The product of two rational numbers
is negative 16 over nine. If one of the numbers is negative
four over three, find the other number.
So the first thing we’re gonna do
is look at a couple of key terms. So we’ve got product, which means
multiply. So if we find the product of two
numbers, that means we’re multiplying them together. And then we’re also looking at the
term, rational. And what this means is a number
that can be written as a fraction with an integer as the numerator and an integer as
the denominator, which is gonna help us when we’re gonna try and find the number
that we’re looking for. So taking the information we’ve got
from the question, what we can do is write it down. And we’ve got negative four over
three multiplied by 𝑎 over 𝑏 equals negative 16 over nine. And it’s this 𝑎 over 𝑏 that we’re
trying to find.
Well, there are, in fact, a couple
of ways we could solve this. So we’re gonna have a look at both
of those. So first of all, what we could do
is divide both sides by negative four over three. So when we do that, we’ll have 𝑎
over 𝑏 equals negative 16 over nine divided by negative four over three. So then what we can do is divide
our fractions. And to do that, we can use our
memory aid, KCF — keep it, change it, flip it — which is gonna give us 𝑎 over 𝑏 is
equal to negative 16 over nine multiplied by negative three over four. So now, before we multiply, what we
can do is divide through by any common factors. Well, first of all, we can divide
numerators and denominators by four and then by three.
So now what we’ve got is negative
four over three multiplied by negative one over one. Well, a negative multiplied by a
negative is a positive. So therefore, what we’re gonna get
is 𝑎 over 𝑏 is equal to four over three. So therefore, we’ve found our
missing number. And what we can do is check this by
using the alternate method. And the alternate method is
equating the numerators and denominators. Well, as we know, we’ve got
negative four over three in the left-hand side and the result is negative 16 over
nine. We know that a negative has to be
multiplied by a positive to give us a negative result. So therefore, we know that 𝑎 over
𝑏 will be positive. So we could ignore the signs when
we’re gonna equate the numerators and denominators.
Well, if you equate the numerators,
we’ve got four 𝑎 cause four multiplied by 𝑎 is equal to 16. So therefore, 𝑎 will be equal to
four. So then if we equate the
denominators, we’re gonna get three 𝑏 is equal to nine. So 𝑏 is equal to three. So therefore, 𝑎 over 𝑏 is gonna
be equal to four over three, which is what we’ve got with the first method.