Question

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: Understand 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, cost is \$2.91, so cost per folder is \( \frac{\$2.91}{3} \). For 2 folders, cost per folder is \( \frac{x}{2} \). Wait, alternatively, the ratio of number of folders to cost: \( \frac{2}{x}=\frac{3}{\$2.91} \) (since number of folders over cost should be proportional). Let's check each option.

Step2: Analyze Option D

Option D: \( \frac{2}{x}=\frac{\$2.91}{3} \). Let's see: Left side is number of folders (2) over cost of 2 folders (x), right side is cost of 3 folders (\$2.91) over number of 3 folders (3). Wait, actually, the unit rate: cost per folder is \( \frac{\$2.91}{3} \) (for 3 folders) and also \( \frac{x}{2} \) (for 2 folders). So setting them equal: \( \frac{x}{2}=\frac{\$2.91}{3} \), which can be rearranged as \( \frac{2}{x}=\frac{3}{\$2.91} \)? Wait no, let's cross - multiply. If \( \frac{x}{2}=\frac{2.91}{3} \), then cross - multiplying gives \( 3x = 2\times2.91 \). Now look at option D: \( \frac{2}{x}=\frac{2.91}{3} \). Cross - multiplying: \( 2\times3=2.91\times x \), which is \( 6 = 2.91x \), same as \( 3x=2\times2.91 \) (wait no, 2*3 = 6 and 2.91*x, while 3x=2*2.91 is 3x = 5.82). Wait, maybe better to think in terms of proportion: number of folders / cost of folders should be equal? No, actually, the ratio of number of folders to number of folders is equal to ratio of cost to cost? Wait, no. Let's think of it as a proportion: If 3 folders cost \$2.91, then 2 folders cost x. So the proportion is \( \frac{3}{\$2.91}=\frac{2}{x} \), which is equivalent to \( \frac{2}{x}=\frac{3}{\$2.91} \)? No, \( \frac{3}{2.91}=\frac{2}{x} \) cross - multiplied is \( 3x = 2\times2.91 \), and option D is \( \frac{2}{x}=\frac{2.91}{3} \), cross - multiplied is \( 2\times3 = 2.91x \), which is \( 6 = 2.91x \), and \( 3x=2\times2.91 \) is \( 3x = 5.82 \), \( x=\frac{5.82}{3}=1.94 \), and from option D: \( x=\frac{6}{2.91}\approx2.06 \), which is wrong. Wait, I made a mistake. Let's start over.

The correct proportion: Let x be the cost of 2 folders. The cost per folder is constant. So cost per folder for 3 folders: \( \frac{2.91}{3} \), cost per folder for 2 folders: \( \frac{x}{2} \). So \( \frac{x}{2}=\frac{2.91}{3} \). Let's rearrange this equation. Cross - multiply: \( 3x=2\times2.91 \). Now, let's look at each option:

Option A: \( \frac{2}{2.91}=\frac{x}{3} \). Cross - multiply: \( 2\times3 = 2.91x \), \( 6 = 2.91x \), \( x=\frac{6}{2.91}\approx2.06 \), which is incorrect.

Option B: \( \frac{2}{x}=\frac{3}{2.91} \). Cross - multiply: \( 2\times2.91 = 3x \), \( 5.82 = 3x \), \( x = 1.94 \). Wait, this is correct? Wait no, option B is \( \frac{2}{x}=\frac{3}{2.91} \), cross - multiplying gives \( 2\times2.91=3x \), \( 5.82 = 3x \), \( x = 1.94 \). But earlier I thought option D, but maybe I messed up. Wait the original problem: Let's check the options again.

Wait, the problem is to find the equation that helps determine the cost of 2 folders. Let's define x as the cost of 2 folders.

The rate (cost per folder) for 3 folders is \( \frac{2.91}{3} \) dollars per folder.

The rate for 2 folders is \( \frac{x}{2} \) dollars per folder.

Since the rate is constant, \( \frac{x}{2}=\frac{2.91}{3} \).

Let's rearrange this equation:

Multiply both sides by 2: \( x=\frac{2\times2.91}{3} \)

Or, we can write the proportion as \( \frac{2}{x}=\frac{3}{2.91} \)? No, wait \( \frac{x}{2}=\frac{2.91}{3} \) can be rewritten as \( \frac{2}{x}=\frac{3}{2.91} \) by taking reciprocals? Wait, if \( \frac{a}{b}=\frac{c}{d} \), then \( \frac{b}{a}=\frac{d}{c} \). So if \( \frac{x}{2}=\frac{2.91}{3} \), then \( \frac{2}{x}=\frac{3}{2.91} \), which is option B? But wait, option D is \( \frac{2}{x}=\frac{2.91}{3} \).

Wait, I think I made a mistake in the reciprocal. Let's do it step by step.

We have two ratios:

1. For 3 folders: number of folders = 3, cost = 2.91. So the ratio of number of folders to cost is \( \frac{3}{2.91} \).

2. For 2 folders: number of folders = 2, cost = x. So the ratio of number of folders to cost is \( \frac{2}{x} \).

Since the cost per folder is the same, the ratio of number of folders to cost should be inversely proportional to the cost per folder. Wait, no, the cost per folder is \( \frac{\text{cost}}{\text{number of folders}} \). So \( \frac{2.91}{3}=\frac{x}{2} \).

Let's solve for x from this equation: \( x=\frac{2\times2.91}{3} \).

Now, let's manipulate this equation to match one of the options.

Starting with \( \frac{2.91}{3}=\frac{x}{2} \), we can cross - multiply to get \( 3x = 2\times2.91 \).

