Question

marie and jake both set up a proportion to find the missing width, w, of the smaller of two similar rectangles. their work is shown below.
marie’s work:
\\(\frac{2}{7} = \frac{w}{6}\\)
\\(7w = 8\\)
\\(w \approx 1.14\\)
jake’s work:
\\(\frac{2}{w} = \frac{7}{6}\\)
\\(7w = 12\\)
\\(w \approx 1.71\\)
image of two rectangles: left rectangle has height 2, length 7; right rectangle has height 6, width w
which student made an error in their work and what error did they make?
\\(\bigcirc\\) marie did not set up the proportion correctly.
\\(\bigcirc\\) jake did not set up the proportion correctly.
\\(\bigcirc\\) marie did not multiply 2*6 correctly.
\\(\bigcirc\\) jake did divide 12 by 7 correctly.

Explanation:

Step1: Identify similar rectangle sides

Small rectangle: width \(w\), height 6; Large rectangle: width 7, height 2. Proportion: \(\frac{\text{small width}}{\text{large width}} = \frac{\text{small height}}{\text{large height}}\) → \(\frac{w}{7} = \frac{6}{2}\) or \(\frac{2}{6} = \frac{7}{w}\) (equivalent).

Step2: Check Marie's proportion

Marie’s \(\frac{2}{7} = \frac{w}{6}\) mismatches side pairs. Correct cross-multiplication for her proportion: \(7w = 2×6 = 12\), but she wrote \(7w=8\) (wrong multiplication).

Step3: Check Jake's proportion

Jake’s \(\frac{2}{w} = \frac{7}{6}\) → cross-multiply: \(7w=12\) (correct setup/multiplication).

Step4: Evaluate options

Marie’s error is incorrect multiplication of \(2×6\) (she got 8 instead of 12).

Answer:

Marie did not multiply 2·6 correctly.