You can put this solution on YOUR website! .
The sum of the reciprocals of two numbers is 7.
The larger reciprocal exceeds the smaller one by 7/3.
Find the numbers
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Since the entire problem is formulated as the relations between reciprocals,
it is natural to solve it for reciprocal.
Let x and y be THE RECIPROCALS.
Then we have these equations
x + y = 7
x - y = 7/3.
Add equations. You will get 2x = = = .
Hence, x = .
Subtract equations. You will get 2y = = = .
Hence, y = .
Thus the numbers are = and = . ANSWER
Solved.
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The lesson to learn from my post is HOW TO select your starting unknowns to make the solution AS SIMPLE as possible.
You can put this solution on YOUR website!
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 = 3/7 is one of the numbers
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 = 3/14 is the other number
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.
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