Question

writing proportions
3 folders cost $2.91.
which equation would help determine the cost of 2 folders?
choose 1 answer:
a $\frac{2}{$2.91} = \frac{x}{3}$
b $\frac{2}{x} = \frac{3}{$2.91}$
c $\frac{x}{2} = \frac{3}{$2.91}$
d $\frac{2}{x} = \frac{$2.91}{3}$
e none of the above

Explanation:

Step1: Define the proportion

Let \( x \) be the cost of 2 folders. The cost per folder should be the same, so the ratio of number of folders to cost should be equal. For 3 folders costing \$2.91, the ratio is \( \frac{3}{\$2.91} \). For 2 folders costing \( x \), the ratio is \( \frac{2}{x} \). Setting them equal gives \( \frac{2}{x} = \frac{3}{\$2.91} \), which is option B. Wait, no, wait. Wait, actually, the unit rate: cost per folder. So cost per folder for 3 folders is \( \frac{\$2.91}{3} \), and for 2 folders is \( \frac{x}{2} \). So setting them equal: \( \frac{x}{2} = \frac{\$2.91}{3} \), which is equivalent to \( \frac{2}{x} = \frac{3}{\$2.91} \) (cross - multiplying: \( 3x = 2\times\$2.91 \) vs \( x\times3 = 2\times\$2.91 \), same thing). Wait, let's check each option:

Option A: \( \frac{2}{\$2.91}=\frac{x}{3} \). Cross - multiply: \( 2\times3=\$2.91\times x \), \( 6 = 2.91x \). Not correct, because we want the ratio of folders to cost to be consistent.

Option B: \( \frac{2}{x}=\frac{3}{\$2.91} \). Cross - multiply: \( 2\times\$2.91 = 3x \), which is \( 5.82 = 3x \), so \( x=\frac{5.82}{3}=1.94 \). Wait, but let's think about the proportion. The number of folders and cost are directly proportional. So \( \frac{\text{number of folders}_1}{\text{cost}_1}=\frac{\text{number of folders}_2}{\text{cost}_2} \)? No, wait, it's \( \frac{\text{number of folders}_1}{\text{number of folders}_2}=\frac{\text{cost}_1}{\text{cost}_2} \)? No, actually, the rate is cost per folder, so \( \frac{\text{cost}_1}{\text{number}_1}=\frac{\text{cost}_2}{\text{number}_2} \). So \( \frac{\$2.91}{3}=\frac{x}{2} \), which can be rewritten as \( \frac{2}{x}=\frac{3}{\$2.91} \) (by cross - multiplying and rearranging). Let's check option D: \( \frac{2}{x}=\frac{\$2.91}{3} \). Cross - multiply: \( 2\times3=x\times\$2.91 \), \( 6 = 2.91x \), which is not correct. Wait, I made a mistake earlier. Let's do it properly.

Let \( x \) be the cost of 2 folders. The cost per folder for 3 folders is \( \frac{2.91}{3} \) dollars per folder. The cost per folder for 2 folders is \( \frac{x}{2} \) dollars per folder. Since the cost per folder should be the same (assuming constant price), we set \( \frac{x}{2}=\frac{2.91}{3} \). If we cross - multiply, we get \( 3x = 2\times2.91 \), or we can rearrange the proportion as \( \frac{2}{x}=\frac{3}{2.91} \) (by taking reciprocals of both sides of \( \frac{x}{2}=\frac{2.91}{3} \): \( \frac{2}{x}=\frac{3}{2.91} \)). So option B is \( \frac{2}{x}=\frac{3}{2.91} \), which is correct. Wait, no, option B is \( \frac{2}{x}=\frac{3}{\$2.91} \), which is the same as \( \frac{2}{x}=\frac{3}{2.91} \). So that's correct.

Wait, let's check option D: \( \frac{2}{x}=\frac{2.91}{3} \). Cross - multiply: \( 2\times3 = x\times2.91 \), \( 6 = 2.91x \), which would give \( x=\frac{6}{2.91}\approx2.06 \), which is wrong. So option B is correct.

Step2: Verify each option

• Option A: \( \frac{2}{2.91}=\frac{x}{3} \). Solving for \( x \), \( x=\frac{2\times3}{2.91}=\frac{6}{2.91}\approx2.06 \), incorrect.
• Option B: \( \frac{2}{x}=\frac{3}{2.91} \). Solving for \( x \), \( 3x = 2\times2.91 \), \( x=\frac{5.82}{3}=1.94 \). Let's check the unit rate: \( 2.91\div3 = 0.97 \) per folder, \( 1.94\div2 = 0.97 \) per folder. Correct.
• Option C: \( \frac{x}{2}=\frac{3}{2.91} \). Solving for \( x \), \( x=\frac{3\times2}{2.91}=\frac{6}{2.91}\approx2.06 \), incorrect.
• Option D: \( \frac{2}{x}=\frac{2.91}{3} \). Solving for \( x \), \( 2.91x = 2\times3 \), \( x=\frac{6}{2.91}\approx2.06 \), incorrect.

Answer:

B. \(\frac{2}{x} = \frac{3}{\$2.91}\)