Which of the following expressions
is rational given 𝑎 equals one and 𝑏 equals 34? Option (A) negative 39 over 𝑎
minus one, option (B) 39𝑏 over 𝑏 minus 34, option (C) 39𝑏 over 𝑎 minus one, or
option (D) 𝑏 over 𝑎.
In order to answer this question,
let’s start by remembering the definition of a rational number. A rational number is a number that
can be expressed as 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to
zero. In the four options here, we can
see that we’ve got four fractions that contain numerical values and also the
algebraic terms 𝑎 and 𝑏. However, we’re given numerical
values for 𝑎 and 𝑏. So, we’ll take each expression in
turn and plug in these values.
Let’s start with the first
expression in option (A). We’ll plug in the value that 𝑎 is
equal to one. We’ll still have 39 on the
numerator, and we’ll have one subtract one on the denominator. This, of course, simplifies to
negative 39 over zero. You might think that this looks
pretty good as a fraction. We’ve got a number on the numerator
and a number on the denominator. But in fact, if you’ve ever tried
to divide a number by zero on your calculator, you’ll get an undefined answer. And importantly, if we look at our
definition of a rational number, the 𝑞, the value on the denominator, cannot be
equal to zero. So, this value of negative 39 over
zero is not a rational number. And so, we can exclude option
In the expression in option (B),
we’ll need to substitute in the value 𝑏 equals 34 twice as 𝑏 occurs twice. We’ll, therefore, have the
calculation 39 times 34 over 34 minus 34. Before we rush to calculate 39
multiplied by 34, you might already notice what’s going to happen on this
denominator. Once again, we’re going to have a
denominator that has a value of zero. So, we know that this expression
when 𝑏 is equal to 34 would not be rational.
We can use the same method of
plugging in the 𝑎- and 𝑏-values into option (C). So, it have 39 times 34 over one
minus one. You may have already noticed that
this denominator will also give a value of zero. So, option (C) is not a rational
number when 𝑎 is one and 𝑏 is 34.
The expression in option (D) is 𝑏
over 𝑎 which will be is 34 over one. Let’s check if this fits the
definition of a rational number. We have it as a fraction 𝑝 over 𝑞
where 𝑝 and 𝑞 are integers. And 34 and one are integers. And of course, the denominator one
is not equal to zero. Therefore, 34 over one is
rational. And so, our answer is (D). 𝑏 over 𝑎 is rational when 𝑎
equals one and 𝑏 equals 34.
Before we finish with this
question, let’s just take a quick look at this expression in option (A). We saw that this expression
negative 39 over 𝑎 minus one is not rational, but that’s not always the case. If we had any other value other
than 𝑎 equals one, for example, 𝑎 equals two, then we’ll work out negative 39 over
two minus one, which would give us a rational value of negative 39 over one. The expressions (A), (B), and (C)
are only irrational because they had a denominator of zero, as they did in fact have
integer values on the numerator and denominator.