Now, let's rearrange this equation to solve for the ratio of 2 to x.

From \( 3x=2\times2.91 \), we can divide both sides by \( x\times3 \):

\( 1=\frac{2\times2.91}{3x} \)

Then, take reciprocals:

\( \frac{3x}{2\times2.91}=1 \)

Wait, maybe a better approach: Let's look at each option's cross - product.

Option D: \( \frac{2}{x}=\frac{2.91}{3} \)

Cross - product: \( 2\times3=2.91\times x \) → \( 6 = 2.91x \) → \( x=\frac{6}{2.91}\approx2.06 \)

Option B: \( \frac{2}{x}=\frac{3}{2.91} \)

Cross - product: \( 2\times2.91 = 3x \) → \( 5.82 = 3x \) → \( x = 1.94 \)

Now, let's calculate the actual cost of 2 folders. The cost of 1 folder is \( \frac{2.91}{3}=0.97 \) dollars. So the cost of 2 folders is \( 2\times0.97 = 1.94 \) dollars.

So the equation that gives x = 1.94 is when we use \( \frac{2}{x}=\frac{3}{2.91} \)? Wait no, the correct equation from \( \frac{2.91}{3}=\frac{x}{2} \) can be rewritten as \( \frac{2}{x}=\frac{3}{2.91} \)? Let's check:

Starting with \( \frac{2.91}{3}=\frac{x}{2} \)

Cross - multiply: \( 2\times2.91=3x \)

Divide both sides by \( x\times2.91 \): \( \frac{2}{x}=\frac{3}{2.91} \)

Ah! So option B is \( \frac{2}{x}=\frac{3}{2.91} \), which is correct? But wait, the original options:

Wait, the user's options:

Option A: \( \frac{2}{2.91}=\frac{x}{3} \)

Option B: \( \frac{2}{x}=\frac{3}{2.91} \)

Option C: \( \frac{x}{2}=\frac{3}{2.91} \)

Option D: \( \frac{2}{x}=\frac{2.91}{3} \)

Option E: None of the above

Wait, I think I made a mistake earlier. Let's recast:

The correct proportion is based on the fact that the cost per folder is the same. So:

Cost per folder for 3 folders: \( \frac{2.91}{3} \)

Cost per folder for 2 folders: \( \frac{x}{2} \)

Setting them equal: \( \frac{x}{2}=\frac{2.91}{3} \)

Let's rearrange this equation to solve for the ratio of 2 to x.

From \( \frac{x}{2}=\frac{2.91}{3} \), we can take reciprocals of both sides: \( \frac{2}{x}=\frac{3}{2.91} \)

Which is option B. But wait, the initial analysis was wrong. Wait, no, let's check the cross - product of option B:

Option B: \( \frac{2}{x}=\frac{3}{2.91} \)

Cross - multiply: \( 2\times2.91 = 3x \)

\( 5.82 = 3x \)

\( x=\frac{5.82}{3}=1.94 \), which is the correct cost of 2 folders (since 1 folder is 0.97, 2*0.97 = 1.94).

Wait, but earlier I thought option D, but that was a mistake. Let's check option D:

Option D: \( \frac{2}{x}=\frac{2.91}{3} \)

Cross - multiply: \( 2\times3=2.91x \)

\( 6 = 2.91x \)

\( x=\frac{6}{2.91}\approx2.06 \), which is incorrect.

Option C: \( \frac{x}{2}=\frac{3}{2.91} \)

Cross - multiply: \( 2.91x = 2\times3 \)

\( 2.91x = 6 \)

\( x=\frac{6}{2.91}\approx2.06 \), incorrect.

Option A: \( \frac{2}{2.91}=\frac{x}{3} \)

Cross - multiply: \( 2\times3 = 2.91x \)

\( 6 = 2.91x \)

\( x=\frac{6}{2.91}\approx2.06 \), incorrect.

So the correct option is B? Wait, but the user's options: Wait, maybe I misread the options. Let me check again.

Wait, the original problem's option D: \( \frac{2}{x}=\frac{2.91}{3} \)

Wait, no, let's re - express the correct proportion.

We have two quantities:

- Quantity 1: Number of folders = 3, Cost = 2.91

- Quantity 2: Number of folders = 2, Cost = x

The ratio of number of folders to cost should be equal if we consider the "number of folders per dollar" ratio.

For quantity 1: number of folders per dollar is \( \frac{3}{2.91} \)

For quantity 2: number of folders per dollar is \( \frac{2}{x} \)

Since the number of folders per dollar should be the same (because the cost per folder is constant), we set \( \frac{3}{2.91}=\frac{2}{x} \), which is equivalent to \( \frac{2}{x}=\frac{3}{2.91} \), which is option B.

Wait, but earlier when I calculated the cost per folder, I got 0.97, and 2 folders cost 1.94. Let's check with option B:

If \( \frac{2}{x}=\frac{3}{2.91} \), then \( x=\frac{2\times2.91}{3}=\frac{5.82}{3}=1.94 \), which is correct.

So the correct option is B. But wait, the user's options: Let me check the original problem again.

Wait, the user's option D is \( \frac{2}{x}=\frac{2.91}{3} \), option B is \( \frac{2}{x}=\frac{3}{2.91} \). So the correct answer is B. But wait, maybe I made a mistake in the initial analysis.

Wait, another way: The proportion can be set up as (number of folders 1)/(cost of folders 1)=(number of folders 2)/(cost of folders 2). Wait, no, that would be \( \frac{3}{2.91}=\frac{2}{x} \), which is the same as \( \frac{2}{x}=\frac{3}{2.91} \), which is option B.

Yes, so the correct answer is B.

Answer:

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