Problem on Numbers
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Solved Examples
Study MaterialSolved Examples – Problem on Numbers
Solved examples help students understand the practical application of Problems on Numbers in competitive examinations. These examples are designed from basic to advanced level and cover important concepts frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement aptitude tests.
Topics Covered in Solved Examples
- Basic Equation Formation
- Consecutive Number Problems
- Even and Odd Number Problems
- Digit-Based Number Problems
- Reversed Number Problems
- Divisibility Concepts
- Algebraic Identity Applications
- Sum and Product Problems
- Age-Based Number Problems
- Logical Number Puzzles
Example 1: Basic Number Problem
Question: A number increased by 15 becomes 48. Find the number.
Solution:
Let the number be x.
According to question:
x + 15 = 48
x = 48 − 15
x = 33
Example 2: Consecutive Numbers
Question: The sum of two consecutive numbers is 41. Find the numbers.
Solution:
Let the numbers be:
x and x + 1
According to question:
x + (x + 1) = 41
2x + 1 = 41
2x = 40
x = 20
Numbers are:
20 and 21
Example 3: Three Consecutive Numbers
Question: The sum of three consecutive numbers is 72. Find the numbers.
Solution:
Let the numbers be:
x, x + 1, x + 2
According to question:
x + (x + 1) + (x + 2) = 72
3x + 3 = 72
3x = 69
x = 23
Numbers are:
23, 24, and 25
Example 4: Consecutive Even Numbers
Question: The sum of two consecutive even numbers is 74. Find the numbers.
Solution:
Let the numbers be:
2x and 2x + 2
According to question:
2x + (2x + 2) = 74
4x + 2 = 74
4x = 72
x = 18
Numbers are:
36 and 38
Example 5: Consecutive Odd Numbers
Question: The sum of three consecutive odd numbers is 57. Find the numbers.
Solution:
Let the numbers be:
2x + 1, 2x + 3, 2x + 5
According to question:
(2x + 1) + (2x + 3) + (2x + 5) = 57
6x + 9 = 57
6x = 48
x = 8
Numbers are:
17, 19, and 21
Example 6: Two-Digit Number Formation
Question: Form the number whose tens digit is 5 and units digit is 8.
Solution:
Two-digit number:
= 10x + y
= 10(5) + 8
= 50 + 8
= 58
Example 7: Reversed Number Problem
Question: Find the reversed form of 73.
Solution:
Original number:
73
After reversing digits:
37
Example 8: Difference Between Number and Reverse
Question: Find the difference between 84 and its reverse.
Solution:
Reverse of 84:
48
Difference:
84 − 48
= 36
Example 9: Divisibility by 3
Question: Check whether 738 is divisible by 3.
Solution:
Sum of digits:
7 + 3 + 8 = 18
18 is divisible by 3.
Therefore:
738 is divisible by 3
Example 10: Divisibility by 9
Question: Check whether 729 is divisible by 9.
Solution:
Sum of digits:
7 + 2 + 9 = 18
18 is divisible by 9.
Therefore:
729 is divisible by 9
Example 11: Algebraic Identity
Question: Simplify:
99 × 101
Solution:
99 × 101
= (100 − 1)(100 + 1)
Using:
(a + b)(a − b) = a² − b²
= 100² − 1²
= 10000 − 1
= 9999
Example 12: Sum of Natural Numbers
Question: Find the sum of first 25 natural numbers.
Solution:
Formula:
n(n + 1)/2
= 25 × 26 / 2
= 25 × 13
= 325
Example 13: Sum of Squares
Question: Find the sum of squares of first 5 natural numbers.
Solution:
Formula:
n(n + 1)(2n + 1)/6
= 5 × 6 × 11 / 6
= 5 × 11
= 55
Example 14: Sum of Cubes
Question: Find the sum of cubes of first 4 natural numbers.
Solution:
Formula:
[n(n + 1)/2]²
= [4 × 5 / 2]²
= 10²
= 100
Example 15: Age-Based Number Problem
Question: A father is 3 times as old as his son. If the son's age is 12 years, find the father's age.
Solution:
Father's age:
= 3 × 12
= 36 years
Example 16: Product-Based Number Problem
Question: The product of two consecutive numbers is 132. Find the numbers.
Solution:
Let numbers be:
x and x + 1
According to question:
x(x + 1) = 132
x² + x − 132 = 0
(x − 11)(x + 12) = 0
x = 11
Numbers are:
11 and 12
Example 17: Missing Digit Problem
Question: Find the missing digit x if 54x is divisible by 9.
Solution:
Sum of digits:
5 + 4 + x = 9 + x
For divisibility by 9:
9 + x must be divisible by 9.
Therefore:
x = 0 or 9
Example 18: Logic-Based Number Problem
Question: A number when divided by 5 leaves remainder 3. What will be the remainder when twice the number is divided by 5?
Solution:
Let the number be:
5x + 3
Twice the number:
= 2(5x + 3)
= 10x + 6
= 5(2x + 1) + 1
Remainder = 1
Important Exam Tips
- Always start by assuming unknown numbers properly.
- Translate statements carefully into equations.
- Use standard representations for consecutive numbers.
- Memorize divisibility rules and algebraic identities.
- Be careful while reversing digits.
- Use elimination techniques in MCQs.
- Practice mental calculations regularly.
Practicing solved examples regularly improves logical reasoning, equation formation skills, and accuracy in solving Problems on Numbers in competitive examinations.