Number system word problems


Word problems in number system help us form equations easily.  This foundation is very important to solve many questions in the upcoming chapters.  The basic rules of forming equations as follow.
1. 'a' exceeds 'b' by 100 ⇒ a – b = 100
2. 'a' is twice that of 'b'  ⇒ \(\dfrac{a}{b} = 2\) or $a = 2b$
3.  'a' is 25% less than 'b' ⇒ \(a = \left( {100 - 25} \right)\% b\) or \(a = 75\% \left( b \right)\)
4.  'a' is 20% more than 'b' ⇒ \(a = 120\% \left( b \right)\)
5. \({a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\)
6. \({\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\)
7. \({\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\)
8. \({\left( {a + b} \right)^2} = {\left( {a - b} \right)^2} + 4ab\)

Solved examples

1. $\displaystyle\frac{4}{5}$ of a number exceeds its $\displaystyle\frac{2}{3}$ by 8. The number is 

a. 30
b. 60
c. 90
d. None of these
Correct Option : B
Explanation:
Let the number be 'a'.
$\displaystyle\frac{4}{5}a - \frac{2}{3}a = 8$
$ \Rightarrow \displaystyle\frac{{12a - 10a}}{{15}} = 8$
$ \Rightarrow 2a = 120 \Rightarrow a = 60$

2. If 1 is added to the denominator of fraction, the fraction becomes $\displaystyle\frac{1}{2}$.  If 1 is added to the numerator, the fraction becomes 1.  The fraction is 

a. $\displaystyle\frac{4}{7}$
b. $\displaystyle\frac{5}{9}{\rm{ }}$
c. $\displaystyle\frac{2}{3}$
d. $\displaystyle\frac{{10}}{{11}}$
Correct Option: C
Explanation:
Let the required fraction be $\displaystyle\frac{a}{b}$. Then
$\Rightarrow \displaystyle\frac{a}{{b + 1}} = \displaystyle\frac{1}{2} \Rightarrow 2a – b = 1$ ---- (1)
$ \Rightarrow \displaystyle\frac{{a + 1}}{b} = 1$  $ \Rightarrow a – b =  – 1$ ------- (2)
Solving (1) & (2) we get a = 2, b=3
Fraction = $\displaystyle\frac{a}{b}{\rm{ = }}\displaystyle\frac{2}{3}$

3. The sum of two numbers is twice their difference.  If one of the numbers is 10, the other number is

a. ${\rm{3}}\displaystyle\frac{1}{3}$
b. 30
c. ${\rm{ - 3}}\displaystyle\frac{1}{3}$
d. a or b
Correct Option:  D
Explanation:
Let the other number be a. 
Given that, ${\rm{10 + a = 2(a - 10)}} \Rightarrow {\rm{a = 30}}$
or if the other number is less than 10, ${\rm{10 + a = 2}}({\rm{10 - a)}} \Rightarrow $${\rm{a = }}\dfrac{{10}}{3} = 3\dfrac{1}{3}$

4. The sum of squares of two numbers is 80 and the square of their difference is 36.  The product of the two numbers is.

a. 22
b. 44
c. 58
d. 116
Correct Option: A
Explanation :
Let the numbers be x and y . Then
${{\rm{x}}^2} + {y^2} = 80$ and ${({\rm{x - y}})^2} = 36$
${(x - y)^2} = 36 \Rightarrow {x^2} + {y^2} - 2xy = 36$
$ \Rightarrow 2xy = ({{\rm{x}}^2} + {y^2}) - 36 = 80 - 36 = 44$
$ \Rightarrow xy = 22$

5. 11 times a number gives 132.  The number is 

a. 11
b. 12
c. 13.2
d. None
Correct Option: B
Explanation:
Let the number be 'N'
$11 \times {\rm{N = 132}} \Rightarrow N = 12$

6. If 1/5th of a number decreased by 5 is 5, then the number is 

a. 25
b. 50
c. 60
d. 75
Correct Option: B
Explanation :
Let the number be a, then
$\displaystyle\frac{a}{5} - 5 = 5 \Rightarrow \frac{a}{5} = 10 \Rightarrow a = 50$

