Question Video: Adding Rational Expressions and Identifying Whether the Outcome Is Rational Mathematics • 7th Grade

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.