Example 1:  Arjun  is twice as old as  Shriya.  Five years ago his age was three times  Shriya’s age. Find their present ages.

Solution:  Let us take Shriya’s present age to bex years. Then Arjun’s present age would be 2 x years.

Shriya’s age five years ago was  ( x  –  5 ) years. Arjun’s  age  five  years  ago  was  ( 2 x   –  5 ) years.

It is given that  Arjun’s age five years ago was three times Shriya’s age.

Thus,     2 x  –  5    =   3( x   –  5 )

or      2 x  –  5    =   3 x   –  15

or      15  –  5     =   3 x   –   2 x

or      10      =     x

So, Shriya’s present age     =   x     =    10 years.

Therefore, Arjun’s present age    =   2 x    =   2 × 10    =   20 years.

Example 2:  The digits of a two    digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. What can be the original number.

Solution: Take, for example, a two    digit number, say, 56. It can be written as     56    =   ( 10 × 5)   +   6.

If the digits in 56 are interchanged, we get  65, which can be written as   (10  ×  6 )  +   5.

Let us take the two digit number such that the digit in the units place is b. The digit in the tens   place differs from b by 3. Let us take it as b  +  3. So the two   digit number is 10 ( b   +   3 )   +   b     =     10 b   +   30   +  b     =     11 b   +   30.

With interchange of digits, the resulting two   digit number will be

10 b   +  ( b   +   3 )     =     11  b   +   3

If we add these two two   digit numbers, their sum is

( 11 b   +   30 )   +   ( 11 b   +   3 )    =   11 b   +   11 b   +   30   +   3     =     22 b   +   33

It is given that the sum is 143. Therefore,

22 b   +   33     =     143    or   22 b     =     143   –   33

or    22 b    =   110

b    =  b    =   5

The units digit is 5  and  therefore  the  tens  digit  is  5   +   3   which  is  8. The number  is  85.

EXERCISE 2.4

1. Amina thinks of a number and subtracts 5 / 2 from it. She multiplies the result by 8. The result now obtained is 3 times the same number she thought of. What is the number.

Solution: Let the number be  x ,

According to the question,

( x  –  5 / 2 ) × 8    =    3 x

⇒      8 x  –  40 / 2    =    3 x

⇒      8 x  –  3 x    =    40 / 2

⇒      5 x    =    20

⇒        x    =    4

Thus, the number is 4.

1. A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?

Solution: Let one of the positive number be ( x ) then other number will be 5 x. According to the question,

5 x  +  21    =    2 ( x   +  21 )

⇒      5 x     +     21     =   2 x   +   42

⇒      5 x     –     2 x    =   42   –   21

⇒      3 x     =    21

⇒      x        =    7

One number      =     x    =   7

Other number    =   5 x     =     5 × 7     =     35    The two numbers are  7  and  35.

1. Sum of the digits of a two  digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two    digit number.

Solution: Let the digit at tens place be  x  then digit at ones place will be ( 9  –   x ).

Original two    digit number    =   10 x   +   ( 9   –   x )

After interchanging the digits, the new number    =   10 ( 9   –   x )   +  x

According to the question,

10 x   +   ( 9   –   x )   +   27     =     10 ( 9   –  x )   +  x

⇒      10 x   +    9    –  x   +  27    =   90   –  10 x   +   x

⇒      9 x     +    36    =   90    –  9 x

⇒      9 x     +    9x    =   90    –  36

⇒      18 x   =   54

⇒        x       =   3

Original number  = 10 x   +   ( 9   –  x )  =  ( 10  ×  3 )   +   ( 9   –  3 )  = 30   +   6  = 36

Thus, the number is 36.

1. One of the two digits of a two  digit number is three times the other digit. If you interchange the digits of this two   digit number and add the resulting number to the original number, you get 88. What is the original number.

Solution: Let the digit at tens place be   x  then digit at ones place will be 3 x.

Original two    digit number    =   10 x   +   3 x

After interchanging the digits, the new number    =   30 x   +  x

According to the question,

( 30 x   +  x )   +   ( 10 x   +   3 x)      =    88

⇒      31 x   +   13 x     =    88

⇒      44 x     =    88

⇒      x     =    2

Original number    =   10 x   +   3 x    =   13 x    =   13  ×  2    =   26

1. Shobo’s mother’s present age is six times Shobo’s present age. Shobo’s age five years from now will be one third of his mother’s present age. What are their present ages.

