<pre>GM of two numbers is (ab)<sup>1/2</sup><br>                           (ab) <sup>1/2</sup>= a+12<br>                           ab = a2+144+24a<br>from 1 b=a+48<br>                          a(a+48)=  a2+144+24a<br>                          a2+48a =  a2+144+24a<br>                                 24a=144=>a=6<br>                Therefore     b=a+48=54.<br><br></pre>Ans (a).<br><br>31.The sum of squares of the digits constituting a positive<br> two digit number is 13,If we subtract 9 from that number <br>we shall get a number written by the same digits in the <br>reverse order. Find the number?<br><br>a.12   &nbsp;&nbsp;   b.32    &nbsp;&nbsp;c.42   &nbsp;&nbsp;  d.52.<br><br>Solution:<br>Let the number be x y.<br>the sum of the squares of the digits of the number is 13<br>                           x2+y2=13<br>If we subtract 9 from the number we get the number<br> in reverse order<br>                        x y-9=y x.<br>                       10x+y-9=10y+x.<br>                       9x-9y=9<br>                        x-y=1<br>               (x-y)2 =x2+y2-2x y<br>                  1    =13-2x y<br>                  2x y = 12<br>                    x y = 6 =>y=6/x<br>                        x-y=1<br>                       x-6/x=1<br>                       x2-6=x<br>                      x2-x-6=0   <br>                     x+2x-3x-6=0<br>                     x(x+2)-3(x+2)=0<br>                          x=3,-2.<br>If x=3 and x-y=1 then y=2.<br>If x=-2 and x-y=1 then y=-3.<br>Therefore the number is 32.<br><br>Ans (b).<br><br>32.If we add the square of the digit in the tens place <br>of the positive two digit number to the product of the <br>digits of that number we get 52,and if we add the square<br> of the digit in the unit's place to the same product <br>of the digits we get 117.Find the two digit number?<br><br>a.18     &nbsp;&nbsp;  b.39     &nbsp;&nbsp; c.49   &nbsp;&nbsp; d.28<br>Solution:<br>Let the digit number be x y<br>Given that if we add square of the digit in the tens place <br>of a number to the product of the digits we get 52.<br><br>                          x2+x y=52.<br>                          x(x+y)=52....................1<br>Given that if we add the square of the digit in the unit's plac<br>e to the product is 117.<br><br>                                 y2+x y= 117 <br>                   y(x+y)=117.........................2<br>dividing 1 and 2  x(x+y)/y(x+y) = 52/117=4/9<br>                                x/y=4/9<br>from the options we conclude that the two digit number is 49 <br>because the condition is satisfied by the third option.<br><br>Ans (c)<br><br>33.The  denominators of an irreducible fraction is greater <br>than the numerator by 2.If we reduce the numerator of the<br> reciprocal fraction by 3 and subtract the given fraction <br>from the resulting one,we get 1/15.Find the given fraction?<br><br>Solution:<br>Let the given fraction be   x / (x+2) because given that <br>denominator of the fraction is greater than the numerator by 2<br>                            1 – [(x – 1/(x+2))/3]      =   1/15.<br>                            1 – (x2+2x-1) /3(x+2)     =   1/15    <br>                                 (3x+6-x2-2x+1)/3(x+2)   =   1/15<br>                             (7-x2+2x)/(x+2)                =   1/5<br>                               -5x2+5x+35     =    x+2   <br>                                                 5x2-4x-33  =  0<br>                                 5x2-15x+11x-33   =   0<br>                                     5x(x-3)+11(x-3)   =   0<br>                                       (5x+11)(x-3)     =   0<br>                             Therefore x=-11/5 or 3<br>  Therefore the fraction is x/(x+2)  =  3/5.<br><br>34.Three numbers are such that the second is as much <br>lesser than the third as    the first is lesser than <br>the second. If the product of the two smaller numbers<br> is 85 and    the product of two larger numbers is 115.<br>Find the middle number?<br><br>Solution:<br>Let the three numbers be x,y,z<br>  Given that z – y = y – x<br>                         2y = x+z.....................1<br>  Given that the product of two smaller numbers is 85<br>                                x y = 85................2 <br>     Given that the product of two larger numbers is 115<br>                                y z = 115...............3<br>Dividing 2 and 3  x y /y z = 85/115<br>                                x / z = 17 / 23<br>From 1<br>                           2y = x+z<br>                          2y = 85/y + 115/y<br>                           2y2 = 200<br>                             y2 = 100<br>                             y = 10<br><br>35.If we divide a two digit number by the sum of its digits<br> we get 4 as a quotient and 3 as a remainder. Now if we <br>divide that two digit number by the product of its digits <br>we get 3 as a quotient and  5 as a remainder .<br>Find the  two digit number?<br><br>Solution:<br>Let the two digit number is x y.<br>  Given that x y / (x+y)   <br>      quotient=4 and remainder = 3<br>  we can write the number as<br>                         x y = 4(x+y) +3...........1<br>  Given that x y /(x*y)     quotient = 3 and remainder = 5<br>    we can write the number as<br>                       x y = 3 x*y +5...............2<br>By trail and error method <br>For example take x=1,y=2<br>1............12=4(2+3)+3<br>                   =4*3+3<br>                  ! =15<br>let us take x=2 y=3<br>1..............23=4(2+3)+3<br>                    =20+3<br>                    =23<br>2.............23=3*2*3+5<br>                   =18+5<br>                   =23 <br> the above two equations are satisfied by x=2 and y=3<br>Therefore the required number is 23.<br><br>36.First we increased the denominator of a positive <br>fraction by 3 and then it by 5.The sum of the <br>resulting fractions proves to be equal to 2/3.<br>Find the denominator of the fraction if its numerator is 2.<br>Solution:<br>Let us assume the fraction is x/y<br>First we increasing the denominator by 3 we get x/(y-3)<br>Then decrease it by 5 we get the fraction as x/(y-5)<br>Given that the sum of the resulting fraction is 2/3<br>               x/(y+3)    +   x/(y-5)   =   2/3<br> Given numerator equal to 2<br>              2*[ 1/y+3   +    1/y-5]   =2/3<br>             (y-5+y+3) / (y-3)(y+5)  =1/3<br>               6y – 6 = y2-5y+3y-15<br>                         y2-8y-9 = 0<br>                        y2-9y+y-9 = 0<br>                        y(y-9)+1(y-9) = 0<br>                         Therefore y =-1 or 9.<br><br>37.If we divide a two digit number by a number consisting <br>of the same digits written in the reverse order,we get 4 <br>as quotient and 15 as a remainder. If we subtract 1 from <br>the given number we get the sum of the squares of the <br>digits constituting that number. Find the number?<br><br>a.71   &nbsp;&nbsp;b.83   &nbsp;&nbsp;c.99   &nbsp;&nbsp; d. none<br><br>Solution:<br>Let the number be x y.<br>If we divide 10x+y by a number in reverse order <br>i e,10y+x we get 4 as quotient and 15 as remainder.<br>   We can write as<br>          10x+y = 4(10y+x)+15......................1<br>  If we subtract 1 from the given number we get square of the digits<br>          10x+y = x2+y2.....................................2<br>By using above two equations and trail and error method<br>we get the required number. From the options also we can<br>solve the problem. In this no option is satisfied so answer is d.<br><br>Ans (d)<br><br> <a href="pnumbers.html"  name="right"><b>BACK</b></a> <br> <br></font></TD></TR></table></html>