2.3    SOME APPLICATIONS:

 

  • We can use the concept of linear equations in our daily routine also. There are some situations where we need to use the variable to find the solution. eg:
  • What number should be added to a certain number  to get a particular number.
  • If the sum of two numbers is 100 and one of the no. is 63 then what will be the other number.

EXERCISE 2.2

If you subtract ½ from a number and multiply the result by ½, you get 1 / 8 what is the number.
Solution: Let the number be x.
According to the question,
( x – 1 / 2 ) × ½ = 1 / 8
x / 2 – ¼ = 1 / 8
x / 2 = 1 / 8 + ¼
x / 2 = 1 / 8 + 2 / 8
x / 2 = ( 1 + 2 ) / 8
x / 2 = 3 / 8
x = ( 3 / 8 ) × 2
x = ¾

2. The perimeter of a rectangular swimming pool is 154 m. Its length is 2 m more than twice its breadth. What are the length and the breadth of the pool.
Solution: Given that, perimeter of rectangular swimming pool = 154 m Let the breadth of rectangle be = x
According to the question,
Length of the rectangle = 2 x + 2 we know that,
Perimeter = 2 ( length + breadth)
⇒ 2 ( 2 x + 2 + x ) = 154 m
⇒ 2 ( 3 x + 2 ) = 154 ⇒ 3 x + 2 = 154 / 2
⇒ 3 x = 77 – 2 ⇒ 3 x = 75
⇒ x = 75 / 3 ⇒ x = 25 m
Therefore, Breadth = x = 25 cm
Length = 2 x + 2
= ( 2 × 25 ) + 2 = 50 + 2 = 52 m

Let the length of equal sides of triangle be x.

According to the question,

4 / 3   +    x  +   x     =      62 / 15

⇒       2     =       (62 / 15   –  4 / 3 )     ⇒    2     =      (62   –   20) / 15

⇒       2     =       42 / 15                      ⇒            =      (42 / 30)   ×   ( ½ )

⇒           =       42 / 30                               ⇒            =      7 / 5 cm

The length of either of the remaining equal sides are 7 / 5 cm.

 

  1. Sum of two numbers is 95. If one exceeds the other by 15, find the numbers.

Solution:  Let one of the numbers be   =   .

Then, the other number becomes  x    +   15   According to the question,

+       +    15    =    95

⇒        2   +  15    =   95         ⇒ 2     =      95   –  15

⇒      2       =      80               ⇒       =      80 / 2         ⇒  x    =   40

First number    =       =      40

And, other number    =     +    15    =   40    +    15    =   55

 

  1. Two numbers are in the ratio 5:3 . If they differ by 18, what are the numbers.

Solution:  Let the two numbers be 5x and 3 x. According to the question,

5   –    3     =      18

⇒   2     =      18           ⇒       =      18 / 2                   ⇒          =      19

The numbers are 5     =      5 × 9      =     45    and   3      =      3 × 9     =     27.

 

  1. Three consecutive integers add up to 51. What are these integers.

Solution:  Let the three consecutive integers be   x , x   +  1 and  x   +   2. According to the question,

+    (  +   1 )    +    (  +   2 )    =   51

⇒   3   +    3    =   51                       ⇒ 3     =      51    –  3

⇒   3     =      48                    ⇒       =      48 / 3                  ⇒       =     16

Thus, the integers are

=       16,           +    1    =   17          and               +    2    =   18

 

  1. The sum of three consecutive multiples of 8 is 888. Find the multiples.

Solution:   Let the three consecutive multiples of   8   be   8 , 8 (x   +   1 )  and             8 (  +   2). According to the question,

8    +    8  (  +   1 )    +    8 (  +   2 )    =    888

⇒      8 (  +     +   1  +    +   2 )    =   888

⇒      8 ( 3   +  3 )    =   888  (Taking  8  as common)

⇒        3   +    3    =   888 / 8            ⇒   3   +   3     =     111

⇒      3     =      111  –  3                          ⇒   3              =      108

⇒          =      108 / 3                       ⇒                    =       36

Thus, the three consecutive multiples of 8 are:

8     =      8 × 36    =   288

8 (  +  1 )    =   8  ×  ( 36   +   1 )    =   8  × 37    =   296

8 (  +   2 )   =   8 ×   ( 36   +   2 )    =   8  ×  38   =   304

 

  1. Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3 and  4 respectively, they add up to 74. Find these numbers.

Solution:  Let the three consecutive integers are   ,    +  1 and    +  2. According to the question,

 

2   +   3 (  +   1 )   +   4 (  +   2 )    =   74

⇒      2   +    3 x    +    3    +    4x    +    8    =   74

⇒        9   +    11    =   74

⇒       9     =      74  –  11

⇒       9     =      63

⇒           =      63 / 9

⇒           =      7

Thus, the numbers are:

   =    7

+    1    =   8

+    2    =   9

 

  1. The ages of Rahul and Haroon are in the ratio 5 :7 . Four years later the sum of their ages will be 56 years. What are their present ages.

Solution:  Let the ages of Rahul and Haroon be  5  and  7 . Four years later,

The ages of Rahul and Haroon will be ( 5   +   4 ) and ( 7   +   4 ) respectively. According to the question,

( 5   +   4 )   +    ( 7   +   4 )     =     56

⇒        5   +    4    +    7   +    4     =     56

⇒        12   +    8    =   56

⇒       12     =      56    –  8

⇒      12     =      48

⇒             =      48 / 12

⇒                =      4

Therefore, Present age of  Rahul    =   5     =      5 × 4    =   20

And, present age of  Haroon    =   7     =     7 × 4     =     28

 

  1. The number of boys and girls in a class are in the ratio 7: 5 . The number of boys is 8 more than the number of girls. What is the total class strength.