7. The difference of two numbers is 2 and the difference of their squares is 12.  The sum of the number is 

a. 6
b. 8
c. 10
d. 22
Correct Option: A
Explanation:
Let the numbers be a, b; $ \Rightarrow $ a-b=2 and
${a^2} - {b^2} = 12$
$a + b = \displaystyle\frac{{{a^2} - {b^2}}}{{a - b}} = \displaystyle\frac{{12}}{2} = 6$

8. The sum of two numbers is 29 and the difference of their squares is 145.  The difference between the number is

a. 13
b. 5
c. 8
d. 11
Correct Option: B
Explanation:
Let the numbers be x and y.
We know that,
$(x - y) = \displaystyle\frac{{{x^2} - {y^2}}}{{(x + y)}}$
$ = \displaystyle\frac{{145}}{{29}} = 5$

9. The difference between the squares of two consecutive numbers is 35.  The numbers are

a. 14,15
b. 15,16
c. 17,18
d. 18,19
Correct Option: C
Explanation:
Let the numbers be a and (a+1)
${(a + 1)^2} - {a^2} = 35$
$ \Rightarrow {a^2} + 2a + 1 - {a^2} = 35$
$ \Rightarrow 2a = 34$ or a = 17
The numbers are 17 & 18.

10.$\displaystyle\frac{{{3^{th}}}}{4}{\rm{ of }}\displaystyle\frac{{{1^{th}}}}{5}$ of a number is 60.  The number is 

a. 300
b. 400
c. 450
d. 1200
Correct Option: B
Explanation:
Let the number be N. Then
$\displaystyle\frac{3}{4} \times \frac{1}{5} \times N = 60 \Rightarrow 3N = 1200 \Rightarrow N = 400$

11. 24 is divided into two parts such that 7 times the first part added to 5 times the second part gives 146. The first part is 

a. 11
b. 13
c. 16
d. 17
Correct Option: B
Explanation:
Let the first and second parts be a and 24  a, then
${\rm{7a + 5(24 − a) = 146}}$
$ \Rightarrow {\rm{7a + 120 − 5a = 146}}$
$ \Rightarrow {\rm{2a = 26}}$ or a = 13


12. The product of two numbers is 120.  The sum of their squares is 289.  The sum of the two numbers is :

a. 20
b. 23
c. 169
d. None
Correct Option: B
Explanation:
Let the number be x and y . We know that,
${(x + y)^2} = ({x^2} + {y^2}) + 2xy = 289 + 2\times120$
$ = 289 + 240 = 529 \Rightarrow x + y = \sqrt {529}  = 23$


13. The sum of squares of two numbers is 68 and the square of their difference is 36.  The product of the two numbers is 

a. 16
b. 32
c. 58
d. 104
Correct Option:a
Explanation:
Let the numbers be x and y. Then
${x^2} + {y^2} = 68$
But ${(x − y)^2} = 36 \Rightarrow {x^2} + {y^2} − 2xy = 36$
$ \Rightarrow 68 − 2xy = 36 \Rightarrow 2xy = 32$
$ \Rightarrow xy = 16$

14. The sum of seven numbers is 235.  The average of the first three is 23 and that of the last three is 42.  The fourth number is 

a. 40
b. 126
c. 69
d. 195
Correct Option: A
Explanation:
Average of the first three is 23.  Therefore their sum = 23 x 3
Average of the last three is 42.  Therefore their sum = 42 x 3
Sum of all number = Sum of first three + Fourth number + Sum of last three
$(23 \times 3 + a + 42 \times 3) = 235 \Rightarrow a = 40$

15. Two numbers are such that the ratio between them is 3:5 but if each is increased by 10, the ratio between them becomes 5 : 7, the numbers are

a. 3, 5
b. 7, 9
c. 13, 22
d. 15, 25
Correct Option: D
Explanation:
Let the numbers be 3a and 5a
Then $\displaystyle\frac{{3a + 10}}{{5a + 10}} = \displaystyle\frac{5}{7}$
$ \Rightarrow 7(3a + 10) = 5(5a + 10) \Rightarrow a = 5$
The numbers are 15 & 25

