Rates for having a manuscript typed at a certain typing service are $5 per page for the first time a...
GMAT Word Problems : (WP) Questions
Rates for having a manuscript typed at a certain typing service are \(\$5\) per page for the first time a page is typed and \(\$3\) per page each time a page is revised. If a certain manuscript has \(100\) pages, of which \(40\) were revised only once, \(10\) were revised twice, and the rest required no revisions, what was the total cost of having the manuscript typed?
- Translate the problem requirements: Break down the manuscript into categories based on revision history - pages with no revisions (typed once), pages revised once (typed twice total), and pages revised twice (typed three times total). Calculate total cost by applying the pricing structure: \(\$5\) for initial typing plus \(\$3\) for each revision.
- Categorize pages by revision status: Organize the 100 pages into three distinct groups based on how many times they were revised, ensuring all pages are accounted for.
- Calculate cost per category: Apply the pricing rules to each category - initial typing cost plus revision costs for each group of pages.
- Sum total costs: Add up the costs from all three categories to get the final manuscript typing cost.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what's happening with this manuscript in everyday terms. We have a 100-page manuscript, and each page has a different "typing history." Think of it like this: every page gets typed once initially, but some pages need to be retyped due to revisions.
The pricing works like this:
- First time any page is typed: \(\$5\) per page
- Each time a page needs revision (retyping): \(\$3\) per page
So if a page is revised once, you pay \(\$5\) (initial) + \(\$3\) (revision) = \(\$8\) total for that page.
If a page is revised twice, you pay \(\$5\) (initial) + \(\$3\) (first revision) + \(\$3\) (second revision) = \(\$11\) total for that page.
Process Skill: TRANSLATE - Converting the problem's language about "revisions" into clear cost calculations
2. Categorize pages by revision status
Let's organize our 100 pages into groups based on their revision history:
- Pages revised twice: 10 pages (given)
- Pages revised only once: 40 pages (given)
- Pages with no revisions: We need to figure this out
Pages with no revisions = Total pages - Pages revised once - Pages revised twice
Pages with no revisions = \(100 - 40 - 10 = 50\) pages
So our three categories are:
- 50 pages: no revisions (typed once only)
- 40 pages: revised once (typed twice total)
- 10 pages: revised twice (typed three times total)
3. Calculate cost per category
Now let's calculate the cost for each group:
Category 1: 50 pages with no revisions
Each page is typed once at \(\$5\) per page
Cost = \(50 \times \$5 = \$250\)
Category 2: 40 pages revised once
Each page is typed initially (\(\$5\)) plus revised once (\(\$3\))
Cost per page = \(\$5 + \$3 = \$8\)
Total cost = \(40 \times \$8 = \$320\)
Category 3: 10 pages revised twice
Each page is typed initially (\(\$5\)) plus revised twice (\(\$3 + \$3 = \$6\))
Cost per page = \(\$5 + \$6 = \$11\)
Total cost = \(10 \times \$11 = \$110\)
4. Sum total costs
Adding up all three categories:
Total cost = Cost of no-revision pages + Cost of once-revised pages + Cost of twice-revised pages
Total cost = \(\$250 + \$320 + \$110 = \$680\)
Let's verify this makes sense: We have 100 pages total, each typed at least once (\(100 \times \$5 = \$500\)), plus 40 pages revised once (\(40 \times \$3 = \$120\)), plus 10 pages revised twice (\(10 \times \$6 = \$60\)).
Total = \(\$500 + \$120 + \$60 = \$680\) ✓
Final Answer
The total cost of having the manuscript typed is \(\$680\), which corresponds to answer choice (D) \(\$680\).
Common Faltering Points
Errors while devising the approach
Faltering Point 1: Misunderstanding what constitutes a "revision"
Students often confuse the total number of times a page is typed with the number of revisions. They may think that a page "revised once" means it was typed once total, when it actually means typed twice (initial + one revision). This leads to undercounting the total typing instances.
Faltering Point 2: Incorrectly interpreting the pricing structure
Students may misread the problem and think that revised pages are charged only \(\$3\) per page total, missing that they still need to pay the initial \(\$5\) typing fee plus \(\$3\) for each revision. This fundamental misunderstanding of the cost structure leads to significantly lower total costs.
Faltering Point 3: Failing to account for all page categories
Students often forget to calculate how many pages had no revisions. They may only work with the explicitly given numbers (40 pages revised once, 10 pages revised twice) and forget that the remaining 50 pages also incur costs, leading to incomplete cost calculations.
Errors while executing the approach
Faltering Point 1: Arithmetic errors in multiplication
Even with the correct approach, students frequently make calculation mistakes when computing costs like \(40 \times \$8 = \$320\) or \(10 \times \$11 = \$110\). These seemingly small errors compound to give incorrect final answers.
Faltering Point 2: Adding revision costs incorrectly
For pages revised twice, students may incorrectly calculate the revision cost as \(\$3\) instead of \(\$6\) (\(2 \times \$3\)). They forget that "revised twice" means two separate \(\$3\) charges, leading to undercounting by \(\$30\) in this problem.
Errors while selecting the answer
No likely faltering points - the calculation directly yields a whole dollar amount that matches one of the given choices exactly, making answer selection straightforward once the arithmetic is completed correctly.