/6
Sum of cubes of first n naturals	[n(n+1)/2]²
Sum of first n odd numbers	n²
Sum of first n even numbers	n(n+1)
Two-digit number	10a + b (a = tens, b = units)
Reversed two-digit number	10b + a

Worked Examples

Example 1: Sum and Difference of Two Numbers

Problem:
The sum of two numbers is 45 and their difference is 9. Find the numbers.

Let the numbers be x and y (x > y).

Step 1: Write equations
  x + y = 45   ... (1)
  x - y = 9    ... (2)

Step 2: Add equations (1) and (2)
  2x = 54
  x = 27

Step 3: Substitute back
  27 + y = 45
  y = 18

Answer: The two numbers are 27 and 18.

Verification: 27 + 18 = 45 ✓ | 27 − 18 = 9 ✓

Example 2: A Number and Its Fraction

Problem:
A number is such that when 3 is subtracted from 5 times the number,
the result is 6 more than 3 times the number. Find the number.

Let the number = n.

Step 1: Translate
  5n - 3 = 3n + 6

Step 2: Solve
  5n - 3n = 6 + 3
  2n = 9
  n = 4.5

Answer: The number is 4.5

Verification: 5(4.5) - 3 = 22.5 - 3 = 19.5
              3(4.5) + 6 = 13.5 + 6 = 19.5 ✓

Example 3: Consecutive Numbers

Problem:
The sum of three consecutive odd numbers is 51. Find the numbers.

Let the three consecutive odd numbers be (n-2), n, (n+2).

Step 1: Sum equation
  (n-2) + n + (n+2) = 51
  3n = 51
  n = 17

Step 2: Find all three
  n-2 = 15
  n   = 17
  n+2 = 19

Answer: The three consecutive odd numbers are 15, 17, and 19.

Verification: 15 + 17 + 19 = 51 ✓

Example 4: Two-Digit Number with Reversed Digits

Problem:
The sum of the digits of a two-digit number is 9. When the digits are
reversed, the new number is 27 more than the original. Find the number.

Let tens digit = a, units digit = b.
Original number = 10a + b
Reversed number = 10b + a

Step 1: Digit sum equation
  a + b = 9   ... (1)

Step 2: Reversed number condition
  (10b + a) - (10a + b) = 27
  9b - 9a = 27
  b - a = 3   ... (2)

Step 3: Solve (1) and (2)
  From (1): a + b = 9
  From (2): b - a = 3
  Adding: 2b = 12 → b = 6
  Then: a = 9 - 6 = 3

Original number = 10(3) + 6 = 36

Answer: The original number is 36.

Verification: Digit sum = 3+6 = 9 ✓ | Reversed = 63 | 63 - 36 = 27 ✓

Example 5: Product and Sum

Problem:
The product of two numbers is 120 and their sum is 22. Find the numbers.

Let the numbers be x and y.

Step 1: Write equations
  x + y = 22   ... (1)
  xy   = 120   ... (2)

Step 2: Use the identity (x - y)² = (x + y)² - 4xy
  (x - y)² = 22² - 4(120)
  (x - y)² = 484 - 480
  (x - y)² = 4
  x - y = 2    ... (3)

Step 3: Solve (1) and (3)
  x + y = 22
  x - y = 2
  2x = 24 → x = 12
  y = 22 - 12 = 10

Answer: The two numbers are 12 and 10.

Shortcut: Think of two numbers whose sum is 22 and product is 120.
Factor pairs of 120: (10, 12) → sum = 22 ✓

Example 6: Number and Its Reciprocal

Problem:
A positive number exceeds its reciprocal by 24/5. Find the number.

Let the number = n.

Step 1: Write equation
  n - 1/n = 24/5

Step 2: Multiply both sides by 5n
  5n² - 5 = 24n
  5n² - 24n - 5 = 0

Step 3: Factorise or use quadratic formula
  5n² - 25n + n - 5 = 0
  5n(n - 5) + 1(n - 5) = 0
  (5n + 1)(n - 5) = 0
  n = 5 or n = -1/5 (rejected, positive number)

Answer: The number is 5.

Verification: 5 - 1/5 = 25/5 - 1/5 = 24/5 ✓

Shortcuts and Tricks

Trick 1: Sum + Difference → Find Numbers Fast

Given sum S and difference D:

Larger number = (S + D) / 2
Smaller number = (S − D) / 2

Example: Sum = 40, Difference = 10 → Larger = 25, Smaller = 15

Trick 2: Consecutive Number Sum

Sum of n consecutive integers starting from a: = na + n(n−1)/2

For consecutive starting from 1 to n: sum = n(n+1)/2

Trick 3: Two-Digit Number Digit-Swap Rule

If reversed number − original = 9k, then the digit difference = k If reversed number + original = S, then sum of digits = S/11

Trick 4: Product and Sum Identity

If you know x + y = S and xy = P:

(x − y)² = S² − 4P
x² + y² = S² − 2P
x³ + y³ = S³ − 3SP

TNPSC Exam Pattern

Question Type	Frequency	Difficulty
Find two numbers given sum and difference	Very common	Easy
Consecutive number sum problems	Very common	Easy
Two-digit number with digit reversal	Common	Medium
Number and its reciprocal	Occasional	Medium
Product and sum given	Common	Medium

Practice Questions

Q1. The sum of two numbers is 60 and their difference is 12. Find the numbers. Answer: Larger = (60+12)/2 = 36, Smaller = (60-12)/2 = 24

Q2. The sum of three consecutive numbers is 84. Find them. Answer: n-1 + n + n+1 = 84 → 3n = 84 → n = 28. Numbers: 27, 28, 29

Q3. A number is 3 times another. Their sum is 48. Find them. Answer: x + 3x = 48 → 4x = 48 → x = 12. Numbers: 12 and 36

Q4. The digits of a 2-digit number sum to 7. If reversed, it is 27 less. Find the number. Answer: a+b=7, (10a+b)-(10b+a)=27 → a-b=3. Solving: a=5, b=2. Number = 52

Q5. The product of two consecutive even numbers is 168. Find them. Answer: n(n+2) = 168 → n²+2n-168=0 → (n+14)(n-12)=0 → n=12. Numbers: 12 and 14

Q6. What is the sum of all natural numbers from 1 to 200? Answer: n(n+1)/2 = 200×201/2 = 20,100

Q7. A number is 4 more than twice its reciprocal. Find the number. Answer: n = 2/n + 4 → n² - 4n - 2 = 0 ... actually: n - 2/n = 4 → n²-4n-2=0 → n = (4+√24)/2 = 2+√6 ≈ 4.45. [Exam version is usually clean: n - 1/n = 8/3 → 3n²-8n-3=0 → n=3]

Q8. Two numbers are in ratio 3:5. Their sum is 96. Find them. Answer: 3x + 5x = 96 → 8x = 96 → x = 12. Numbers: 36 and 60

Q9. Sum of squares of two consecutive numbers is 145. Find them. Answer: n² + (n+1)² = 145 → 2n²+2n+1=145 → 2n²+2n-144=0 → n²+n-72=0 → (n+9)(n-8)=0 → n=8. Numbers: 8 and 9

Q10. The sum of a number and its square is 72. Find the number. Answer: n + n² = 72 → n²+n-72=0 → (n+9)(n-8)=0 → n=8