Question

A rectangle is reduced by a scale factor of One-fourth.

A large rectangle has a length of 16 and width of 12. A smaller rectangle has length of 4 and width of 3.

Which choices show the ratio of the area of the smaller rectangle to the area of the larger rectangle? Select three options.
StartFraction 4 over 16 EndFraction
(StartFraction 4 over 16 EndFraction) squared
StartFraction 12 over 192 EndFraction
StartFraction 4 squared over 12 squared EndFraction
(StartFraction 3 over 12 EndFraction squared
helpppp

Answer 1

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.

1. **Area of the larger rectangle**:
- Length = 16, Width = 12
- Area = Length × Width = \( 16 \times 12 = 192 \)

2. **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:

1. **Choice A**: \( \frac{4}{16} \)
- This simplifies to \( \frac{1}{4} \), not equal to \( \frac{1}{16} \).

2. **Choice B**: \( \left( \frac{4}{16} \right)^2 \)
- \( \left( \frac{1}{4} \right)^2 = \frac{1}{16} \) (This choice is valid).

3. **Choice C**: \( \frac{12}{192} \)
- This simplifies to \( \frac{1}{16} \) (This choice is valid).

4. **Choice D**: \( \frac{4^2}{12^2} \)
- \( \frac{16}{144} = \frac{1}{9} \) (This choice is not valid).

5. **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