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The product of the reciprocal of 2 consecutive integers is 1/30. What are the numbers?

Answer 1

Isaiah Anthony

There are two possibilities:

#5# and #6#
#-6# and #-5#

#1/5*1/6 = 1/30#

#1/(-6)*1/(-5) = 1/30#

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Answer 2

Hazel Anglin

There are two possibilities: #-6,-5# and #5,6#

Call the two integers #a# and #b#.

The reciprocals of these two integers are #1/a# and #1/b#.

The product of the reciprocals is #1/axx1/b=1/(ab)#.

Thus, we know that #1/(ab)=1/30#.

Multiply both sides by #30ab# or cross-multiply to show that #ab=30#.

However, this doesn't really solve the problem: we have to address that fact that the integers are consecutive. If we call the first integer #n#, then the next consecutive integer is #n+1#. Thus, we can say that instead of #ab=30# we know that #n(n+1)=30#.

To solve #n(n+1)=30#, distribute the left-hand side and move the #30# to the left hand side as well to obtain #n^2+n-30=0#. Factor this into #(n+6)(n-5)=0#, which implies that #n=-6# and #n=5#.

If #n=-6# then the next consecutive integer is #n+1=-5#. We see here that the product of their reciprocals is #1/30#:

#1/(-6)xx1/(-5)=1/30#

If #n=5# then the next consecutive integer is #n+1=6#.

#1/5xx1/6=1/30#

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Answer 3

Kennedy Adams

The numbers are 5 and 6.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.