Word Problems
Several types of everyday problems expressed in language form can be solved by translating the language sentences into equations.
Problems on Numbers
One number is half the other. Sum of two numbers is equal to 6. Find the numbers.
Let one number be
x. The other number =Sum of two numbers = 6.
Hence, x + = 6
Example :
The numbers are 4 and 2.
Check : The sum of two numbers = 6. Hence the answer is correct.
The numerator of a fraction is 3 less than its denominator. If the numerator is increased by 1 and the denominator is increased by 3, the fraction becomes equal to . Find the original fraction.
Let the denominator of the original fraction be x.
If the numerator is 3 less than the denominator, then numerator =
x â€“ 3.Original fraction = .
The denominator of the new fraction =
x + 3The numerator of the new fraction = (
x3) + 1 = x2. The new fraction =
The new fraction becomes.
Hence ,


2 (x  2) = 1 (x +3)

Apply cross multiplication

2 x  4 = x + 3

Simplify LHS and RHS

2 x  x = 3 + 4

Transpose x to LHS and ( 4) to RHS

x = 7
.^{.}. original fraction = Check : Original numerator + 1 Original denominator + 3 = which is correct. 
Simplify LHS and RHS

The sum of two numbers is 35. Their difference is 13. Find the numbers.
Let one number be x. Then, the other number is (35  x).
The difference of the numbers is 13.
(35  x)  x = 13
(35  x)  x = 13


35  2 x = 13.

Simplify LHS

( 2) x = 13  35

Transpose 35 to RHS

( 2) x = ( 22)

Simplify RHS

x =

Transpose (  2) to RHS

x = 11

The two numbers are 11 and 24.
Check: Sum of numbers = 11+24= 35;
difference of numbers = 24  11= 13, hence the answer is correct.
Two number are in the ratio 3 : 4. sum of two numbers is equal to 56. Find the numbers?
Let the numbers be 3x and 4x.
Their sum is equal to 56.
7x = 56
x = 8
.^{.}. the numbers are 3 x 8 = 24, and 4 x 8 = 32.
The sum of three consecutive numbers is 243. Find the numbers.
Method 1: Let the numbers be x, x+1, x+2.
The sum is x + (x +1) + (x +2) = 243
3x + 3 = 243
.^{.}.3x = 240 or x = 80.
The numbers are 80, 81, 82.
Problems Involving Place Values
Remember that if the digits of a 2 digit number are a (units) and b (tens) then the number is 10 b + a.
In a 2digit number, the tens digit is 4 times the units digit. When the digits are reversed, the new number formed is 54 less than the original number. Find the original number.
Let the units digit of the number be x.
The tens digit = 4x
Tens  Units  Number  
Old number  4 x  x  (4 x Â´ 10) + (x Â´ 1) = 41 x 
New number Difference between the two = 41x  14x = 54 (given) 
x  4 x  (x Â´ 10) + (4 x Â´ 1) = 14 x 
Example :
27x =54
x = 2
The old number = (8 Ã— 10) + (2 Ã— 1) = 82
The new number =(2 Ã— 10) + (8 Ã— 1) = 28
Check: Difference between the old and new number
= 82  28 = 54, which is correct.
Age Problems
Anil is 9 years older than Ajith. In 10 years, Anil will be twice as old as Ajith was 10 years ago. Find their present ages.
Let Ajith's present age be x. Look at the table given here.
YearsAnil AjithPresentx+9x10 years ago
x  1010 years hencex+9+10 = x + 19
The problem says 10 years hence, Anil will be twice as old as Ajith was 10 years ago.
Anil's age 10 years hence = x+19 â€¦â€¦â€¦(1)
Ajith's age 10 years ago = x10 â€¦â€¦â€¦.(2)
Therefore (1) = 2 Ã— (2)
x + 19 = 2(x  10)
x  2x = 20  19 or x = 39
.^{.}. x = 39
Their present ages are: Anil = 48 years; Ajith = 39 years.
Check: 10 years hence Anil will be 58 years. Ajith was 29 years old, 10 years ago
Checking 58 = 2 Ã— 29 Here L.H.S = R.H.S.