Question 1187207
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x and y are the two numbers
1/x and 1/y are their respective reciprocals
Let x > y
The reciprocal operation will flip the inequality sign
1/x < 1/y


Example: 5 > 2 leads to 1/5 < 1/2 aka 0.2 < 0.5


The larger reciprocal (1/y) exceeds the smaller one (1/x) by 7/3
So,
1/y = 1/x + 7/3
1/y = 3/(3x) + (7x)/(3x)
1/y = (3+7x)/(3x)


The sum of the reciprocals is 7
1/x + 1/y = 7
1/x + (3+7x)/(3x) = 7
3/(3x) + (3+7x)/(3x) = 7
(3+3+7x)/(3x) = 7
(6+7x)/(3x) = 7
6+7x = 3x*7
6+7x = 21x
6 = 21x-7x
6 = 14x
14x = 6
x = 6/14
x = (2*3)/(2*7)
x = <font color=red>3/7 is one of the numbers</font>
1/x = 7/3


We'll use that value to find the following
1/y = (1/x) + 7/3
1/y = 7/3 + 7/3
1/y = 14/3
y = <font color=red>3/14 is the other number</font>


Note that,
1/x + 1/y = 7/3 + 14/3 = (7+14)/3 = 21/3 = 7
which helps confirm we have the correct x & y values.


Answers: <font color=red>3/7 and 3/14</font>
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