One positive number is 3 more than another positive number.

Question

If 24 times the reciprocal of the smaller number is added to 13 times the reciprocal of the larger number, the sum is 1. Find the two numbers.

Answer 1

Greetings! Lets solve this shall we ?

Then, let the equation be equivalent to

(24/x) + (13/y) = 1 .................Lets solve for x shall we? So let the one positive number be 3 more than the other such that y = x + 3. Then we substitute it into the equation we translated as

(24/x) + (13/(x+3)) = 1 ................The common denominator is x(x+3) such that it simplifies into

24(x+3) + 13x = x(x+3)................Now we solve for x !

24x + 72 + 13x = x² + 3x...........Adding like terms

37x + 72 = x² + 3x .................Adding more like terms

72 = x² - 34x..............Subtract 72 to both sides

0 = x² - 34x - 72................This equation is factorable such that

0 = (x-36)(x+2).............Using the zero product property such that

x = 36 and x = -2...........We can use -2 because we must find positive numbers so 36 is one of the numbers.

Now we plug 36 back into the equation to find the other number such that

(24/36) + (13/y) = 1.................Now we must solve for y! The common denominator is 36y and simplifies to

24y + 13(36) = 36y.................Now we solve for y by subtracting 24y to both sides such that

468 = 12y...............Then divide 12 such that

y = 39

So the two numbers are 36 and 39

I hope this helped!