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Question 1

Solve the equation and check your solution:

Solution:
    

 

Question 2

Solve the equation and check your solution:

Solution:
 

y = -1



Question 3

Solve the equation and check your solution:

 


Solution:

Cross multiplying, we have:


45z - 16z = 27 (Transposing 16z and - 27)




 

Question 4

Solve the equation and check your solution:


Solution:
 
Cross multiplying , we have:




Question 5

Solve the equation:

Solution:


By cross multiplying , we have




Question 6

Solve the equation:


Solution:


By cross multiplying , we have



Question 7

Solve the equation: 

Solution:



Multiplying both sides of the above equation by (1 - 7x), we have:



Question 8

Solve the equation:


Solution:


Question 9

Solve the equation:

Solution:



By cross multiplying , we have



Question 10

The sum of two numbers is 45 and their ratio is 7:8. Find the numbers.


Solution:
Let one of the numbers be x.
Then the other number will be 45 - x

By the given condition,


By cross multiplying , we have


 


Question 11

The sum of the digits of a two digit number is 12. If the new number formed by reversing the digits is greater than the original number by 18, find the original number. Check your solution.


Solution:
Let the digit in the ones place be x.

Then the digit in the tens place will be 12 – x.

Therefore, the original number = 10(12 - x) + x  = 120 – 10x + x = 120 - 9x.

And, the new number = 10
x + (12 - x) = 10x + 12 – x = 9x + 12.

By the given condition,

New number = original number + 18

     9x + 12   = 120 – 9x + 18 

       9x + 12 = 138 – 9x

       9x + 9x = 138 – 12                     (Transposing 9x and 12)

             18x = 126


 
 (Dividing both sides by 18)


x = 7

Thus, ones digit is 7 and tens digit is 12 - 7 = 5.  Hence, the required number is 57.


Check:    (1)  5 + 7 = 12 is the sum of the digit.
             (2)   New number = 75. 

Difference between the original number and new number = 75 – 57 = 18.


\
The new number is 18 more than the original number.
 

Question 13

Meera’s mother is four times as old as Meera.After five years,her mother will be three times as old as she will be then.What are their present ages?


Solution:
Let present age of Meera be x years.
Then the present age of Meera's mother  =  4x years.
                   Meera's age after five years  =  (x + 5) years.
        Meera's mother age after five years  =  (4x + 5) years.

By the given condition, we have:
4x + 5   =  3(x + 5)
4x  + 5  = 3x + 15
4x - 3x  = 15 - 5(Transposing 3x and 5)
         x  = 10
Thus, age of Meera  = 10 years and that of Meera's mother  =  4 x 10  =  40 years.

Check: 
   
(1) Meera's age after five years  = 10 + 5 = 15 years.
Meera's mother age after five years = 4 (10) + 5  =  40 + 5  =  45 years.
Therefore, Meera's mother age after five years is three times Meera's age after five years. Let present age of Meera be x years.
Then the present age of Meera's mother  =  4x years. 
                   Meera's age after five years = (x + 5) years.
       Meera's mother age after five years  =  (4x + 5) years.

By the given condition, we have:
4x + 5  = 3(x + 5)
4x  + 5  = 3x + 15
4x - 3x  = 15 - 5(Transposing 3x and 5)
           x = 10
Thus, age of Meera = 10 years and that of Meera's mother = 4 x 10 = 40 years.


Question 14

The length of a rectangle exceeds its breadth by 4 cm. If length and breadth are each increased by 3 cm, the area of the new rectangle will be 81 sq. cm more than that of the given rectangle.  Find the length and breadth of the given rectangle.  Check your solution.   

Solution:
Let the breadth of the rectangle be x cm.
Then the length of the rectangle will be (x + 4) cm.
Therefore, the area of the given rectangle = length
´ breadth
                                                             = (x + 4)x
                                                             = (x2 + 4x) cm2
By the given condition, we have:
   Area of the new rectangle = Area of the given rectangle + 81 cm2

\ Area of the new rectangle = New length ´ New breadth
(x2 + 4x)  +  81   =  [(x + 4) + 3] (x + 3)
     x2 + 4x + 81   =   (x +7)(x +3)
     x2 + 4x + 81   =  x (x +3)  + 7 (x + 3)
      x2 + 4x + 81  =   x2 + 3x + 7x + 21
     


Check:  
(1) (14-10)cm  = 4cm, i.e. the length of rectangle exceeds the breadth by 4 cm.

(2)     New length = 14 + 7 = 17 cm and new breadth = 10 + 3 = 13 cm.

Therefore, area of the new rectangle – area of the given rectangle 
 = 17
´ 13 – 14 ´ 10 
 = 221 – 140 
 = 81 cm.
 

Question 15

An altitude of a triangle is five-thirds the length of its corresponding base. If the altitude were increased by 4 cm and the base decreased by 2 cm, the area of the triangle would remain the same. Find the base and altitude of the triangle.  

Solution:
Let the base of the triangle be x cm.


                                                     
By the given condition, we have:
Area of the new triangle = Area of the given triangle


                                          0 = (Transposing to the L.H.S.)
                                        (Multiplying by 3 on both sides)


                                    =120 cm2





Solution:
Let one number be x
Then the other number will be 2490 – x.
By the given condition, we have:
 


Thus, one number is = 1411 and the other number = 2490 – 1411 = 1079.

