SOLUTION: The sum of the reciprocals of two numbers is 7 and the difference of the reciprocals is 3. What are the numbers?

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Question 1111018: The sum of the reciprocals of two numbers is 7 and the difference of the reciprocals is 3. What are the numbers?
Found 3 solutions by mananth, Theo, greenestamps:
Answer by mananth(16946)   (Show Source):
You can put this solution on YOUR website!

add the two equations
we get
2/x = 10
x= 2/10
x=1/5
If x= 1/5 then
1/y =7-5
1/y = 2
y= 1/2

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
you have 2 equations that need to be solved simultaneously.

they are:

1/A + 1/B = 7

1/A - 1/B = 3

add the 2 equations together to get:

2/A = 10

solve for A to get A = 2/10 = 1/5

when A = 1/5, 1/A + 1/B = 7 becomes:

1/(1/5) + 1/B = 7

simplify to get 5 + 1/B = 7

subtract 5 from both sides to get 1/B = 2

solve for B to get B = 1/2

you have A = 1/5 and B = 1/2.

your first equation is 1/A + 1/B = 7

replace A with 1/5 and B with 1/2 to get:

1/(1/5) + 1/(1/2) = 7

simplify to get 5 + 2 = 7, which is true.

your second equaton is 1/A - 1/B = 3

replace A with 1/5 and B with 1/2 to get:

1/(1/5) - 1/(1/2) = 3

simplify to get 5 - 2 = 3, which is true.

both equations are true when A = 1/5 and B = 1/2, therefore the solution can be assumed to be good.

how do you get 5 out of 1/(1/5)?

it's a simple matter of multiplying both numerator and denominator by 5.

1/(1/5) * 5/5 = (5*1) / (5*1/5) = 5/1 = 5

you can do this without changing the fraction because multiplying by 5/5 is the same as multiplying by 1 which keeps the value of the original fraction the same.

to confirm, used your calculator to divide 1 by (1/5) and you will see that the answer is 5.

the same procedure to solve these simultaneous equations is used to solve simultaneous equations that do not include fractions.

since the first equation had + 1/B and the second equation had - 1/B, adding the equations together eliminated B from the equation and allowed you to solve for A.

once you solved for A, it was a matter of replacing A with it's value to solve for B in either of the original equations.

here's a reference on how to solve simultaneous equations if you think you might need it.

http://www.purplemath.com/modules/systlin1.htm

here's a reference on how to add fractions with different denominators if you think you need it.

http://www.purplemath.com/modules/fraction4.htm

Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!

You have received two good solutions, very similar to each other, from other tutors.

Here is a suggestion for a different approach you might find easier.

Let the two numbers we are looking be x and y. We know and .

Working with fractions is more difficult than working with whole numbers. So

let a = 1/x
let b = 1/y

then the two equations are
and

That pair of equations is easily solved to find a=5 and b=2.

Then, to finish the original problem,
x = 1/a = 1/5
y = 1/b = 1/2

The two numbers are 1/5 and 1/2.