Question 483727
<pre>
let x = the number of lemons
let y = the number of limes

$0.52x + $0.21y = $3.75

      52x + 21y = 375

Write 52 and 375 in terms of their nearest multiple of 21
So we write 52 as (42+10) and 375 as 378 - 3

(42 + 10)x + 21y = 378 - 3

 42x + 10x + 21y = 378 - 3

Divide through by 21

  2x + {{{10x/21}}} + y = 18 - {{{3/21}}}

Isolate fractions on the left:

  {{{10x/21}}} + {{{3/21}}} = 18 - 2x - y


The right side is an integer and the left side is positive, so
both sides equals some positive integer A

  {{{10x/21}}} + {{{3/21}}} = A and  18 - 2x - y = A

Clear of fractions:

        10x + 3 = 21A

Write 21 in terms of its nearest multiple of 10

        10x + 3 = (20 + 1)A

        10x + 3 = 20A + A

Divide through by 10

          x + {{{3/10}}} = 2A + {{{A/10}}}

Isolate fractions on the left

          {{{3/10}}} - {{{A/10}}} = 2A - x

The right side is an integer, so the left side is too, say B

{{{3/10}}} - {{{A/10}}} = B and  2A - x = B

Clear of fractions:

3 - A = 10B
   -A = -3 - 10B
    A = 3 + 10B
    A = 10B + 3

2A - x = B

2(10B + 3) - x = B

20B + 6 - x = B

19B + 6 = x

and since

18 - 2x - y = A,

18 - 2(19B + 6) - y = 10B + 3

18 - 38B - 12 - y = 10B + 3

               -y = 48B - 3
  
                y = -48B + 3

                y = 3 - 48B

we must buy more than -1 but less than 18 limes

            -1 < y < 18

         -1 < 3 - 48B < 18

          -4 < -48B < 18

Divide through by -48 which reverses inequalities:

          {{{(-4)/(-48)}}} > B > {{{18/(-48)}}}

          {{{1/12}}} > B > {{{-3/8}}}

Since B is an integer then B = 0 because 0 is
the only integer between those values.

Therefore since 

         y = 3 - 48B
         y = 3 - 48(0)
         y = 3 - 0
         y = 3     

and since 

       19B + 6 = x
     19(0) + 6 = x
         0 + 6 = x
             6 = x

Therefore 6 lemons and 3 limes is the only possibility.

Edwin</pre>