Adam and Brianna plan to install a new tile floor in a classroom. Adam works at a constant rate of...

1. Translate the problem requirements: We need to find how long it takes for Adam and Brianna working together to install 1400 tiles. Adam's rate is 50 tiles/hour, Brianna's rate is 55 tiles/hour.
2. Calculate combined work rate: Add their individual rates to find how many tiles they can install per hour when working together.
3. Apply time formula using combined rate: Use the relationship that time = total work ÷ rate to find the total hours needed.
4. Convert to hours and minutes format: Transform the decimal hours into the hours and minutes format that matches the answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find.

What we know:

• Adam installs tiles at 50 tiles per hour
• Brianna installs tiles at 55 tiles per hour
• The classroom needs exactly 1400 tiles total

What we need to find:

• How long it takes when Adam and Brianna work together to complete all 1400 tiles

Process Skill: TRANSLATE - Converting the problem's everyday language into clear mathematical understanding

2. Calculate combined work rate

When two people work together on the same task, we can think of it like this: if Adam can install 50 tiles in one hour, and Brianna can install 55 tiles in that same hour, then together they can install \(50 + 55 = 105\) tiles in one hour.

This makes intuitive sense - their work rates simply add up when they're working simultaneously.

Combined rate = Adam's rate + Brianna's rate
Combined rate = 50 tiles/hour + 55 tiles/hour = 105 tiles/hour

3. Apply time formula using combined rate

Now we know that working together, Adam and Brianna can install 105 tiles per hour. We need to install 1400 tiles total.

We can think of this as: "If they install 105 tiles every hour, how many hours do they need to install 1400 tiles?"

Time needed = Total tiles ÷ Tiles per hour
Time needed = \(1400 ÷ 105\)

Let's calculate this:
\(1400 ÷ 105 = 13.333...\) hours

4. Convert to hours and minutes format

We got 13.333... hours, but the answer choices are in hours and minutes format. Let's convert the decimal part to minutes.

The whole number part is 13 hours.

For the decimal part: 0.333... hours
To convert to minutes, we multiply by 60 (since there are 60 minutes in an hour):
\(0.333... × 60 = 20\) minutes

Therefore: 13 hours and 20 minutes

Final Answer

Adam and Brianna working together will take 13 hours and 20 minutes to complete the classroom floor.

This matches answer choice C: 13 hrs. 20 mins.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding combined work rates
Students might think that when two people work together, they should use some complex formula or find the harmonic mean of the rates, rather than simply adding the individual rates. This leads to unnecessary complications when the straightforward approach is to add 50 + 55 = 105 tiles per hour.

2. Setting up incorrect time relationships
Some students may try to find how long each person would take individually first, then attempt to combine those times rather than combining the rates directly. This backward approach makes the problem more complex than needed.

Errors while executing the approach

1. Division calculation errors
When calculating \(1400 ÷ 105\), students might make arithmetic mistakes, especially since this doesn't result in a clean whole number. Common errors include getting 13.23 instead of 13.333... or miscalculating the division entirely.

2. Decimal to minutes conversion mistakes
Students often struggle with converting 0.333... hours to minutes. They might forget to multiply by 60, or incorrectly calculate \(0.333... × 60\), getting values like 33 minutes instead of 20 minutes.

Errors while selecting the answer

1. Rounding errors in time conversion
Students might round 13.333... hours to 13.33 hours and then convert \(0.33 × 60 = 19.8 ≈ 20\) minutes, getting the right answer by luck. However, they could also round to 13.3 hours, leading to \(0.3 × 60 = 18\) minutes and selecting answer choice D (13 hrs. 18 mins.) instead of the correct answer C.