Question

the rectangle on the right is a scaled copy of the rectangle on the left. identify the scale factor. express your answer as a whole number or fraction in simplest form.

Explanation:

Step1: Recall scale factor formula

The scale factor is the ratio of a corresponding side length in the scaled copy to the original side length. Let's take the vertical sides (heights) for calculation. Original height: \(12\), Scaled height: \(60\).

Step2: Calculate scale factor

Scale factor \(=\frac{\text{Scaled length}}{\text{Original length}}=\frac{60}{12}\)
Simplify \(\frac{60}{12}\) by dividing numerator and denominator by their greatest common divisor, which is \(12\). \(\frac{60\div12}{12\div12} = 5\)? Wait, no, wait. Wait, maybe I took the wrong sides? Wait, original rectangle: length \(30\), height \(12\). Scaled rectangle: length \(24\), height \(60\)? Wait, no, maybe the horizontal and vertical. Wait, maybe the corresponding sides: original height \(12\), scaled height \(60\); or original length \(30\), scaled length \(24\)? Wait, no, scaled copy: so the right is scaled from left. So left is original, right is scaled. So we need to check which sides correspond. Let's see: original height is \(12\), scaled height is \(60\). So \(60\div12 = 5\)? But original length is \(30\), scaled length is \(24\). \(24\div30=\frac{4}{5}\). Wait, that's a problem. Wait, maybe I mixed up which is original. Wait, the problem says "the rectangle on the right is a scaled copy of the rectangle on the left". So left is original, right is scaled. So we need to find the ratio of right's side to left's corresponding side. Wait, maybe the vertical sides: left height \(12\), right height \(60\). So \(60/12 = 5\). But horizontal: left length \(30\), right length \(24\). \(24/30 = 4/5\). That can't be. Wait, maybe I got the sides wrong. Wait, maybe the left rectangle has length \(30\) and height \(12\), and the right rectangle has length \(24\) and height \(60\). Wait, that would mean it's a scaled copy, but maybe the sides are swapped? Wait, no, rectangles have length and width, so maybe the corresponding sides are height to height and length to length, but maybe I made a mistake. Wait, no, let's check again. Wait, maybe the left rectangle: height \(12\), length \(30\). Right rectangle: height \(60\), length \(24\). Wait, that would be a scale factor of \(60/12 = 5\) for height, but \(24/30 = 4/5\) for length. That's inconsistent. Wait, maybe I flipped the scale. Wait, maybe the right is the original and left is the scaled? No, the problem says right is scaled copy of left. Wait, maybe the sides are not length and height but width and height. Wait, maybe the left rectangle has width \(12\) and length \(30\), and the right rectangle has width \(24\) and height \(60\). Wait, no, that doesn't make sense. Wait, maybe I misread the numbers. Let me check again. Left rectangle: vertical side \(12\), horizontal side \(30\). Right rectangle: vertical side \(60\), horizontal side \(24\). Wait, so if we take vertical sides: \(60\) (scaled) / \(12\) (original) = \(5\). Horizontal sides: \(24\) (scaled) / \(30\) (original) = \(4/5\). That's a contradiction. Wait, maybe the right rectangle's horizontal side is \(24\) and vertical is \(60\), and left's horizontal is \(30\) and vertical is \(12\). Wait, maybe the corresponding sides are horizontal to vertical? No, that's not how scaling works. Scaling should be uniform, so all sides are scaled by the same factor. So maybe I made a mistake in identifying corresponding sides. Wait, maybe the left rectangle has length \(30\) and height \(12\), and the right rectangle has length \(24\) and height \(60\). Wait, no, that would mean that length is scaled by \(24/30 = 4/5\) and height by \(60/12 = 5\), which is impossible. So maybe the numbers are different. Wait, maybe the left rectangle's height is \(12\) and length is \(30\), and the right rectangle's height is \(60\) and length is \(150\)? No, the problem says right has length \(24\). Wait, maybe I misread the right rectangle's length. Wait, the problem shows right rectangle with length \(24\) and height \(60\), left with length \(30\) and height \(12\). Wait, maybe the scale factor is \(24/30 = 4/5\) for length, but \(60/12 = 5\) for height. That can't be. So there must be a mistake in my understanding. Wait, no, maybe the right rectangle is a scaled copy, so maybe the left is the original, and the right is scaled, but maybe the sides are swapped. Like, left rectangle: width \(12\), length \(30\); right rectangle: width \(60\), length \(24\). No, that's not a rectangle. Wait, no, rectangles have opposite sides equal. So left: \(12\) (height) and \(30\) (length). Right: \(60\) (height) and \(24\) (length). So if we scale the left to get the right, the height is multiplied by \(5\) (12*5=60) and length is multiplied by \(24/30 = 4/5\). That's impossible. So maybe the problem has a typo, or I misread the numbers. Wait, wait, maybe the right rectangle's length is \(150\) instead of \(24\)? No, the user provided the image with right rectangle length \(24\) and height \(60\), left with length \(30\) and height \(12\). Wait, maybe the scale factor is \(60/12 = 5\), but then the length should be \(30*5=150\), but it's \(24\). So that's not possible. Wait, maybe the scale factor is \(24/30 = 4/5\), but then the height should be \(12*(4/5)=48/5=9.6\), but it's \(60\). So that's a problem. Wait, maybe I got the original and scaled reversed. If right is original and left is scaled, then scale factor would be \(12/60 = 1/5\) for height, and \(30/24 = 5/4\) for length. Also inconsistent. Wait, this must be a mistake. Wait, maybe the numbers are different. Wait, maybe the left rectangle has height \(12\) and length \(30\), and the right rectangle has height \(24\) and length \(60\). Then scale factor would be \(24/12 = 2\) and \(60/30 = 2\), which is consistent. But the user's image shows right height \(60\) and length \(24\), left height \(12\) and length \(30\). Wait, maybe the user made a mistake in the image. Alternatively, maybe I misread the length of the right rectangle. Wait, maybe the right rectangle's length is \(150\) and height \(60\), left length \(30\) and height \(12\). Then scale factor is \(60/12 = 5\) and \(150/30 = 5\), which is consistent. But the user's image shows right length \(24\). Wait, maybe the problem is correct, and I need to check again. Wait, the problem says "the rectangle on the right is a scaled copy of the rectangle on the left". So scaled copy means that all corresponding linear measurements are multiplied by the same scale factor. So let's denote scale factor as \(k\). Then, if left has height \(h_1 = 12\) and length \(l_1 = 30\), right has height \(h_2 = 60\) and length \(l_2 = 24\). Then \(h_2 = k \times h_1\) and \(l_2 = k \times l_1\). So \(60 = k \times 12\) and \(24 = k \times 30\). Solving first equation: \(k = 60/12 = 5\). Solving second equation: \(k = 24/30 = 4/5\). Contradiction. So this is impossible. Therefore, maybe the corresponding sides are not height to height and length to length, but height to length and length to height? Like, left height \(12\) corresponds to right length \(24\), and left length \(30\) corresponds to right height \(60\). Then scale factor would be \(24/12 = 2\) and \(60/30 = 2\). Ah! That makes sense. So maybe the rectangles are rotated? So the left rectangle has height \(12\) and length \(30\), and the right rectangle is rotated, so its length is \(24\) (corresponding to left's height \(12\)) and height is \(60\) (corresponding to left's length \(30\)). So then scale factor is \(24/12 = 2\) (since \(12 \times 2 = 24\)) and \(60/30 = 2\) (since \(30 \times 2 = 60\)). That works. So the scale factor is \(2\)? Wait, no, \(24/12 = 2\), \(60/30 = 2\). Yes, that's correct. So I must have misidentified the corresponding sides. So the left rectangle's height (12) corresponds to the right rectangle's length (24), and left's length (30) corresponds to right's height (60). So scale factor is \(24/12 = 2\) or \(60/30 = 2\). Wait, but \(24/12 = 2\), \(60/30 = 2\). So that's the scale factor. So my initial mistake was assuming the sides were aligned vertically, but maybe the right rectangle is rotated, so the horizontal and vertical sides are swapped. So that's the key. So let's redo the step.

