Video Transcript
Consider two rational numbers 𝑎
over 𝑏 and 𝑐 over 𝑑. Find an expression equivalent to 𝑎
over 𝑏 plus 𝑐 over 𝑑 and state whether this equivalent expression belongs to the
set of rational numbers.
For the first part of the question,
we need to find an expression equivalent to 𝑎 over 𝑏 plus 𝑐 over 𝑑. To do this, we will need to
simplify the given expression. Typically, we would do this by
finding the lowest common multiple of the denominators, 𝑏 and 𝑑. However, we do not know the values
of 𝑏 or 𝑑 and whether they have any shared factors. So, instead of finding the lowest
common multiple of these constants, we can instead use their product, 𝑏𝑑, for the
denominator. We can multiply the first fraction
by 𝑑 over 𝑑 and the second fraction by 𝑏 over 𝑏. Since 𝑑 over 𝑑 and 𝑏 over 𝑏 are
both equal to one, multiplying by these fractions will not change the value of the
expression.
After completing the
multiplication, we can see that both fractions now have a common denominator of
𝑏𝑑. We are now ready to add the
fractions. Adding the fractions together, we
have found an equivalent expression. So our solution to the first part
of the question is that 𝑎 over 𝑏 plus 𝑐 over 𝑑 is equivalent to 𝑎𝑑 plus 𝑏𝑐
over 𝑏𝑑.
Next, we need to consider whether
this expression belongs to the rational numbers. Recall that the rational numbers
can be defined as any number 𝑥 over 𝑦, where 𝑥 and 𝑦 are integers and 𝑦 is not
equal to zero. Since 𝑎 over 𝑏 and 𝑐 over 𝑑 are
rational numbers, we have that 𝑎, 𝑏, 𝑐, and 𝑑 are integers. Also, 𝑏 and 𝑑 are both
nonzero.
Using these facts, we need to
determine whether 𝑎𝑑 plus 𝑏𝑐 over 𝑏𝑑 is a rational number. Recall the closure property of
addition and multiplication of the integers, which tell us that if 𝑥 and 𝑦 are
integers, then 𝑥 plus 𝑦 is an integer and 𝑥𝑦 is also an integer. Using these properties, we can say
that 𝑎𝑑, 𝑏𝑐, and 𝑏𝑑 are all integers, and 𝑎𝑑 plus 𝑏𝑐 is also an
integer. We have shown that both the
numerator and denominator are integers. Since both 𝑑 and 𝑏 are nonzero,
their product, 𝑑𝑏, will also be nonzero. Hence, we have shown that 𝑎𝑑 plus
𝑏𝑐 over 𝑏𝑑 is a rational number, which completes our solution to this
question.
