7 Problem With Numbers

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7. PROBLEMS ON NUMBERS

In this section, questions involving a set of numbers are put

in the form of a puzzle. You have to analyze the givenconditions, assume the unknown numbers and form

equations accordingly, which on solving yield the unknown

numbers.

SOLVED EXAMPLES

Ex.1. A number is as much greater than 36 as is less

than 86. Fin the number.

S!l. Let the number be x. Then, x !" # $" x #% &x #

$" ' !" # (&& #% x # "(. )ence, the required

number is "(.

Ex. ". Fin a number such that #hen 1$ is subtracte

%r!m & times the number' the

Result is 1( m!re than t#ice the number. *)otel

+anagement, &&-

S!l. Let the number be x. Then, x (/ # &x ' ( #% /x #

&/ #%x # /. )ence, the required number is /.

Ex. 3. )he sum !% a rati!nal number an its reci*r!cal is

13+6. Fin the number.

*0.0.1. &-

S!l. Let the number be x.

Then, x ' *(2x- # (!2" #% *x& ' (-2x # (!2" #% "x& 3 (!x ' " #



#% "x& 3 4x 3 5x ' " # #% *!x 3 &- *&x 3

!- #

x # &2! or x # !2&

)ence the required number is &2! or !2&.

Ex. ,. )he sum !% t#! numbers is 18,. -% !nethir !% the

!ne excees !nese/enth

!% the !ther b0 8' %in the smaller number.

S!l. Let the numbers be x and *($5 x). Then,

*62!- **($5 3 x-2- # $ #% x 3 !*($5 3 x- # ("$ #%

(x # & #% x # &. 0o, the numbers are & and ((&. )ence, smaller

number # &.

Ex. $. )he i%%erence !% t#! numbers is 11 an !ne%i%th

!% their sum is . Fin the numbers.

S!l. Let the number be x and y. Then, x 3 y # (( *i- and (2/ (x ' y) # 4 #% x ' y

# 5/ *ii-

7dding *i- and *ii-, we get8 &x # /" or x # &$. 9utting

x # &$ in *i-, we get8 y # (.

Hence, the numbers are &$ and (.

Ex. 6. -% the sum !% t#! numbers is ," an their *r!uctis ,3&' then %in the

absolute i%%erence bet#een the numbers.*0.0.1. &!-

S!l. Let the numbers be x and y. Then, x ' y # 5& and xy

# 5!

x y # sqrt:*x ' y)2 5xy; # sqrt:*5&-& 5 x 5! ; #

sqrt:("5 3 (5$; # sqrt:("; # 5. <equired difference # 5.



Ex. &. )he sum !% t#! numbers is 16 an the sum !%

their s2uares is 113. Fin the

numbers.

S!l. Let the numbers be x and *(/ x). Then, x& ' *(/ x-& # ((! #%x& ' &&/ ' 6&

!x # ((!

#% 2x2 !x ' ((& # #% x& 15x ' /"

#

#% (x - (x $- # #% x # or x #

$.

So, the numbers are and $.

Ex. 8. )he a/erage !% %!ur c!nsecuti/e e/en numbers is

"&. Fin the largest !% these

numbers.

S!l. Let the four consecutive even numbers be x, x ' &, x

' 5 and x ' ". Then, sum of these numbers # *& x 5- # ($.

So, x ' (x ' &- ' (x ' 5- ' (x ' "- # ($ or 4x # 4"

or x # &5.

8. Largest number # (x ' "- # !.

Ex. . )he sum !% the s2uares !% three c!nsecuti/e !

numbers is "$31.Fin thenumbers.

S!l. Let the numbers be x, x ' & and x ' 5.

Then, 6& ' (x ' &-& ' (x ' 5-& # &/!( #% 3x2 ' 12x

&/(( #

#% 6& ' 4x $! # #% (x &- (x ' !(- #

#% x # &. )ence, the required numbers are &, &4 and !(.



Ex. 1(. O% t#! numbers' , times the smaller !ne is less

then 3 times the 1arger !ne b0 $. -% the sum !% the

numbers is larger than 6 times their i%%erence b0 6' %inthe t#! numbers.

S!l. Let the numbers be x and y, such that x % y

Then, 3x 5y # / ...*i- and (x ' y) " (x y) # " #%

5x ' y # " =*ii-

0olving *i- and (ii), we get8 x # /4 and y # 5!.

)ence, the required numbers are /4 and 5!.

Ex. 11. )he rati! bet#een a t#!igit number an the

sum !% the igits !% that

number is , 1.-% the igit in the unit4s *lace is 3 m!re

than the igit in the ten5s *lace' #hat is the number

S!l. Let the ten>s digit be x. Then, unit>s digit # (x ' !-.

0um of the digits # x ' (x ' !- # 2x ' !. ?umber #lx ' (x ' !- # llx ' !.

((x'! 2 &x ' ! # 5 2 ( #% (lx ' ! # 5 *&x ' !- #%

!x # 4 #% x # !.

)ence, required number # 11x ' ! # !".

Ex. 1". A number c!nsists !% t#! igits. )he sum !% the

igits is . -% 63 is subtracte%r!m the number' its igits are interchange. Fin the

number.

S!l. Let the ten>s digit be x. Then, unit>s digit # *4 x).

?umber # lx ' *4 x- # 9x ' 4.

?umber obtained by reversing the digits # ( *4 x-

' x # 4 4x.therefore, (9x ' 4- "! # 4 9x #% 18x # (55



#% x # $.

So, ten>s digit # $ and unit>s digit # (.

)ence, the required number is $(.

Ex. 13. A %racti!n bec!mes "+3 #hen 1 is ae t! b!th'its numerat!r an en!minat!r.

An 'it bec!mes 1+" #hen 1 is subtracte %r!m b!th the

numerat!r an en!minat!r. Fin the %racti!n.

S!l. Let the required fraction be x2y. Then,

x'( 2 y'( # & 2 ! #% !x 3 &y # ( =*i- and x 3 ( 2

y 3 ( # ( 2 &

&x 3 y # ( =*ii-0olving *i- and *ii-, we get 8 x # ! , y # /

therefore, <equired fraction# ! 2 /.

Ex. 1,. $( is i/ie int! t#! *arts such that the sum !%

their reci*r!cals is 1+ 1".Fin the t#! *arts.

S!l. Let the two parts be x and */ x).

Then, ( 2 x ' ( 2 */ 3 x- # ( 2 (& #% */ 3 x ' x- 2 x

* / 3 x- # ( 2 (&

#% x& 3 /x ' " # #% *x 3 !- * x 3 &- #

#% x # ! or x # &.

0o, the parts are ! and &.

Ex. 1$. -% three numbers are ae in *airs' the sums

e2ual 1(' 1 an "1. Fin the

numbers 7S!l. Let the numbers be x, y and z. Then,

x' y # ( ...(i)y ' z # (4...(ii) x ' z # &(

=*iii-7dding *i- ,(ii) and (iii), we get8 2 (x ' y + z ) # / or



*x ' y ' z- # &/.

Thus, x# *&/ (4- # "@ y # *&/ &(- # 5@ z # *&/ (-

# (/.

)ence, the required numbers are ", 5 and (/.

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7 Problem With Numbers