Question

The four, five, and sixth question

Answer 1

 present age of rahaman is x yrs 
3 yrs ago age of rahman was x-3 yrs
5 yrs from now age of rahman will be x+5 yrs
as per given conditions 
1/x-3 +1/x+5 =1/3
by LCM
x+5+x-3/(x+5)(x-3) =1/3
2x+2/x^2 +2x -15 =1/3 
by cross multiplication we can find 
6x+6 =x^2 +2x-15 
x^2+2x -15 -6x -6 =0
x^2 -4x-21=0
(x-7)(x+3) =0
x-7=0 or x=3 =0
x=7 or x=-3





Let take Shefali’s marks in Mathematics = x

Given that

the sum of Shefali’s marks in Mathematics and English is 30

so that marks of Shefali’s in English = 30 – x

given that . she got 2 marks more in Mathematics and 3 marks less in English,

so that marks in mathematics  = x  +2

marks in English = 30 – x -3  = 27 – x

 the product of their marks would have been 210.

So that

(x+2)(27-x) = 210

27 x  - x2 + 54 – 2x  = 210

25 x - x2  + 54 – 210 = 0

- x2 + 25 x – 156  =0


x2-25x +156 =0

x2-25x +156 =0

x2 – 12 x – 13 x  + 156 = 0

x(x-12) -13(x-12) =0

(x-12)(x-13)=0

x-12= 0 then x  = 12

x-13 = 0 then x = 13

if Shefali marks in mathematics  = 12

then marks in English  = 30 – 12  = 18

if marks of Shefali in mathematics is 13

then marks in English  = 30 – 13  = 17


6. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field. 7. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

 Let shorter side  = x m

Given that diagonal of a rectangular field is 60 metres more than the shorter side

So diagonal will  = x + 60 m

Given that the longer side is 30 metres more than the shorter side

Longer side = x + 30  m

All angles in rectangle is always = 90 so that we can Use Pythagoras theorem



Diagonal 2  =  shorter side 2  +  longer side 2


Plug the values In this formula we get


(x+60)2 = (x)2   + (x+30)2


Now use the formula (a+b)2 =  a2 +b2 +2ab


We get


x2 +  120 x + 3600 = x2 +  x2 +  60 x +  900


subtract x`2  + 120 x  + 3600  both side we get




0 = x2  - 60 x - 2700


Factorize it now


x 2 -  90 x + 30 x – 2700  = 0


x ( x - 90 ) +  30 (x – 90 )=0


( x – 90 ) (x + 30 ) = 0


x- 90 = 0 or x + 30 = 0 


x = 90   or x  = - 30 any side of rectangle cannot negative

  

so shorter side  = x + 30 = 90 + 30 = 120

Diagonal is  = x + 60 = 90 + 60 = 150