Olivia, Jacob, and Ethan are laying down tile flooring in a large hallway. Working alone at their respective rates, Olivia can tile the hallway in 6 hours, Jacob can tile the hallway in 4 hours, and Ethan can tile the hallway in 5 hours. To speed up the work, exactly two of them will work together to tile the hallway. Select for Shortest Time the time closest to the least amount of time required for any 2-person team to complete the hallway, and select for Longest Time the time closest to the greatest amount of time required for any 2-person team to complete the hallway. Make only two selections, one in each column.

Official Solution: Individual rates: • Olivia — 1/6 hallway per hour • Jacob — 1/4 hallway per hour • Ethan — 1/5 hallway per hour Fastest pair is Jacob and Ethan. • Combined rate = 1/4 + 1/5 = 9/20 hallway per hour • Time = 1/(9/20) = 20/9 hours • 20/9 hours = 2 hours and 2/9 of an hour • 2/9 * 60 minutes ≈ 13 minutes So, time ≈ 2 hours 13 minutes Slowest pair is Olivia and Ethan. • Combined rate = 1/6 + 1/5 = 11/30 hallway per hour • Time = 1/(11/30) = 30/11 hours • 30/11 hours = 2 hours and 8/11 of an hour • 8/11 * 60 minutes ≈ 44 minutes So, time ≈ 2 hours 44 minutes Correct answer: Shortest Time "2 hours 13 minutes" Longest Time "2 hours 44 minutes"

This is the correct solution. Here is the official solution: The first thing to note is: Jacob and Ethan working together will always be the fastest. This is because Jacob has the highest individual rate (1 hallway per 4 hours), and Ethan’s rate (1 hallway per 5 hours) is the second-highest (or in the middle). Olivia and Ethan will always be the slowest. This is because Olivia has the lowest rate (1 hallway per 6 hours), and with Ethan’s second-highest rate, the combined speed is still slower compared to other pairs. This is a Work problem. These questions are always solved by looking at the rate of work which is expressed per unit of time such as 1/4th of hallway per hour can be done by Jacob. 1. Rate of Work Olivia's rate : She can complete the hallway in 6 hours, so her rate is 1 hallway per 6 hours. Jacob's rate : He can complete the hallway in 4 hours, so his rate is 1 hallway per 4 hours. Ethan's rate : He can complete the hallway in 5 hours, so his rate is 1 hallway per 5 hours. 2. Combined Rates When two people work together, their combined rate is the sum of their individual rates. The total time they take together to complete the hallway is calculated as: Time = 1 / (combined rate) A. Jacob and Ethan (Shortest Time) Jacob's rate: 1 hallway per 4 hours Ethan's rate: 1 hallway per 5 hours The combined rate for Jacob and Ethan is: Combined rate = (1/4) + (1/5) = 9/20 of a hallway per hour Time taken for Jacob and Ethan together: Time = 1 / (9/20) = 20/9 ≈ 2.22 hours (or about 2 hours 13 minutes) B. Olivia and Ethan (Longest Time) Olivia's rate: 1 hallway per 6 hours Ethan's rate: 1 hallway per 5 hours The combined rate for Olivia and Ethan is: Combined rate = (1/6) + (1/5) = 11/30 of a hallway per hour Time taken for Olivia and Ethan together: Time = 1 / (11/30) = 30/11 ≈ 2.73 hours (or about 2 hours 44 minutes)

Olivia, Jacob, and Ethan are laying down tile flooring in a large : Two-part Analysis (TPA)

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catinabox

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13 Sep 2024, 07:32

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siddhantvarma

Let w: work to tile the hallway
work = rate*time => w = rate*time
From this we can find the individual rate of all three:
Olivia's rate (\(r_o\)) = w/6
Jacob's rate (\(r_j\)) = w/4
Ethan's rate (\(r_e\)) = w/5

For shortest time, we need the largest values of rate and for the longest time, we need the smallest values of rate. This is because time is inversely proportional to rate.
We know: 1/4 > 1/5 > 1/6
This means \(r_j\) > \(r_e\) > \(r_o\)

For shortest time, we need the largest values of rate so we'll take Jacob and Ethan.
w = (\(r_j\) + \(r_e\))*time => w = (w/4 + w/5)*t => t = 20/9 hours
Converting to minutes we get 2 hours 13 minutes.

For longest time, we need the smallest values of rate so we'll take Olivia and Ethan.
w = (\(r_o\) + \(r_e\))*time => w = (w/6 + w/5)*t => t = 30/11 hours
Converting to minutes we get 2 hours 47 minutes.

This is the correct solution.

Here is the official solution:

The first thing to note is:
Jacob and Ethan working together will always be the fastest. This is because Jacob has the highest individual rate (1 hallway per 4 hours), and Ethan’s rate (1 hallway per 5 hours) is the second-highest (or in the middle).

Olivia and Ethan will always be the slowest. This is because Olivia has the lowest rate (1 hallway per 6 hours), and with Ethan’s second-highest rate, the combined speed is still slower compared to other pairs.

This is a Work problem. These questions are always solved by looking at the rate of work which is expressed per unit of time such as 1/4th of hallway per hour can be done by Jacob.

1. Rate of Work

Olivia's rate: She can complete the hallway in 6 hours, so her rate is 1 hallway per 6 hours.
Jacob's rate: He can complete the hallway in 4 hours, so his rate is 1 hallway per 4 hours.
Ethan's rate: He can complete the hallway in 5 hours, so his rate is 1 hallway per 5 hours.

2. Combined Rates
When two people work together, their combined rate is the sum of their individual rates. The total time they take together to complete the hallway is calculated as:

Code:

Time = 1 / (combined rate)

A. Jacob and Ethan (Shortest Time)

Jacob's rate: 1 hallway per 4 hours
Ethan's rate: 1 hallway per 5 hours

The combined rate for Jacob and Ethan is:

Code:

Combined rate = (1/4) + (1/5) = 9/20 of a hallway per hour

Time taken for Jacob and Ethan together:

Code:

Time = 1 / (9/20) = 20/9 ≈ 2.22 hours (or about 2 hours 13 minutes)

B. Olivia and Ethan (Longest Time)

Olivia's rate: 1 hallway per 6 hours
Ethan's rate: 1 hallway per 5 hours

The combined rate for Olivia and Ethan is:

Code:

Combined rate = (1/6) + (1/5) = 11/30 of a hallway per hour

Time taken for Olivia and Ethan together:

Code:

Time = 1 / (11/30) = 30/11 ≈ 2.73 hours (or about 2 hours 44 minutes)

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12 Feb 2025, 06:18

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assume total work done 120 hrs and than it is easy to find rate and combined rate

catinabox

Olivia, Jacob, and Ethan are laying down tile flooring in a large hallway. Working alone, Olivia can tile the hallway in 6 hours, Jacob can tile the hallway in 4 hours, and Ethan can tile the hallway in 5 hours. To speed up the work, two of them will work together to tile the hallway, with each person working at his or her respective rate.

Select the value closest to the shortest time in which a 2-person team could tile the hallway, and select the value closest to the longest time in which a 2-person team could tile the hallway. Make only two selections, one in each column.

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