To find the ratio of the area of the smaller rectangle to the area of the larger rectangle, we first need to calculate the areas of both rectangles.
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Area of the larger rectangle:
- Length = 16, Width = 12
- Area = Length × Width = \( 16 \times 12 = 192 \)
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Area of the smaller rectangle:
- Length = 4, Width = 3
- Area = Length × Width = \( 4 \times 3 = 12 \)
Now we can find the ratio of the area of the smaller rectangle to the area of the larger rectangle:
\[ \text{Ratio} = \frac{\text{Area of smaller rectangle}}{\text{Area of larger rectangle}} = \frac{12}{192} \]
To simplify this ratio, we divide both the numerator and the denominator by 12:
\[ \frac{12}{192} = \frac{1}{16} \]
Now we will check the choices given to see which ones represent this ratio:
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Choice A: \( \frac{4}{16} \)
- This simplifies to \( \frac{1}{4} \), not equal to \( \frac{1}{16} \).
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Choice B: \( \left( \frac{4}{16} \right)^2 \)
- \( \left( \frac{1}{4} \right)^2 = \frac{1}{16} \) (This choice is valid).
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Choice C: \( \frac{12}{192} \)
- This simplifies to \( \frac{1}{16} \) (This choice is valid).
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Choice D: \( \frac{4^2}{12^2} \)
- \( \frac{16}{144} = \frac{1}{9} \) (This choice is not valid).
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Choice E: \( \left( \frac{3}{12} \right)^2 \)
- \( \left( \frac{1}{4} \right)^2 = \frac{1}{16} \) (This choice is valid).
Thus, the three valid choices that show the ratio of the area of the smaller rectangle to the area of the larger rectangle \( \left( = \frac{1}{16} \right) \) are:
- (StartFraction 4 over 16 EndFraction) squared
- StartFraction 12 over 192 EndFraction
- (StartFraction 3 over 12 EndFraction) squared
1 answer