Gene can proofread 3 essays in 1.7 hours. Working with his daughter Lauren, they can proofread 2 essays in 0.75 hours. Using a rational equation, which of the following correctly calculates how long it would take Lauren to proofread 1 essay by herself?
1. 1.11 hours
2. 0.90 hours
3. 1.275 hours
4. 2.45 hours
Let L be the amount of time it takes for Lauren to proofread 1 essay by herself.
Considering their individual rates, we can set up the equation: 1/L + 1/1.7 = 1/0.75.
Multiplying through by L*1.7*0.75, we get 1.7*0.75 + L*0.75 = L*1.7.
Simplifying the left side gives 1.275 + 0.75L = 1.7L.
Subtracting 0.75L from both sides, we get 1.275 = 0.95L.
Dividing both sides by 0.95 gives L ≈ 1.34.
Therefore, it would take Lauren approximately 1.34 hours to proofread 1 essay by herself.
The correct answer is not among the options given.
Let's assume that Lauren can proofread 1 essay in x hours.
From the given information, we know that Gene can proofread 3 essays in 1.7 hours. This means that Gene can proofread 1 essay in (1.7 hours / 3 essays) = 0.567 hours per essay.
Working together, Gene and Lauren can proofread 2 essays in 0.75 hours. So their combined rate is (2 essays / 0.75 hours) = 2.667 essays per hour.
Using the concept of rates, we can form the following equation:
Lauren's rate + Gene's rate = Combined rate
1/x + 1/0.567 = 2.667
To solve this rational equation, we can find a common denominator:
0.567 / (x * 0.567) + 1 / (0.567) = 2.667
Simplifying the equation:
1 / x + 1 / 0.567 = 2.667
Multiplying through by x * 0.567:
0.567 + x = 2.667 * x
Subtracting x from both sides:
0.567 = 2.667 * x - x
Combining like terms:
0.567 = 1.667 * x
Dividing both sides by 1.667:
x = 0.567 / 1.667
x ≈ 0.34
Therefore, Lauren can proofread 1 essay by herself in approximately 0.34 hours or 0.34 * 60 ≈ 20.4 minutes.
The correct answer is not provided in the given options.
To find out how long it would take Lauren to proofread 1 essay by herself, we can set up a rational equation using the given information. Let's assume that Lauren's proofreading speed is represented by "L essays per hour."
From the problem, we know that:
Gene can proofread 3 essays in 1.7 hours, so Gene's proofreading speed is 3 essays / 1.7 hours = 1.76 essays per hour.
When Gene and Lauren work together, they can proofread 2 essays in 0.75 hours, so their combined proofreading speed is 2 essays / 0.75 hours = 2.67 essays per hour.
Now, let's set up the rational equation:
L (Lauren's proofreading speed) + 1.76 (Gene's proofreading speed) = 2.67 (combined proofreading speed)
L + 1.76 = 2.67
Subtract 1.76 from both sides:
L = 2.67 - 1.76
L = 0.91
Therefore, Lauren's proofreading speed is 0.91 essays per hour.
Finally, to find out how long it would take Lauren to proofread 1 essay by herself, we can use the formula: Time = Number of essays / Speed.
Time = 1 essay / 0.91 essays per hour = 1.1 hours.
Thus, the correct answer is option 1: 1.11 hours.