There were 36,000 hardback copies of a certain novel sold before the paperback version was issued. From the time the...

1. Translate the problem requirements: Identify the three phases of book sales: initial hardback sales (36,000), additional sales after paperback launch (hardback and paperback together), and understand that paperback sales were 9 times the additional hardback sales during the second phase.
2. Set up the relationship equation: Express the \(9\)-to-\(1\) ratio between paperback and additional hardback sales using simple variables.
3. Create the total sales equation: Combine initial hardback sales, additional hardback sales, and paperback sales to equal 441,000 total copies.
4. Solve using substitution: Use the ratio relationship to eliminate one variable and solve for paperback copies directly.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what happened with this novel's sales in plain English:

Phase 1 (Before paperback launch): 36,000 hardback copies were sold. This is straightforward - we know this number exactly.

Phase 2 (After paperback launch): Two things happened simultaneously:

- More hardback copies continued to be sold (let's call this unknown amount)
- Paperback copies started being sold for the first time

The key relationship we're told is that during Phase 2, paperback sales were 9 times the additional hardback sales.

So if we think of it like this: for every 1 additional hardback copy sold after the paperback launch, 9 paperback copies were sold.

Total sales constraint: Everything together (initial hardbacks + additional hardbacks + paperbacks) equals 441,000 copies.

Process Skill: TRANSLATE - Converting the problem's language about "before" and "after" paperback launch into distinct phases

2. Set up the relationship equation

Now let's use a simple variable to represent what we don't know.

Let's say \(\mathrm{x}\) = the number of additional hardback copies sold after the paperback version was issued.

Since paperback sales were 9 times this amount:
Paperback copies sold = \(9\mathrm{x}\)

This makes sense because the ratio is 9 paperbacks for every 1 additional hardback.

So we have:

- Additional hardback copies = \(\mathrm{x}\)
- Paperback copies = \(9\mathrm{x}\)
- The ratio between them is \(9\mathrm{x}\) to \(\mathrm{x}\), which simplifies to \(9\) to \(1\) ✓

3. Create the total sales equation

Now we can add up all the sales to equal the total:

Total copies = Initial hardbacks + Additional hardbacks + Paperback copies

In plain English: \(441,000 = 36,000 + \mathrm{x} + 9\mathrm{x}\)

Let's simplify the right side:

- We have \(\mathrm{x}\) additional hardbacks and \(9\mathrm{x}\) paperbacks
- \(\mathrm{x} + 9\mathrm{x} = 10\mathrm{x}\)
- So: \(441,000 = 36,000 + 10\mathrm{x}\)

4. Solve using substitution

Now we can solve for \(\mathrm{x}\):
\(441,000 = 36,000 + 10\mathrm{x}\)

Subtract 36,000 from both sides:
\(441,000 - 36,000 = 10\mathrm{x}\)
\(405,000 = 10\mathrm{x}\)

Divide both sides by 10:
\(\mathrm{x} = 40,500\)

So \(\mathrm{x} = 40,500\) additional hardback copies were sold after the paperback launch.

Since paperback copies = \(9\mathrm{x}\):
Paperback copies = \(9 \times 40,500 = 364,500\)

Let's verify this makes sense:

- Initial hardbacks: 36,000
- Additional hardbacks: 40,500
- Paperbacks: 364,500
- Total: \(36,000 + 40,500 + 364,500 = 441,000\) ✓
- Ratio check: \(364,500 \div 40,500 = 9\) ✓

Final Answer

The number of paperback copies sold was 364,500.

This matches answer choice C. 364,500.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the timing of the 9:1 ratio
Students often misread the problem and think that 9 times as many paperback copies were sold compared to ALL hardback copies (including the initial 36,000). However, the problem clearly states this ratio applies only to the copies sold "from the time the first paperback copy was sold until the last copy." This means the \(9:1\) ratio is between paperback copies and only the additional hardback copies sold after paperback launch, not the total hardback copies.

2. Confusion about what counts as "total paperback copies"
Some students might think they need to find paperback copies sold only during a specific period, when the question asks for the total number of paperback copies sold throughout the entire novel's sales period. Since paperbacks were only sold after the paperback version was issued, this is simply \(9\mathrm{x}\).

3. Setting up incorrect total equation
Students may forget to include all three components in their total: initial hardbacks (36,000) + additional hardbacks (\(\mathrm{x}\)) + paperbacks (\(9\mathrm{x}\)) = 441,000. They might accidentally exclude the initial 36,000 hardbacks or double-count some sales.

Errors while executing the approach

1. Arithmetic errors in solving the linear equation
When solving \(441,000 = 36,000 + 10\mathrm{x}\), students might make calculation errors such as: incorrectly subtracting (\(441,000 - 36,000 = 405,000\)), incorrectly dividing by 10, or making errors when multiplying \(\mathrm{x} = 40,500\) by 9 to get the final paperback count.

2. Forgetting to multiply by 9 for the final answer
After correctly solving for \(\mathrm{x} = 40,500\), students might mistakenly think this represents the number of paperback copies, when \(\mathrm{x}\) actually represents additional hardback copies. They need to calculate \(9\mathrm{x} = 9 \times 40,500 = 364,500\) to get paperback copies.

Errors while selecting the answer

1. Selecting the value of x instead of 9x
Students who correctly solve for \(\mathrm{x} = 40,500\) but forget that the question asks for paperback copies (which equals \(9\mathrm{x}\)) might look for 40,500 among the answer choices. Since this exact value isn't listed, they might pick the closest option like A. 45,000 instead of the correct C. 364,500.