Question

jose and nancy both set up a proportion to find the missing width, w, of the smaller of two similar rectangles. their work is shown below.
jose’s work:
\\(\frac{3}{5} = \frac{w}{6}\\)
\\(5w = 9\\)
\\(w \approx 1.8\\)
nancy’s work:
\\(\frac{3}{w} = \frac{5}{6}\\)
\\(5w = 18\\)
\\(w \approx 3.6\\)
(image of two rectangles: first rectangle has height 3, length 5; second rectangle has height 6, width w)
which student made an error in their work and what error did they make?
\\(\circ\\) nancy did divide 18 by 5 correctly.
\\(\circ\\) jose did not multiply 3×6 correctly.
\\(\circ\\) jose did not set up the proportion correctly.
\\(\circ\\) nancy did not set up the proportion correctly

Explanation:

Step1: Analyze similar rectangles proportion

For similar rectangles, corresponding sides are proportional. The first rectangle has height 3 and width 5, the second has height 6 and width \( w \). So the correct proportion should relate height to width: \( \frac{3}{5}=\frac{w}{6} \) (Jose) or \( \frac{3}{w}=\frac{5}{6} \) (Nancy)? Wait, no—wait, the first rectangle: height 3, width 5. Second: height 6, width \( w \). So height of first over height of second should equal width of first over width of second? Wait, no, similar figures: ratio of corresponding sides. So first rectangle: height 3, width 5. Second: height 6, width \( w \). So \( \frac{\text{height of first}}{\text{height of second}}=\frac{\text{width of first}}{\text{width of second}} \) → \( \frac{3}{6}=\frac{5}{w} \)? Wait, no, maybe I mixed up. Wait, the first rectangle is horizontal: height 3, width 5. The second is vertical: width \( w \), height 6. So corresponding sides: first's height (3) corresponds to second's width (\( w \)), first's width (5) corresponds to second's height (6). So proportion should be \( \frac{3}{w}=\frac{5}{6} \) (Nancy) or \( \frac{3}{5}=\frac{w}{6} \) (Jose)? Wait, let's check cross - multiplication.

For Jose's proportion: \( \frac{3}{5}=\frac{w}{6} \). Cross - multiply: \( 3\times6 = 5\times w \) → \( 18 = 5w \)? Wait, but Jose wrote \( 5w = 9 \). Oh! Wait, Jose's cross - multiplication: \( 3\times6 = 18 \), but he wrote \( 5w = 9 \). Wait, no, wait the options: let's re - examine.

Wait, the options: "Jose did not multiply \( 3\times6 \) correctly." Let's check Jose's work. His proportion is \( \frac{3}{5}=\frac{w}{6} \). Cross - multiplying: \( 3\times6 = 5\times w \) → \( 18 = 5w \). But Jose wrote \( 5w = 9 \). So he made a mistake in multiplying \( 3\times6 \) (he got 9 instead of 18). Wait, but let's check the proportion setup. Wait, maybe the correct proportion is \( \frac{3}{w}=\frac{5}{6} \) (Nancy) or \( \frac{3}{5}=\frac{w}{6} \) (Jose). Wait, let's see the rectangles: first rectangle: height 3, width 5. Second: width \( w \), height 6. So if they are similar, the ratio of height to width should be the same. So for first: \( \frac{\text{height}}{\text{width}}=\frac{3}{5} \). For second: \( \frac{\text{width}}{\text{height}}=\frac{w}{6} \). Wait, no, maybe orientation. Alternatively, first rectangle's height (3) and second's width (\( w \)) are corresponding, first's width (5) and second's height (6) are corresponding. So \( \frac{3}{w}=\frac{5}{6} \) (Nancy) or \( \frac{3}{5}=\frac{w}{6} \) (Jose). Let's check cross - multiplication for both.

Jose: \( \frac{3}{5}=\frac{w}{6} \). Cross - multiply: \( 3\times6 = 5\times w \) → \( 18 = 5w \). But Jose wrote \( 5w = 9 \). So he made a mistake in \( 3\times6 \) (he calculated \( 3\times6 = 9 \) instead of 18). So the option "Jose did not multiply \( 3\times6 \) correctly" is correct? Wait, but let's check the options again.

Wait the options are:

1. Nancy did divide 18 by 5 correctly. (Nancy's \( 5w = 18 \), \( w=\frac{18}{5}=3.6 \), which is correct. So this is wrong.)

2. Jose did not multiply \( 3\times6 \) correctly. (Jose's \( 3\times6 \) should be 18, but he got 9, so this is correct.)

3. Jose did not set up the proportion correctly. (Is his proportion setup wrong? Let's see: if the first rectangle has height 3 and width 5, second has width \( w \) and height 6, the proportion \( \frac{3}{5}=\frac{w}{6} \) is a valid proportion (relating height of first to width of first, and width of second to height of second). So setup is correct, but cross - multiplication is wrong. So this option is wrong.)

4. Nancy did not set up the proportion correctly. (Nancy's proportion \( \frac{3}{w}=\frac{5}{6} \): if 3 is height of first, \( w \) is width of second; 5 is width of first, 6 is height of second. So ratio of height to width for first: \( \frac{3}{5} \), ratio of height to width for second: \( \frac{6}{w} \). Wait, no, maybe I messed up the correspondence. But Nancy's cross - multiplication is correct: \( 3\times6 = 5\times w \) → \( 18 = 5w \), and she solved \( w=\frac{18}{5}=3.6 \), which is correct. So her proportion setup is correct? Wait, maybe the correct proportion is \( \frac{3}{6}=\frac{w}{5} \), but no, let's go back to the options.

The key is Jose's cross - multiplication: \( 3\times6 = 18 \), but he wrote \( 5w = 9 \), so he made a mistake in multiplying \( 3\times6 \). So the correct option is "Jose did not multiply \( 3\times6 \) correctly."

Step2: Evaluate each option

- Option 1: Nancy's \( 5w = 18 \), \( w = 3.6 \) (18÷5 = 3.6), so she divided correctly. Eliminate.
- Option 2: Jose's proportion \( \frac{3}{5}=\frac{w}{6} \). Cross - multiply: \( 3\times6=18 \), but he got \( 5w = 9 \), so he messed up \( 3\times6 \). This is correct.
- Option 3: Jose's proportion \( \frac{3}{5}=\frac{w}{6} \) is a valid proportion for similar rectangles (relating corresponding sides). So he set up the proportion correctly. Eliminate.
- Option 4: Nancy's proportion \( \frac{3}{w}=\frac{5}{6} \) is also a valid proportion (relating corresponding sides). So she set up the proportion correctly. Eliminate.

Answer:

Jose did not multiply \( 3\times6 \) correctly.