One number is 7 times another. The sum of their reciprocals is 48/7. Find the numbers.

Question

 One number is 7 times another. The sum of their reciprocals is 48/7. Find the numbers. 

Answer 1

Hi Carolina!

 

There are a number of ways to solve this problem, but I think the most straight forward is to first write two equations.  Let's let x be the first number and y be the second number.  A lot of times, I find that students have trouble translating English into Math, so I tell them translate one word at a time just like they would translate a sentence into French.  The first sentence says "One number" ... that means x ... "is" ... that means equals ... "7 times" ... that means 7 multiplied by ... "another" ... that means y. So, the first sentence produces the equation: x=7y . The second sentence produces the equation 1/x + 1/y = 48/7.  Now, you have two equations with two unknowns.  You can solve this!!  Although there are lots of paths up the mountain, so to speak, I think the easiest is to use the first equation to substitute into the second equation.  (Since we know from the first equation that x=7y, we can plug in 7y for x in the second equation.)  That gives us 1/(7y) + 1/y = 48/7.  We need to get a common denominator to add the two fractions, so multiply 1/y by 7/7.  Now we have 1/(7y) + 7/(7y) = 48/7.  Add the two fractions to get 8/(7y) = 48/7.  Now you can cross-multiply and solve to get y.  You should get y=1/6.  You can substitute back in to either equation to get x, so I'd recommend the first one ... it's simpler!  You should get x=7/6.  Now substitute both x and y values into the second equation to check that it works.  :-)

Answer 2

Hi Carolina,

 

So, you have two numbers. One is 7 times the other. Let one number be n, then the other number is 7n.

 

The sum of their reciprocals is 48/7. So,

 

1/n + 1/(7n) = 48/7

 

A common denominator is 7n. So,

 

7/(7n) + 1/(7n) = 48/7

 

Or

 

8/(7n) = 48/7

 

Cancel out the 7's on each side.

 

8/n = 48

 

n/8 = 1/48

 

n = 8/48 or n = 1/6

 

So, the two numbers are 1/6 and 7/6