Solution: Let the present age of Shobo be ( x ) then age of her mother will be 6x.

Shobo’s age after  5 years      =     x    +    5

According to the question,

( x   +   5 )    =   ( 1 / 3 )  ×  6 x

⇒        x   +   5     =    2 x

⇒       2 x   –x   =    5

⇒         x     =    5

Present age of  Shobo      =     x    =   5 years

Present age of  Shobo’s  mother    =   6 x    =   30 years.

1. There is a narrow rectangular plot, reserved for a school, in Mahuli village. The length and breadth of the plot are in the ratio 11:4. At the rate Rs 100 per metre it will cost the village panchayat Rs 75000 to fence the plot. What are the dimensions of the plot.

Solution: Let the length of the rectangular plot be  11 x and breadth be  4 x.

Rate of fencing per metre    =    Rs 100

Total cost of fencing    =    Rs. 75000

Perimeter of the plot    =    2 ( l  +  b )    =    2 (11 x  +  4 x )    =    2 × 15 x    =    30 x

Total amount of fencing    =    (30 x × 100)

According to the question,

(30 x  ×  100)    =    75000

⇒      3000 x    =    75000

⇒        x    =    75000 / 3000

⇒        x    =    25

Length of the plot    =    x    =    11 × 25    =    275 m

Breadth of the plot    =    4 × 25    =    100 m.

1. Hasan buys two kinds of cloth materials for school uniforms, shirt material that costs him Rs 50 per metre and trouser material that costs him Rs90 per metre. For every 3 meters of the shirt material he buys 2 metres of the trouser material. He sells the materials at 12% and 10% profit respectively. His total sale is Rs 36,600. How much trouser material did he buy.

Solution: Let 2 x m of trouser material and 3 x m of shirt material be bought by him

Selling price of shirt material per meter    =    Rs 50   +   50  × ( 12 / 100 )    =    Rs 56

Selling price of trouser material per meter    =    Rs 90  +  90 × ( 10 / 100 )   =    Rs 99

Total amount of sale    =    Rs. 36,600

According to the question,

( 2 x × 99 )   +   ( 3 x  ×  56 )    =    36600

⇒      198 x   +   168 x    =    36600

⇒      366 x    =    36600

⇒        x    =    36600 / 366

⇒        x    =    100

Total trouser material he bought    =    2 x    =    2 × 100    =    200 m.

1. Half of a herd of deer are grazing in the field and three fourths of the remaining are playing nearby. The rest 9 are drinking water from the pond. Find the number of deer in the herd.

Solution:  Let the total number of deer be  x .

Deer grazing in the field    =   x / 2

Deer playing nearby    =   x / 2 × ¾    =    3 x  / 8

Deer drinking water    =    9

According to the question,

/  2  +  3 x / 8  +  9    =   x

( 4 x  +  3 x ) / 8  +  9    =   x

⇒        7 x / 8  +  9    =   x

⇒        x  –  7 x / 8    =    9

⇒      ( 8 x  –  7 x ) / 8    =    9

⇒        x    =    9 × 8

⇒        x    =    72

1. A grandfather is ten times older than his granddaughter. He is also 54 years older than her. Find their present ages.

Solution: Let the age of granddaughter be ( x ) and grandfather be 10x.

Also, he is 54 years older than her.

According to the question, 10x    =   x  +  54

⇒      10 x –  x    =    54

⇒        9 x    =    54

⇒           x    =    6

Age of grandfather    =    10 x    =    10 ×  6    =    60 years.

Age of granddaughter    =   x    =    6 years.

1. Aman’s age is three times his son’s age. Ten years ago he was five times his son’s age. Find their present ages.

Solution: Let the age of Aman’s son be  x  then age of Aman will be 3 x.

According to the question,

5 ( x  –  10)    =    3 x  –  10

⇒        5 x  –  50    =    3 x  –  10

⇒      5 x  –  3 x    =   –  10  +  50

⇒      2 x    =    40

⇒        x    =    20

Aman’s son age    =   x    =    20 years

Aman age    =    3 x    =    3  ×  20    =    60 years