Question

problem 5
write a situation that could be represented by the expression $3 \div \frac{2}{3}$. explain what the quotient means in your situation.

Explanation:

1. Situation Creation: We need to create a real - life scenario where we are dividing a quantity (3 of something) by a fraction ($\frac{2}{3}$ of something). For example, in the context of food (like pizza or cake) or length (like rope). Let's take the example of rope.

• Suppose we have a rope that is 3 meters long. We want to cut it into pieces, and each piece is $\frac{2}{3}$ of a meter long. The expression $3\div\frac{2}{3}$ represents the number of such $\frac{2}{3}$ - meter - long pieces we can get from the 3 - meter - long rope.

1. Calculating the Quotient and Its Meaning:

• First, recall the rule for dividing by a fraction: dividing by a fraction is the same as multiplying by its reciprocal. So, $3\div\frac{2}{3}=3\times\frac{3}{2}=\frac{9}{2} = 4.5$.
• In our rope - cutting situation, the quotient 4.5 means that we can get 4 full pieces of length $\frac{2}{3}$ meters and a half - piece (since 0.5 of a $\frac{2}{3}$ - meter - long piece is $\frac{1}{2}\times\frac{2}{3}=\frac{1}{3}$ meters) from the 3 - meter - long rope. Or, in other words, if we consider the number of pieces, we can get 4.5 pieces of length $\frac{2}{3}$ meters from a 3 - meter - long rope.

Another example: Suppose there is a cake that weighs 3 pounds. We want to serve slices, and each slice is $\frac{2}{3}$ of a pound. The expression $3\div\frac{2}{3}$ represents the number of $\frac{2}{3}$ - pound slices we can get from the 3 - pound cake. Calculating the quotient: $3\div\frac{2}{3}=3\times\frac{3}{2}=\frac{9}{2}=4.5$. The quotient 4.5 means we can serve 4 full slices (each of $\frac{2}{3}$ pounds) and a half - slice (which would be $\frac{1}{2}\times\frac{2}{3}=\frac{1}{3}$ pounds) from the 3 - pound cake.

Answer:

• Situation: There is a 3 - meter - long rope. We want to cut it into pieces where each piece has a length of $\frac{2}{3}$ meters. The expression $3\div\frac{2}{3}$ represents the number of such pieces we can obtain.
• Quotient Meaning: First, calculate $3\div\frac{2}{3}=3\times\frac{3}{2}=\frac{9}{2} = 4.5$. In this situation, the quotient 4.5 means that we can get 4 full pieces of length $\frac{2}{3}$ meters and a half - piece (where the half - piece has a length of $\frac{1}{3}$ meters) from the 3 - meter - long rope. (Other valid situations and explanations are also acceptable as long as they follow the division of a whole number by a fraction concept.)