The difference between two natural
numbers is five and the difference between their reciprocals is one tenth. Find the two numbers.
Let’s start by defining what the
natural numbers are. The natural numbers are simply the
positive integers. Now, we have two natural numbers,
which we can call 𝑎 and 𝑏.
Now, let’s choose our natural
numbers 𝑎 and 𝑏 such that 𝑎 is greater than 𝑏. And now, we’re told in the question
that the difference between our two natural numbers is five. And so what this means is that 𝑎
minus 𝑏 must be equal to five. We’re also told that the difference
between the reciprocals of 𝑎 and 𝑏 is one tenth.
Now, before we write down an
equation for this, let’s first think about what a reciprocal is. And now the reciprocal of a whole
number is simply one divided by that number. And so the reciprocal of, for
example, 𝑎 is simply one over 𝑎 and the reciprocal of 𝑏 is one over 𝑏. And now, since we’ve already stated
that 𝑎 is greater than 𝑏, this must mean that one over 𝑎 is less than one over 𝑏
since the fraction one over 𝑎 will have a larger denominator than the fraction one
over 𝑏 making it smaller than one over 𝑏.
So now, we have found that one over
𝑏 is greater than one over 𝑎, we can use the fact that the difference between the
reciprocals is one tenth to form an equation. And we obtain that one over 𝑏
minus one over 𝑎 is equal to one tenth. The reason why we are subtracting
one over 𝑎 from one over 𝑏 is simply because one over 𝑏 is larger and the
difference between the reciprocals is a positive number. So therefore, when we subtract one
from the other, it must be positive. So we must subtract the smaller
number from the larger number.
Now, we have formed two equations
in 𝑎 and 𝑏. And so we can solve these equations
simultaneously in order to find the values of 𝑎 and 𝑏. If we first consider 𝑎 minus 𝑏
equals five and we add 𝑏 to both sides of the equation, we obtain that 𝑎 is equal
to 𝑏 plus five. And we can call this equation
one. And let’s label our other equation
one over 𝑏 minus one over 𝑎 is equal to one tenth equation two.
And now, what we can do is
substitute equation one into equation two. And what this will give us is one
over 𝑏 minus one over 𝑏 plus five is equal to one tenth. And now we have formed an equation
involving only 𝑏, which we can solve in order to find 𝑏.
On the left-hand side of the
equation, we notice that we have two denominators: one which contains just 𝑏 and
one which contains 𝑏 plus five. So let’s multiply both sides of the
equation by 𝑏 and 𝑏 plus five. And this will give us 𝑏 plus five
minus 𝑏 is equal to one tenth timesed by 𝑏 timesed by 𝑏 plus five. And now, we noticed we can simplify
the left-hand side of the equation since we have a 𝑏 minus 𝑏. So the left-hand side of the
equation becomes five.
And next, we can get rid of the
tenth by multiplying both sides of the equation by 10. And this gives us that 50 is equal
to 𝑏 multiplied by 𝑏 plus five. Expanding the brackets gives us
that 50 is equal to 𝑏 squared plus five 𝑏. Next, we want to move all of the
terms onto one side of the equation. So we can subtract 50 from both
sides, which leaves us with zero is equal to 𝑏 squared plus five 𝑏 minus 50.
And the next step here is to
factorize this equation. In order to do this, we want to
find factor pairs of negative 50, which sum to give five. And we find that the factor pair of
negative five and 10 sum to give five as required. Therefore, the factorized form of
this equation becomes zero is equal to 𝑏 minus five multiplied by 𝑏 plus 10.
And from here, we obtain that 𝑏
minus five is equal to zero which gives us that 𝑏 is equal to five or else 𝑏 plus
10 is equal to zero, which gives us that 𝑏 is equal to negative 10.
Now, although we have two solutions
here, only one of them is correct. If we remember back to the start,
we have that 𝑎 and 𝑏 are both natural numbers. And a natural number is defined as
a positive integer. And since one of these solutions is
negative, we can ignore this solution. And so we obtained that 𝑏 is equal
Now that we have found 𝑏, it’s
very easy to find 𝑎. We simply substitute 𝑏 into
equation one. And this gives us that 𝑎 is equal
to five plus five or 𝑎 is equal to 10. Now that we found our values for 𝑎
and 𝑏, let’s quickly check that they match the conditions given in the
We are told that the difference
between the two natural numbers is five. And the difference between our two
numbers is 10 minus five, which is also equal to five. So this is correct. The second condition is that the
difference in their reciprocals is one tenth. So that’s one over 𝑏 minus one
over 𝑎 or one-fifth minus one tenth.
And now, we can simply multiply
one-fifth by two over two which is just one. And this gives us two tenths minus
one tenth which is equal to one tenth. And this is also correct. And so therefore, we can be sure
that our solution of 10 and five is correct.