Question

which model shows \\(\frac{5}{8} \times \frac{2}{3}\\)? multiply. \\(\frac{5}{8} \times \frac{2}{3} = \frac{\square}{\square}\\)

Explanation:

Step1: Recall fraction multiplication rule

To multiply fractions, we use the rule: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.

Step2: Apply the rule to given fractions

For $\frac{5}{8} \times \frac{2}{3}$, multiply the numerators: $5 \times 2 = 10$, and multiply the denominators: $8 \times 3 = 24$.

Step3: Simplify the result

Simplify $\frac{10}{24}$ by dividing both numerator and denominator by their greatest common divisor, which is 2. So, $\frac{10 \div 2}{24 \div 2} = \frac{5}{12}$. Wait, wait, no—wait, earlier calculation: $5\times2 = 10$, $8\times3 = 24$, then $\frac{10}{24}$ simplifies to $\frac{5}{12}$? Wait, no, wait the initial wrong step. Wait, no, the problem's first part is about the model, but the second part is multiplying $\frac{5}{8} \times \frac{2}{3}$. Let's re - do:

Wait, no, let's focus on the multiplication. $\frac{5}{8} \times \frac{2}{3}$: numerator $5\times2 = 10$, denominator $8\times3 = 24$. Then simplify $\frac{10}{24}$ by dividing numerator and denominator by 2: $\frac{10\div2}{24\div2}=\frac{5}{12}$. Wait, but the box has 1 in the numerator, maybe a mistake in the user's input, but let's do the correct calculation.

Wait, maybe the model part: To represent $\frac{5}{8} \times \frac{2}{3}$, first, a rectangle is divided into 8 rows (to represent the denominator 8 of the first fraction) and 3 columns (to represent the denominator 3 of the second fraction), so total $8\times3 = 24$ small rectangles. Then, $\frac{5}{8}$ means shading 5 out of 8 rows, and $\frac{2}{3}$ means shading 2 out of 3 columns. The overlapping shaded area (the product) should be $5\times2 = 10$ small rectangles out of 24, which simplifies to $\frac{5}{12}$. But for the multiplication:

Step1: Multiply numerators and denominators

$\frac{5}{8} \times \frac{2}{3}=\frac{5\times2}{8\times3}=\frac{10}{24}$

Step2: Simplify the fraction

Divide numerator and denominator by their GCD (2): $\frac{10\div2}{24\div2}=\frac{5}{12}$

Answer:

$\frac{5}{12}$ (For the multiplication part. For the model, we need to check the two models: The first model: Let's see, if we have a grid with 8 rows and 3 columns (total 24 cells). $\frac{5}{8}$ would be 5 rows shaded, and $\frac{2}{3}$ would be 2 columns shaded. The overlapping area (the product) should be 52 = 10 cells. Let's check the two models. The left model: Let's count the shaded rows (vertical or horizontal?). Wait, maybe the left model has 8 rows (vertical) and 3 columns (horizontal). The shaded part: 5 rows (vertical) and 2 columns (horizontal)? Wait, the left model: the blue - shaded (darker) part: let's see, the first model (left) has a grid with, say, 8 horizontal rows (since the first fraction is $\frac{5}{8}$, so 5 out of 8 rows) and 3 vertical columns (since the second fraction is $\frac{2}{3}$, 2 out of 3 columns). So the darker shaded area should be 5 rows and 2 columns, so 52 = 10 cells. The right model: let's see, the number of rows and columns. If the left model has 8 rows and 3 columns, and the right model has a different division. But for the multiplication, the correct result of $\frac{5}{8} \times \frac{2}{3}$ is $\frac{5}{12}$ (after simplifying $\frac{10}{24}$).