## Words Problem Based on Numbers

Numbers play an important role in our day to day life. Puzzles based on these numbers are known as word problems. To solve such problems you have to extract the information correctly and form equations based on given information.

The equations formed can be single variable, multi variables, linear, quadratic etc., depending on the type of problem asked.

### Types of Word Problems Based on Numbers :-

#### Based on Operation with Numbers :-

These types of questions include the operations like : addition, subtraction, multiplication, division of number with other number, calculation of average of consecutive numbers, calculation of sum or difference of reciprocal of numbers etc.

**Important Points :- **

**1**.Consecutive natural numbers x - 1, x, x + 1, x + 2, x + 3,……**Example-** Find the two consecutive numbers whose sum is 75. **Solution**:- Let the first number be x.

then other number = x + 1

according to the question,

x + x + 1 = 75

or, 2x = 75 - 1

or, 2x = 74

∴ x = | = 37 | |

2 |

**2**. Consecutive odd natural numbers x - 3, x - 1, x + 1, x + 3,……….**Example- ** The sum of three odd consecutive number whose sum is 81. What is the largest number? **Solution**:- Let the three odd consecutive number is x + 1, x + 3 and x + 5. According to the question,

x + 1 + x + 3 + x + 5 = 81

or, 3x + 9 = 81

or, 3x = 81 - 9

or, 3x = 72

or, x = | = 24 | |

3 |

Largest number = x + 5 = 24 + 5 = 29

**3**. Consecutive even natural numbers x - 2, x, x + 2, x + 4,…….**Example- **The sum of three even consecutive number whose sum is 72. Find the numbers.

**Solution**:- Let the three even consecutive number be x, x + 2 and x + 4. According to the question,

x + x + 2 + x + 4 = 72

or, 3x + 6 = 72

or, 3x = 66

∴ x = | = 22 | |

3 |

**4**. If the sum of two numbers is given as P, then take one number as x and other number as ( P - x )**Example-** The sum of two numbers is 4500. If 10% of first number is equal to 12.5% of the other, Find the numbers.

**Solution**:- Let the first number be x.

then other number = 4500 - x

According to the question,

10 % of x = 12.5 % of ( 4500 - x )

or, | × x = | × ( 4500 - x ) | ||

100 | 100 |

or, | = | × ( 4500 - x ) | ||

10 | 1000 |

or, | = | × 4500 - | |||

10 | 1000 | 1000 |

or, | = | × 45 - | |||

10 | 10 | 1000 |

or, | = | - | |||

10 | 10 | 1000 |

or, | + | = | |||

10 | 1000 | 10 |

or, | = | ||

1000 | 10 |

or, | = | ||

1000 | 10 |

or, x = 5625 × | × 10 = 2500 | |

225 |

∴ First number = 2500

other other number = 2000

and other as (x + d)

**5**. If the difference of numbers is d, then take one number as xand other as (x + d)

**Example**- The difference between the numerator and the denominator of a fraction is 5. If 5 is added to its denominator, the fraction is decreased by

1^{1}/_{4} Find the value of the fraction. **Solution**:- Let the denominator be x.

then, numerator = x + 5

fraction = | ||

x |

- | = | ||||

x | ( x + 5 ) | 4 |

or, | = | + 1 | ||

x | 4 |

or, | = | |||

x | 4 |

or, 9x = 4x + 20

or, 5x = 20

or, x = | = 4 | |

5 |

∴ fraction = | = | |||

x | 4 |

Hence , Fraction = 2 | ||

4 |

**Example**-If three-fourth of a number is 141,then find the number.

**Solution**:-Let the number be x.

then, according to the question,

of x = 141 | ||

4 |

or , | × x = 141 | |

4 |

or ,x = 141 × | ||

4 |

∴ x = 188

**Example**- A number is 35 more than three-fifth of that number. Find the number.**Solution**:- Let the number be x.

then, according to the question,

x = 35 + x × | ||

5 |

or, x = 35 + | ||

5 |

or , x - | = 35 | |

5 |

or , | = 35 | |

5 |

or, 2x = 175

∴ x = | ||

2 |

#### Based on Formation of Number with Digits :-

These types of questions include:

a. formation of a number with digits and its difference with reciprocal of the same number

b. calculation of a number if a number is added or subtracted to it and subsequently the digits get reversed etc.

**Important Points :-**

**1**.A two-digit number with x as unit digit and y as ten's digit is formed as ( 10y + x ) and if the digits are reversed, then number is represented as ( 10x + y ).

**2**. A three-digit number with x as unit digit, y as ten's digit and z as hundred's digit is formed as ( 100z + 10y + x ).

**Example-** A number consists of two digits whose sum is 9. If 27 is added to the number, the digits interchange their places, then find the number. **Solution**:- Let the unit's digit be x and ten's digit be y.

then, number = 10y + x

when digits are interchanged, then the number = 10x + y

According to the question,

x + y = 9 … (1)

and

10y + x + 27 = 10x + y

or, 10y - y + 27 = 10x - x

or, 9y + 27 = 9x

or, 9x - 9y = 27

or, 9 ( x - y ) = 27

or, x - y = | ||

9 |

on adding Eq. (1) and (2), we get

x + y + x - y = 9 + 3

or, 2x = 12

∴ x = | = 6 | |

2 |

6 + y = 9

∴ y = 3

∴ number = 10y + x

= 10 × 3 + 6 = 36

**Example-** The difference between a two-digit number and the number obtained by interchanging the position of its digits is 36. What is the difference between the two digits of that number?

**Solution**:- Let the unit's digit be x and ten's digit's be y.

then, number = 10y + x

when the digits are interchanged, then the number = 10x + y

according to the question,

10y + x - ( 10x + y ) = 36

or, 10y + x - 10x - y = 36

or, 9y - 9x = 36

or, 9 ( y - x ) = 36

∴ y - x = | = 4 | |

9 |

#### Question Regarding calculation of Heads and Feet or Animals

If a group of animals having either two feet ( like ducks, hens etc ) or four feet ( like horses cows etc ) is there and total number of heads in the group are H and number of feet of these animals are L, then

a. Number of animals with four feet = | ||

2 |

**b. Number of animals with two feet = Total number of heads - Total number of four feted animals**

**Example**- A man has some hens and cows. If the number of heads be 48 and number of feet equals 140, then the number of hens will be? **Solution**:- Let the number of hens be x.

and number of cows = y

then, Total number of heads = total number of hens and cows

x + y = 48 …..(1)

Total numbers of feet = ( 2 × Number of hens ) + ( 4 × Number of cows )

2x + 4y = 140

or, x + 2y = 70 ….(2)

On subtracting Eq. (1) from Eq.(2), we get

x + 2y - ( x + y ) = 70 - 48

or, x + 2y - x - y = 22

or, y = 22

∴ x = 26

∴ number of hens = 26

**2nd Method, **

Number of animals with four feet = | ||

2 |

= | ||

2 |

Number of animals with four feet = | = | = 22 | ||

2 | 2 |