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Question Video: Finding the Missing Rational Number in an Equation Mathematics • 7th Grade

Fill in the blank: (5/7 Γ— 8/3) + (5/7 Γ— 2/9) = οΌΏ Γ— (2/9 + 8/3).

03:08

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 π‘Ž, 𝑏, and 𝑐 are rational numbers, then π‘Ž times 𝑏 plus π‘Ž times 𝑐 is equal to π‘Ž multiplied by 𝑏 plus 𝑐.

Let’s apply this result to the left-hand side of the equation. We will set π‘Ž 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.

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