What multiplication problem is represented by the following model?
[Fraction model image]
$\frac{1}{4} \times \frac{1}{4}=\frac{1}{16}$
$\frac{1}{4} \times \frac{3}{4}=\frac{3}{8}$
$\frac{3}{4} \times \frac{2}{3}=\frac{6}{12}$
$\frac{3}{4} \times \frac{3}{4}=\frac{9}{16}$
Answer (2)
The multiplication problem represented by the model is 4 3 × 4 3 = 16 9 . This is derived from realizing that 3 out of 4 rows and 3 out of 4 columns were shaded, covering 9 out of 16 total sections in the grid. Hence, the result matches the fraction of the grid that is shaded.
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The model represents a 4x4 grid, with 9 cells shaded.
The fraction of the grid that is shaded is 16 9 .
The shaded region occupies 4 3 of the columns and 4 3 of the rows.
The multiplication problem represented by the model is 4 3 × 4 3 = 16 9 .
Explanation
Understanding the Grid The model is a 4x4 grid, which means there are 4 rows and 4 columns. The total number of cells in the grid is 4 × 4 = 16 .
Determining the Shaded Fraction By counting the shaded cells, we find that there are 9 shaded cells. Therefore, the fraction of the grid that is shaded is 16 9 .
Expressing Shaded Area as a Product We need to express the shaded area as a product of two fractions. The shaded region occupies 3 out of 4 columns and 3 out of 4 rows. Thus, the multiplication problem represented by the model is 4 3 × 4 3 .
Verification Let's verify that the product of the two fractions equals the shaded area: 4 3 × 4 3 = 4 × 4 3 × 3 = 16 9 . This matches the shaded area we calculated earlier.
Final Answer The multiplication problem represented by the model is 4 3 × 4 3 = 16 9 .
Examples
Fractions are used in baking to measure ingredients. For example, if a recipe calls for 4 3 cup of flour and you want to double the recipe, you need to multiply 4 3 by 2. Understanding fraction multiplication helps you adjust recipes accurately. Similarly, if you are tiling a floor and 4 3 of the tiles are blue and you use 4 3 of the blue tiles, you can calculate the fraction of the total tiles that are used by multiplying 4 3 × 4 3 .
