# Number Problems – Fraction, Natural and Prime Numbers, Sum and Difference (Aptitude, Formulas, Examples, Explanation)

## Formulas for Numbers

1. When sum and difference of two numbers (X and Y) are given, then
X = (sum + difference)/2
Y = (sum + difference)/2
2. Difference between two digits of two digit number is = (Difference in original and interchanged number)/9
3. Sum of first n odd numbers is n2
4. Sum of first n even numbers is n(n+1)
5. Sum of squares of first n natural numbers is n(n+1)(2n+1)/6
6. Sum of cubes of first n natural numbers is [n(n+1)/2]2

## Questions and Solved Examples on Numbers

Q.1: A number consists of two digits. If 3/5 of 1/5 of the number is 9. Find the sum of its two digits?
A. 8
B. 10
C. 12
D. 14
Explanation: x * 3/5 * 1/5 = 9
x = 75 => 7 + 5 = 12

Q.2: [50 – {20 + 60% of (26/13 + 24/12 + 21) – 2} + 8] = 625/?
A. 25
B. 241/5
C. 253/5
D. 125
Explanation: [50 – {20 + 60% of (26/13 + 24/12 + 21) – 2} + 8] = 625/x
=> [50 – {20 + 3/5 of (25) – 2} + 8] = 625/x
=> [50 – 33 + 8] = 625/x
=> x = 25

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Q.3: If the sum and difference of 2 numbers are 20 and 8 respectively, then the difference of their square is:
A. 120
B. 140
C. 160
D. 180
Explanation: Let the numbers be x and y.
Then, x + y = 20 and x – y = 8
Now, x^2 – y^2 = (x + y) (x – y) = 20 * 8 = 160

Q.4: The average of first 10 prime numbers is?
A. 11.9
B. 12.7
C. 12.9
D. 13.7
Explanation: The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29 (being not divisible by any other number other than the number itself),
The sum of these number will be 129 and hence the average will be 129/10 = 12.9

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Q.5: How many figures are required to number the pages of a book containing 365 pages?
A. 897
B. 967
C. 977
D. 987
Explanation: 1 to 9 = 9 * 1 = 9
10 to 99 = 90 * 2 = 180
100 to 365 = 266 * 3 = 798
So, 798 + 180 + 9 = 987

Q.6: The numerator of a certain fraction is 8 less than the denominator. If 3 is added to the numerator and 3 is subtracted from the denominator, the fraction becomes 3/4. Find the original fraction?
A. 2/5
B. 7/9
C. 3/11
D. 8/5
Explanation: The denominator be P, the numerator will be (P – 8)
The fraction will be (P – 8)/P
Adding 3 to the numerator and subtracting 3 from the denominator, (P – 8 + 3)/(P – 3) = 3/4
(P – 5)/(P – 3) = 3/4
P = 20 – 9 => P = 11
The fraction is: 3/11

Q.7: What is the least number, which is a perfect square but contains 2700 as its factor?
A. 2700
B. 5400
C. 8100
D. 10800
Explanation: 900 * 3 * 3 = 8100

Q.8: If 3/p = 6 and 3/q = 15 then p – q =?
A. 1/3
B. 2/5
C. 3/10
D. 5/6
Explanation: Given, 3/p = 6 and 3/q = 15
=> p=3/6 = 1/2
=> q=3/15 = 1/5
Hence, p-q = 1/2 – 1/5 = 5 – 2/10 = 3/10

Q.9: When the numerator of a fraction is decreased by 25% and its denominator is decreased by 20%, the new fraction obtained is 3/4. Find the original fraction?
A. 4/7
B. 4/5
C. 6/7
D. None of these
Explanation: Let the fraction be x/y.
When the numerator decreased by 25% and the denominator decreased by 20%
(x * 3/4)/(y * 4/5) = 3/4
=> x/y = 3/4 * (4/5)/(3/4) = 4/5

Q.10: If the numerator of a fraction is increased by 20% and its denominator is diminished by 25% value of the fraction is 2/15. Find the original fraction.
A. 1/12
B. 1/8
C. 1/6
D. 1/4