Solution:  Let the number of boys be 7x and girls be 5x.

According to the question,

7     =      5   +    8

⇒      7   –  5     =      8

⇒      2     =      8

⇒          =      8 / 2

⇒          =      4

Therefore, Number of boys     =    7  ×  4    =   28

And, Number of girls    =   5   ×   4    =   20

Total number of students    =   20    +    28    =   48

 

  1. Baichung’s father is 26 years younger than Baichung’s grandfather and 29 years older than Baichung. The sum of the ages of all the three is 135 years. What is the age of each one of them.

Solution:  Let the age of Baichung’s father be  x.

Then, the age of Baichung’s grandfather    =   (  +   26 )

and, Age of Baichung    =   ( 29 ) According to the question,

+   (  +   26  )    +    (  29)     =    135

⇒      3   +    26    –    29    =   135

⇒      3    –  3     =    135

⇒      3     =       135    +    3

⇒      3     =       138

⇒             =    138 / 3

⇒               =       46

Age of Baichung’s father     =        =       46

Age of Baichung’s grandfather     =    (   +   26 )    =    46    +    26    =   72

Age of Baichung     =   ( 29 )    =   46    –   29    =   17

  1. Fifteen years from now Ravi’s age will be four times his present age. What is Ravi’s present age.

Solution:  Let the present age of Ravi be x.

Fifteen years later, Ravi age will be      +   15  years. According to the question,

+    15     =    4

⇒       4  –x     =    15

⇒      3     =       15

⇒              =       15 / 3

⇒              =       5

Therefore, Present age of Ravi      =    5   years.

 

  1. A rational number is such that when you multiply it by 5 / 2 and add    2 / 3 to the product, you get 7 / 12. What is the number.

Solution:  Let the rational be .

According to the question,

x ( 5 / 2 )   +   2 / 3    =   7 / 12

⇒      5  / 2   +   2 / 3     =    7 / 12

⇒      5  / 2     =    7 / 12  –  2 / 3

⇒      5  /   =   (7 8 ) / 12

⇒      5  / 2     =    15 / 12

⇒        5   /   =    5 / 4

⇒            =       (5 / 4 )  ×  ( 2 / 5 )

⇒            =        –  10 / 20

⇒           =       ½

Therefore, the rational number is   –  ½.

 

  1. Lakshmi is a cashier in a bank. She has currency notes of denominations Rs 100 , Rs 50 and Rs 10 , respectively. The ratio of the number of these notes is       2 : 3 : 5.  The total cash with Lakshmi is Rs 4,00,000. How many notes of each denomination does she have.

Solution:  Let the numbers of notes of Rs 100,  Rs 50  and Rs 10  be  2 x,  3 x  and  5x  respectively.

Value of Rs 100     =    2  ×   100    =   200

Value of Rs 50       =    3  ×   50    =   150

Value of Rs 10       =    5  ×   10     =   50 x  According to the question,

200 x    +    150   +    50     =       4,00,000

⇒      400    =       4,00,000

⇒                =       400000 / 400

⇒                  =       1000

Numbers of Rs 100  notes     =    2     =       2000

Numbers of Rs 50  notes       =    3     =       3000

Numbers of Rs 10  notes       =    5     =       5000

 

  1. I have a total of Rs 300 in coins of denomination Rs 1, Rs 2 and Rs 5. The number of Rs 2 coins is 3  times the number of Rs 5 coins. The total number of coins is 160. How many coins of each denomination are with me.

Solution:  Let the number of Rs 5 coins be x.

Then,

number Rs 2 coins     =    3 x

and, number of Rs1 coins     =    ( 160    –  4x ) Now,

Value of Rs5 coins     =   x    ×   5    =    5x

Value of Rs2 coins     =    3 x  ×   2     =    6x

Value of Rs1 coins     =    ( 160    –  4 x )  ×  1   =   ( 160    –  4 x )

According to the question,

5 x   +   6 x   +   ( 160   –  4 x )     =    300

⇒      11 x   +   160   –  4 x    =   300

⇒      7 x    =   140

⇒      x      =    140 / 7

⇒      x      =    20

Number of  Rs  5 coins     =   x       =    20

Number of  Rs  2 coins     =    3 x    =   60

Number of  Rs  1 coins     =    ( 160   –  4 x )    =   160    –  80    =   80

  1. The organisers of an essay competition decide that a winner in the competition gets a prize of Rs 100 and a participant who does not win gets a prize of Rs 25. The total prize money distributed is Rs 3,000. Find the number of winners, if the total number of participants is 63.

Solution:  Let the numbers of winner be x.

Then, the number of participants who didn’t win    =   63    –  x

Total money given to the winner     =   x  ×  100     =   100 x

Total money given to participant who didn’t win    =    25  ×  ( 63 )

According to the question,

100 x    +    25  ×  ( 63   )    =   3,000

⇒      100 x    +    1575    –  25 x     =    3,000

⇒      75 x     =    3,000    –  1575

⇒      75 x     =    1425

⇒       x         =    1425 / 75

⇒       x         =    19

Therefore, the numbers of winners are 19.