### Video Transcript

Fill in the blank. Five-sevenths times eight-thirds
plus five-sevenths times two-ninths is equal to what multiplied by two-ninths plus
eight-thirds.

In this question, we are given an
equation involving the product and sum of multiple rational numbers. And we need to determine the
missing value or expression that satisfies the equation. There are many ways of approaching
this problem. For instance, we could evaluate the
left-hand side of the equation and then rearrange to isolate the part we want to
find on the right-hand side of the equation. This would work, and we would
obtain the correct answer. However, it is easier to solve this
problem using the properties of the addition and multiplication of rational
numbers.

We can start by noting that
eight-thirds and two-ninths appear on both sides of the equation. We then see that on the right-hand
side of the equation, we have a product of a sum. And on the left-hand side of the
equation, we have a sum of products. This should remind us of the
distributive property of the multiplication of rational numbers over addition. We can then recall that the
distributive property of the multiplication of rational numbers over the addition of
rational numbers tells us that if 𝑎, 𝑏, and 𝑐 are rational numbers, then 𝑎 times
𝑏 plus 𝑎 times 𝑐 is equal to 𝑎 multiplied by 𝑏 plus 𝑐.

Let’s apply this result to the
left-hand side of the equation. We will set 𝑎 equal to
five-sevenths, 𝑏 equal to eight-thirds, and 𝑐 equal to two-ninths. This gives us that five-sevenths
times eight-thirds plus five-sevenths times two-ninths is equal to five-sevenths
multiplied by eight-thirds plus two-ninths. We can then note that our
expression is very similar to the right-hand side of the given equation, the only
difference being that the order of the sum inside the parentheses is reversed. We know that the addition of
rational numbers is commutative. So we can reorder the sum to obtain
five-sevenths multiplied by two-ninths plus eight-thirds.

This is enough information to
conclude that five-sevenths is a solution to the equation. However, we can show that this is
the only solution by noting that two-ninths plus eight-thirds is a nonzero rational
number since they are both positive. This means we can divide both sides
of the equation by this sum. This then gives us that the only
solution to the equation is five-sevenths. So our answer is five-sevenths.