Step1: Identify corresponding sides

Left rectangle: height \(12\), length \(30\). Right rectangle: length \(24\) (corresponds to left's height \(12\)), height \(60\) (corresponds to left's length \(30\)).

Step2: Calculate scale factor

Scale factor \(k = \frac{\text{Right length}}{\text{Left height}} = \frac{24}{12} = 2\) or \(k = \frac{\text{Right height}}{\text{Left length}} = \frac{60}{30} = 2\). Both give \(2\). Wait, but earlier I thought \(60/12 = 5\), but that was wrong because I misidentified the corresponding side. The correct corresponding sides are left height (12) to right length (24), and left length (30) to right height (60). So scale factor is \(2\). Wait, but let's check again. If left is height 12, length 30, and right is length 24 (which is 12*2) and height 60 (which is 30*2), then yes, scale factor is 2. So that makes sense. So the scale factor is 2. Wait, but the user's image: left rectangle has 12 (vertical) and 30 (horizontal), right rectangle has 60 (vertical) and 24 (horizontal). So if we rotate the right rectangle 90 degrees, then its horizontal side is 60 and vertical is 24? No, that's not. Wait, maybe the right rectangle's horizontal side is 24 and vertical is 60, so when scaled from left (horizontal 30, vertical 12), the scale factor for horizontal is 24/30 = 4/5, and vertical is 60/12 = 5. Which is impossible. So there must be a mistake in the problem or my understanding. Wait, maybe the numbers are different. Wait, maybe the left rectangle has height 12 and length 30, and the right rectangle has height 24 and length 60. Then scale factor is 2. But the user's image shows right height 60 and length 24. So maybe the user made a typo, or I misread. Alternatively, maybe the scale factor is 5, but then the length should be 30*5=150, but it's 24. So that's not. Wait, maybe the scale factor is 4/5, but then height should be 12*(4/5)=48/5=9.6, but it's 60. So this is a problem. Wait, maybe the problem is correct, and I need to check again. Wait, the problem says "the rectangle on the right is a scaled copy of the rectangle on the left". So scaled copy means that the ratio of corresponding sides is the same. So let's set up the equation: let the scale factor be \(k\). Then, if left has length \(l_1 = 30\) and height \(h_1 = 12\), right has length \(l_2 = 24\) and height \(h_2 = 60\). Then \(l_2 = k \times l_1\) and \(h_2 = k \times h_1\). So \(24 = 30k\) and \(60 = 12k\). Solving \(24 = 30k\) gives \(k = 24/30 = 4/5\). Solving \(60 = 12k\) gives \(k = 60/12 = 5\). Contradiction. Therefore, the only way this works is if the corresponding sides are swapped. So left length \(30\) corresponds to right height \(60\), and left height \(12\) corresponds to right length \(24\). Then \(60 = 30k\) => \(k=2\), and \(24 = 12k\) => \(k=2\). So that's consistent. So the scale factor is 2. So the mistake was in assuming the orientation of the rectangles. So the scale factor is 2.

Answer:

\(2\)