SOLUTION: The difference of 2 numbers is 4. The difference between their reciprocals is 4/21. Find the 2 numbers

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Question 131457: The difference of 2 numbers is 4. The difference between their reciprocals is 4/21. Find the 2 numbers
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Call the larger number a
Call the smaller number b
a+-+b+=+4
a+=+b+%2B+4
%281%2Fb%29+-+%281%2Fa%29+=+4%2F21
Notice that the reciprocal of a smaller
number will be larger than the reciprocal
of a larger number
Substituting:
%281%2Fb%29+-+%281%2F%28b%2B4%29%29+=+4%2F21
multiply both sides by b%2A%28b%2B4%29%2A21
21%2A%28b%2B4%29+-+21%2Ab+=+4%2Ab%2A%28b%2B4%29
21b+%2B+84+-+21b+=+4%2Ab%5E2+%2B+16b
4%2Ab%5E2+%2B+16b+-+84+=+0
b%5E2+%2B+4b+-+21+=+0
solve by completing the square
b%5E2+%2B+4b+%2B+%284%2F2%29%5E2+=+21+%2B+%284%2F2%29%5E2
b%5E2+%2B+4b+%2B+4+=+25
%28b+%2B+2%29%5E2+=+25
b+%2B+2+=+5 (only the + square root makes sense here)
b+=+3
a+-+b+=+4
a+-+3+=+4
a+=+7
The numbers are 7 and 3
check:
%281%2Fb%29+-+%281%2Fa%29+=+4%2F21
%281%2F3%29+-+%281%2F7%29+=+4%2F21
%287%2F21%29+-+%283%2F21%29+=+4%2F21
7+-+3+=+4
4+=+4
OK