16.  A fraction becomes 4 when 1 is added to both the numerator and denominator, and it becomes 7 when 1 is subtracted from both the numerator and denominator.  The numerator of the given fraction is :

a. 2
b. 3
c. 7
d. 15
Correct Option: D
Explanation:
Let the required fraction be $\displaystyle\frac{a}{b}$
Then $\displaystyle\frac{{a + 1}}{{b + 1}} = 4 \Rightarrow a − 4b = 3$
and $\displaystyle\frac{{a − 1}}{{b − 1}} = 7 \Rightarrow a − 7b = - 6$
Solving these equations we get,
a=15
b=3

17. A number exceeds 20% of itself by 40.  The number is

a. 50
b. 60
c. 80
d. 320
Correct Option: A
Explanation:
Let the answer be 'a'
Then $a - \displaystyle\frac{{20}}{{100}}a = 40 \Rightarrow 5a − a = 200$
$ \Rightarrow a = 50$

18. Three numbers are in the ratio 3:4:5.  The sum of the largest and the smallest equals the sum of the third and 52.The smallest number is :

a. 20
b. 27
c. 39
d. 52
Correct Option: C
Explanation:
Let the numbers be 3N, 4N and 5N
Then 5N+3N = 4N +52
$ \Rightarrow 4N = 52 \Rightarrow N = 13$
The smallest number = 3N = 3 x 13=39

19. If 16% of 40% of a number is 8, then the number is 

a. 200
b. 225
c. 125
d. 320
Correct Option: C
Explanation:
Let $\displaystyle\frac{{16}}{{100}} \times \displaystyle\frac{{40}}{{100}} \times a = 8$
a = $\displaystyle\frac{{8 \times 100 \times 100}}{{16 \times 40}} = 125$

20. If 3 is added to the denominator of a fraction, it becomes $\displaystyle\frac{1}{3}$ and if 4 be added to its numerator, it becomes $\displaystyle\frac{3}{4}$ ; the fraction is :

a. $\displaystyle\frac{4}{9}$
b. $\displaystyle\frac{3}{{20}}$
c. $\displaystyle\frac{7}{{24}}$
d. $\displaystyle\frac{5}{{12}}$
Correct Option: D
Explanation:
Let the required fraction be $\displaystyle\frac{a}{b}$
Then $\displaystyle\frac{a}{{b + 3}} = \displaystyle\frac{1}{3}$
$ \Rightarrow 3a - b = 3$ and
$\displaystyle\frac{{a + 4}}{b} = \displaystyle\frac{3}{4} \Rightarrow 4a - 3b = -16$ solving,we get
a = 5, b = 12; Required answer = 5/12

21. Of the three numbers, the first is twice the second and is half of the third.  If the average of three numbers is 56, then the smallest number is

a. 24
b. 36
c. 40
d. 48
Correct Option: A
Explanation:
Let the second number is a. Then the first number is 2a and third number is 4a.
$\displaystyle\frac{{2a + a + 4a}}{3} = 56 \Rightarrow 7a = 3 \times 56$ or
a= $\displaystyle\frac{{3 \times 56}}{7} = 24$
Smallest number is 24

22. The difference of two numbers is 8 and $\displaystyle\frac{{{1^{th}}}}{{12}}$ of the sum is 1.  The numbers are

a. 10, 2
b. 18, 26
c. 10, 18
d. 26, 34
Correct Option: A
Explanation :
Let the numbers be a and (a+8). Then
$\displaystyle\frac{1}{{12}}\left[ {a + (a + 8)} \right] = 1 \Rightarrow 2a + 8 = 12$
$ \Rightarrow a = 2,a + 8 = 10$

23. A number is 25 more than its $\displaystyle\frac{2}{5}{\rm{th}}$. The number is 

a. 60
b. 80
c. $\displaystyle\frac{{125}}{3}$
d. $\displaystyle\frac{{125}}{7}$
Correct Option: C
Explanation:
Let the number be N. Then
$N - 25 = \displaystyle\frac{2}{5}{\rm{ }}N{\rm{ or  5N - 125 = 2N}}$
or N = 125/3

24. The sum of three numbers is 68.  If the ratio between first and second be 2 : 3 and that between second and third be 5 : 3, then the second number is 

a. 30
b. 20
c. 58
d. 48


25. The sum of two numbers is 100 and their difference is 37. The difference of their squares is

a. 37
b. 100
c. 63
d. 3700