SOLUTION: The difference of the reciprocals of two numbers is 11⁄6. If large number is divided by smaller, the result is 1⅞ . Find the numbers.

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Question 203183: The difference of the reciprocals of two numbers is 11⁄6. If large number is divided by smaller, the result is 1⅞ . Find the numbers.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Let's call our two numbers x and y, with x being the larger number. Their reciprocals would be 1/x and 1/y, respectively.
"The difference of the reciprocals of two numbers is 11/6": Since the difference is postive, then the larger number must come first. Since x is larger than y, its reciprocal is smaller that the reciprocal of y! So the equation is: 1%2Fy+-+1%2Fx+=+11%2F6
(Reworded) "the larger number divided by the smaller is 1 and 7/8" translates to:
x%2Fy+=+1%267%2F8

We now have a system of two equations in two variables. We should be able to solve this. As usual there are several ways. Here is one:
Rewrite the second equation with an improper fraction:
x%2Fy+=+15%2F8
This is a proportion (one fraction equals another). Cross multiply:
8x+=+15y
Divide by 8"
x+=+%2815y%29%2F8
So the reciprocal of x is:
1%2Fx+=+8%2F%2815y%29
Substituting this into the first equation:
1%2Fy+-+8%2F%2815y%29+=+11%2F6
Multiply both sides by the Lowest Common Denominator (LCD) of all three fractions:
%2830y%29%281%2Fy+-+8%2F%2815y%29%29+=+%2830y%29%2811%2F6%29
Simplifying:
30+-+16+=+55y
14+=+55y
Divide by 55:
14%2F55+=+y
Substituting this for y into x+=+%2815y%29%2F8:
x+=+%2815%2814%2F55%29%29%2F8+=+%2815%2A14%29%2F%2855%2A8%29+=+21%2F44
So the two numbers are: 21/44 and 14/55.