SOLUTION: 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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Question 1187207: The sum of the reciprocals of two numbers is 7. The larger reciprocal exceeds the smaller one by 7/3. Find the numbers
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52786)   (Show Source): 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.



Answer by math_tutor2020(3817)   (Show Source): 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.

Answers: 3/7 and 3/14
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