In order to complete a reading assignment on time, Terry planned to read 90 pages per day. However, she read...

1. Translate the problem requirements: Terry had a total assignment with a planned reading rate of \(90 \text{ pages/day}\). She actually read \(75 \text{ pages/day}\) for some initial period, then had 690 pages remaining for the final 6 days. We need to find the total days available for the entire assignment.
2. Set up the reading scenario with variables: Define the total days available and identify the two reading periods - initial period at \(75 \text{ pages/day}\) and final 6 days.
3. Express total pages using both reading periods: The total pages equals pages read in initial period plus pages read in final 6 days (690 pages).
4. Create equation using planned reading rate: The same total pages should equal \(90 \text{ pages/day}\) times total days available, giving us an equation to solve.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday terms:

Terry has a reading assignment that she needs to finish on time. She made a plan to read 90 pages every day to complete it. However, things didn't go according to plan - she only managed to read 75 pages per day for most of the time. When she realized she was behind, she had 690 pages left to read and only 6 days remaining to finish the assignment.

Our goal is to find out how many total days Terry had from the very beginning to complete this entire assignment.

Process Skill: TRANSLATE - Converting the story into clear mathematical relationships

2. Set up the reading scenario with variables

Let's think about this step by step using plain English:

Let's call the total number of days Terry had for the assignment "d" days.

Now, Terry's reading happened in two distinct periods:

- Period 1: She read 75 pages per day for some number of days

- Period 2: She had exactly 6 days left with 690 pages to read

Since Period 2 is the last 6 days, Period 1 must have lasted for \((d - 6)\) days.

So Terry read 75 pages per day for \((d - 6)\) days, then had 690 pages remaining for the final 6 days.

3. Express total pages using both reading periods

Now let's figure out the total number of pages in the assignment:

The total pages equals:

- Pages read in Period 1: \(75 \text{ pages/day} \times (d - 6) \text{ days} = 75(d - 6) \text{ pages}\)

- Pages read in Period 2: 690 pages

Therefore: Total pages = \(75(d - 6) + 690\)

Let's simplify this:

Total pages = \(75d - 450 + 690 = 75d + 240\)

4. Create equation using planned reading rate

Here's the key insight: Terry's original plan was to read 90 pages per day for d days total. This means the total assignment should have been:

Total pages = \(90d\) pages

Since both expressions represent the same total assignment, we can set them equal:

\(90d = 75d + 240\)

Now let's solve for d:

\(90d - 75d = 240\)

\(15d = 240\)

\(d = 240 \div 15 = 16\)

Process Skill: MANIPULATE - Solving the equation systematically

Final Answer

Terry had 16 days in total to complete the assignment.

Let's verify this makes sense:

- Original plan: \(90 \text{ pages/day} \times 16 \text{ days} = 1,440 \text{ total pages}\)

- What actually happened: \(75 \text{ pages/day} \times (16-6) \text{ days} + 690 \text{ pages} = 75 \times 10 + 690 = 750 + 690 = 1,440 \text{ pages}\) ✓

The answer is (B) 16.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the timeline structure

Students often struggle to understand that Terry's reading happened in two distinct periods: \((d-6)\) days at \(75 \text{ pages/day}\), followed by exactly 6 days with 690 pages remaining. They may incorrectly think she read 75 pages/day for some unspecified time and then switched to a different rate for the last 6 days, missing that the 690 pages is what was LEFT to read, not what she actually read during those 6 days.

2. Confusing what the 690 pages represents

A critical misunderstanding occurs when students think the 690 pages is what Terry read during the last 6 days, rather than what remained to be read. This leads to setting up the wrong equation structure and getting an incorrect total page count.

3. Forgetting to use the original plan as a constraint

Students may set up equations based on what actually happened but forget that Terry's original plan \((90 \text{ pages/day for } d \text{ days})\) must equal the same total assignment. Without this key insight, they cannot create the equation needed to solve for d.

Errors while executing the approach

1. Algebraic manipulation errors when expanding

When expanding \(75(d-6)\), students commonly make errors like getting \(75d - 75\) instead of \(75d - 450\), or incorrectly combining terms when simplifying \(75d - 450 + 690\) to get the wrong coefficient or constant term.

2. Sign errors when solving the equation

When solving \(90d = 75d + 240\), students may incorrectly subtract terms or make sign errors, such as getting \(90d + 75d = 240\) instead of \(90d - 75d = 240\), leading to drastically wrong answers.

3. Arithmetic errors in final division

Even with the correct equation \(15d = 240\), students may make basic division errors, such as calculating \(240 \div 15\) incorrectly, especially if they don't recognize this as a simple fraction that reduces to 16.

Errors while selecting the answer

1. Selecting the number of initial reading days instead of total days

Students who correctly calculate \(d = 16\) may mistakenly think this represents only the initial period of reading and select (A) 15, confusing the \((d-6) = 10\) days of initial reading with the total assignment duration.

2. Choosing the total pages instead of total days

Some students may calculate correctly but then select an answer choice that seems close to their calculated total pages (1,440) instead of the total days (16), potentially choosing (E) 46 if they see similarities in the numbers.