Question

look at this rectangle: if both dimensions are halved, then which of the following statements about its area will be true? the ratio of the new area to the old area will be 1:2. the ratio of the new area to the old area will be 9:1.

Explanation:

Step1: Calculate the old area

The area formula for a rectangle is \(A = l\times w\). Given \(l = 92\) ft and \(w=93\) ft, the old area \(A_{old}=92\times93\).

Step2: Calculate the new dimensions and area

If both dimensions are halved, the new length \(l_{new}=\frac{92}{2} = 46\) ft and new width \(w_{new}=\frac{93}{2}\) ft. Then the new area \(A_{new}=46\times\frac{93}{2}\).

Step3: Calculate the ratio of new - to - old area

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Wait, let's use the formula \(A = lw\). Let the original length be \(l\) and original width be \(w\), so \(A_{old}=lw\). After halving, \(l_{new}=\frac{l}{2}\), \(w_{new}=\frac{w}{2}\), and \(A_{new}=\frac{l}{2}\times\frac{w}{2}=\frac{lw}{4}\). So \(\frac{A_{new}}{A_{old}}=\frac{\frac{lw}{4}}{lw}=\frac{1}{4}\). But if we assume there was a mis - read (maybe the options were cut off in the problem statement). Let's re - check with the formula \(A = lw\).

Let \(A_{old}=l\times w\), \(A_{new}=\frac{l}{2}\times\frac{w}{2}\). The ratio \(\frac{A_{new}}{A_{old}}=\frac{1}{4}\). But if we consider the general case of area scaling. If length and width are scaled by a factor \(k\) (here \(k = \frac{1}{2}\)), the area is scaled by \(k^{2}\).

Answer:

None of the given options (The ratio of the new area to the old area should be \(1:4\)). But if we assume it's a formatting error and we use the formula \(A = lw\), and recalculate:

Let \(A_{old}=92\times93\), \(A_{new}=\frac{92}{2}\times\frac{93}{2}= 46\times46.5\)

\(\frac{A_{new}}{A_{old}}=\frac{46\times46.5}{92\times93}=\frac{1}{4}\)

If we assume the problem was about a square (but it's a rectangle in the image). If it was a square with side \(s\), \(A_{old}=s^{2}\), \(A_{new}=(\frac{s}{2})^{2}=\frac{s^{2}}{4}\), ratio \(1:4\)

If we check the given options:

• For the first option \(\frac{1}{2}\) (incorrect, as area scales with the product of length and width scaling factors)
• For the second option \(\frac{9}{1}\) (incorrect)

So there is an error in the provided options. But if we follow the formula \(A = lw\) and scaling: If length and width are halved (\(x=\frac{1}{2}\) scaling factor for each dimension), area scaling factor \(x\times x=\frac{1}{4}\)