Check:
(1) 1411+1079=2490 i.e the sum of the two numbers

Thus, 6.5% of one number is equal to 8.5% of the other.
 


Solution:
 
 Let one of the angle be ÐA and the other two angles are ∠B and ∠C
Given one of the angles of a triangle is equal to the sum of the other two angles
i.e., ∠A = ∠B + ∠C
Also it is given the ratio of the other two angles is 4:5
Therefore ,the other two angles B and C be 4x and 5x  

        By the given condition, we have

Thus, the angles of the triangle are 9
´10, 4´10, 5´10 i.e. 90°, 40°, 50°.

Check:  
(1) 40° + 50°  = 90° i.e. one of the angles of a triangle is equal to the sum of the other two angles.

(2)   40:50  = 4:5 is the ratio of the other two angles.


Question 19

Solve for x.
         (i) 3x - 5 = x + 5
         (ii) 9x - 3 = 7x + 3
         (iii)
         (iv)

Solution:
 (i)3x - 5 = x + 5
    3x - x = 5 + 5
         2x = 10
           x = 5

(ii) 9x - 3 = 7x + 3
     9x - 7x = 3 + 3
          2x = 6
            x = 3

(iii)
       3x + 2 = 8(2x - 3)
       3x + 2 = 16x - 24
    3x - 16x = -24 - 2
          -13x = -26
               x = 2


(iv)
   28(3x - 8)=9(4x + 4)
   84x - 224 = 36x + 36
   84x - 36x = 36 + 224
           48x = 260

              x = 65/12

Question 20

In a students' hostel, one third of the total number of girls and four more take vegetarian food only, one fourth and three more take non-vegetarian food only. The other 103 take both vegetarian and non-vegetarian food. How many girls are there in the hostel?

Solution:
Let the total number of girls be x.
                                No. of girls who take vegetarian food only  =   x + 4
                      No. of girls who take non - vegetarian food only  =  x + 3
No. of girls who take both vegetarian and non - vegetarian food  =  103.

By the given condition,

x + 4 + x + 3 + 103 = x

        x + x - x + 110 = 0

Multiply through out by 12,

4x + 3x - 12x + 1320 = 0

               -5x + 1320 = 0

                          -5x = -1320

                             x =
                             x = 264∴ There are 264 girls in the hostel.


Question 21

The sum of the digits of a two digit number is 9. If the digits are reversed,the number is 63 more than the original. Find the number.


Solution:
Let the units digit be x.
Then the tens digit is 9 - x.
              Therefore the original number is 10(9 - x) + x = 90 - 10x + x = 90 - 9x.
On reversing the order of the digits the number obtained is 10x + 9 - x = 9x + 9.

By the given condition,

 9x + 9 = 63 + (90 - 9x)

(9x + 9) - (90 - 9x) = 63

    9x + 9 - 90 + 9x = 63

                   18x = 144 (Transposing 9 and -90 to the R.H.S.)

                         x = 8

Therefore the original number is 90 - 9× 8 = 90 - 72 = 18.

Question 22

A mother is 46years old and her son is 21 years old in the year 1997. In what year was the mother six times as old as the son?


Solution:
Age of the mother in the year 1997  =  46years
      Age of her son in the year 1997 = 21years
        x years ago, age of the mother  =  46 - x.
             x years ago, age of the son  =  21 - x.
Hence by the given condition,
46 - x  =  6(21 - x)
46 - x  =  126 - 6x
-x + 6x  =  126 - 46
      5x  =  80
       x  =  16
Thus the year was 1997-16 = 1981.

Question 23

A man sold a bicycle for an amount, which was greater than Rs.988 by half the price he paid for it, and made a profit of Rs.300. How much did he buy the bicycle for?     

Solution:
Let the CP be Rs.x
Then SP  =  988 + x
      Profit  = Rs. 300
By the given condition,
           SP  =  CP + Profit
988 + x  =  x + 300
Multiplying by 2 on both sides
2× 988 + x  =  2x + 600
   1976 + x  = 2x + 600
1976 - 600  =  2x - x
       ∴ x  = 1376

Question 24

The unequal side of an isosceles triangle is 3cm more than one of its equal sides. If the perimeter of the triangle is 18cm, find the length of the three sides.

Solution:
Let the length of the equal sides be x cm.
Then the length of the unequal sides will be (x + 3) cm.
Perimeter of an isosceles triangle is 18.∴ x + x + 3 + x  = 18
               3x + 3  = 18
                     3x  = 15            ∴ x  =  5
Therefore the equal sides of the triangle is 5cm and unequal side is 8cm.

Question 25

The sum of 5 consecutive numbers is 140. Find the numbers.  

Solution:
Let the 5 consecutive numbers be x, x+1, x+2, x +3, x + 4.
x + x + 1+ x + 2 + x + 3 + x + 4 = 140
                                     5x + 10  = 140
                                             5x  = 130
                                               x  =  26
Therefore the numbers are 26, 27, 28, 29, 30.

Question 26

The tens and units digits of a number are the same. When the number is added to its reverse, the sum is 110. What is the number?

Solution:
Let the units digit be x. 

Then the tens digit is also x.

Therefore the number is 10x + x = 11x.

On reversing the order of the digit the number is 10x + x = 11x.

Hence by the given condition we have,

11x + 11x = 110

         22x = 110

            x = 5

Therefore the required number is 55.





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