A length of rope 7.2 feet long is cut into three pieces. The lengths of two of the pieces are...

1. Translate the problem requirements: We need to find the third piece length by subtracting the two given pieces from the total rope length, then identify which piece is longest and shortest to form their ratio.
2. Calculate the length of the third piece: Use subtraction to find the remaining length after cutting the two known pieces.
3. Compare all three pieces to identify extremes: Determine which piece is the longest and which is the shortest among the three pieces.
4. Express the ratio in simplest form: Create the ratio of longest to shortest and match it with the given 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. We have a rope that's \(\mathrm{7.2}\ \mathrm{feet}\) long total. Someone cut it into three pieces, and we know the lengths of two pieces: \(\mathrm{1.8}\ \mathrm{feet}\) and \(\mathrm{2.4}\ \mathrm{feet}\). We need to find out how long the third piece is, then figure out which piece is the longest and which is the shortest so we can create a ratio.

Think of it like this: if you had a pizza cut into three slices and you knew the sizes of two slices, you could figure out the third slice by seeing what's left over. That's exactly what we're doing here with the rope.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical steps we need to take

2. Calculate the length of the third piece

To find the third piece, we simply subtract the two known pieces from the total length. It's like figuring out how much money you have left after making two purchases.

Total rope length = \(\mathrm{7.2}\ \mathrm{feet}\)
First piece = \(\mathrm{1.8}\ \mathrm{feet}\)
Second piece = \(\mathrm{2.4}\ \mathrm{feet}\)

Third piece = Total length - First piece - Second piece
Third piece = \(\mathrm{7.2 - 1.8 - 2.4}\)
Third piece = \(\mathrm{7.2 - 4.2 = 3.0}\ \mathrm{feet}\)

So our third piece is exactly \(\mathrm{3.0}\ \mathrm{feet}\) long.

3. Compare all three pieces to identify extremes

Now let's line up all three pieces to see which is longest and which is shortest:

First piece: \(\mathrm{1.8}\ \mathrm{feet}\)
Second piece: \(\mathrm{2.4}\ \mathrm{feet}\)
Third piece: \(\mathrm{3.0}\ \mathrm{feet}\)

Looking at these numbers:

- The shortest piece is \(\mathrm{1.8}\ \mathrm{feet}\) (the first piece)
- The longest piece is \(\mathrm{3.0}\ \mathrm{feet}\) (the third piece)
- The middle piece is \(\mathrm{2.4}\ \mathrm{feet}\) (the second piece)

This is straightforward since \(\mathrm{1.8 < 2.4 < 3.0}\).

4. Express the ratio in simplest form

We need the ratio of the longest piece to the shortest piece:

Ratio = Longest : Shortest = \(\mathrm{3.0 : 1.8}\)

To simplify this ratio, let's convert these decimals to fractions and then reduce:
\(\mathrm{3.0} = \frac{30}{10}\) and \(\mathrm{1.8} = \frac{18}{10}\)

So our ratio is \(\frac{30}{10} : \frac{18}{10}\), which simplifies to \(\mathrm{30 : 18}\)

Now we need to find the greatest common factor of 30 and 18:
\(\mathrm{30 = 2 \times 3 \times 5}\)
\(\mathrm{18 = 2 \times 3^2}\)
\(\mathrm{GCD = 2 \times 3 = 6}\)

Dividing both numbers by 6:
\(\mathrm{30 \div 6 = 5}\)
\(\mathrm{18 \div 6 = 3}\)

Therefore, the ratio is \(\mathrm{5 : 3}\), which can also be written as "5 to 3".

5. Final Answer

The ratio of the longest piece to the shortest piece is 5 to 3, which corresponds to answer choice D.

Verification: Longest piece (\(\mathrm{3.0}\ \mathrm{feet}\)) to shortest piece (\(\mathrm{1.8}\ \mathrm{feet}\)) = \(\frac{3.0}{1.8} = \frac{5}{3}\), confirming our answer is correct.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what constitutes "three pieces"
Students may assume they need to find ratios between just the two given pieces (\(\mathrm{1.8}\) and \(\mathrm{2.4}\ \mathrm{feet}\)) rather than recognizing that there's a third piece they must calculate first. This leads them to work with incomplete information and miss the actual longest piece.

2. Confusing which ratio to find
The question asks for "longest to shortest" but students may set up the ratio as "shortest to longest" instead. This reversal in the setup phase will lead to selecting an incorrect answer even if all calculations are done correctly.

Errors while executing the approach

1. Arithmetic errors in subtraction
When calculating the third piece (\(\mathrm{7.2 - 1.8 - 2.4}\)), students may make basic arithmetic mistakes, such as getting \(\mathrm{2.0}\) instead of \(\mathrm{3.0}\), or incorrectly computing \(\mathrm{1.8 + 2.4 = 4.0}\) instead of \(\mathrm{4.2}\).

2. Decimal to ratio conversion errors
When converting the ratio \(\mathrm{3.0:1.8}\) to its simplest form, students may struggle with the decimal arithmetic or make errors when finding the greatest common factor, potentially arriving at ratios like \(\mathrm{6:3}\) or \(\mathrm{10:6}\) instead of the correct \(\mathrm{5:3}\).

3. Incorrect identification of longest/shortest pieces
After finding all three pieces (\(\mathrm{1.8, 2.4, 3.0}\)), students may misidentify which is longest or shortest, especially if they rush through the comparison step, leading to using wrong values in their ratio calculation.

Errors while selecting the answer

1. Selecting the inverted ratio
Even after correctly calculating that the longest piece is \(\mathrm{3.0}\ \mathrm{feet}\) and shortest is \(\mathrm{1.8}\ \mathrm{feet}\), students may accidentally select the answer choice that represents \(\mathrm{3:5}\) instead of \(\mathrm{5:3}\), choosing option A (8 to 5) when the correct answer is D (5 to 3).