Question

select the correct answer.
jay and kevin are shoveling the snow off a driveway. working together, they can clear the driveway of snow in 14 minutes. working alone, it would take kevin 21 minutes longer to clear the driveway of snow than it would take jay working alone. when ( j ) is the number of minutes it would take jay to clear the driveway of snow when working alone, the situation is modeled by this rational equation:
( \frac{1}{j} + \frac{1}{j + 21} = \frac{1}{14} ).
how long would it take jay to clear the driveway of snow working alone?
a. 7 minutes
b. 21 minutes
c. 35 minutes
d. 42 minutes

Explanation:

Step1: Define Variables and Equation

Let \( j \) be Jay's time (minutes) alone. Kevin's time is \( j + 21 \). The work - rate equation is \( \frac{1}{j}+\frac{1}{j + 21}=\frac{1}{14} \).

Step2: Find a Common Denominator

The common denominator of \( j \), \( j + 21 \), and \( 14 \) is \( 14j(j + 21) \). Multiply each term by this denominator:
\( 14(j + 21)+14j=j(j + 21) \)

Step3: Expand and Simplify

Expand the left - hand side: \( 14j+294 + 14j=j^{2}+21j \)
Combine like terms: \( 28j+294=j^{2}+21j \)
Rearrange to form a quadratic equation: \( j^{2}+21j-28j - 294 = 0\), so \( j^{2}-7j - 294 = 0 \)

Step4: Solve the Quadratic Equation

Factor the quadratic equation. We need two numbers that multiply to \( - 294 \) and add to \( - 7 \). The numbers are \( -21 \) and \( 14 \).
So, \( j^{2}-7j - 294=(j - 21)(j + 14)=0 \)
Set each factor equal to zero: \( j-21 = 0\) or \( j + 14=0 \)
Since time cannot be negative, \( j=21 \) is not a solution (we will check later). Wait, let's re - check the factoring.
Wait, using the quadratic formula \( j=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for \( ax^{2}+bx + c = 0 \), here \( a = 1 \), \( b=-7 \), \( c=-294 \)
\( \Delta=b^{2}-4ac=(-7)^{2}-4\times1\times(-294)=49 + 1176 = 1225 \)
\( \sqrt{\Delta}=35 \)
\( j=\frac{7\pm35}{2} \)
We have two solutions: \( j=\frac{7 + 35}{2}=\frac{42}{2}=21 \) and \( j=\frac{7-35}{2}=\frac{-28}{2}=-14 \)
We discard \( j=-14 \) because time cannot be negative. But wait, if \( j = 21 \), Kevin's time is \( j + 21=42 \)
Let's check the original equation: \( \frac{1}{21}+\frac{1}{42}=\frac{2 + 1}{42}=\frac{3}{42}=\frac{1}{14} \), which is correct. Wait, but let's check the answer options. Wait, maybe I made a mistake in the initial assumption. Wait, the equation is \( \frac{1}{j}+\frac{1}{j + 21}=\frac{1}{14} \)
Wait, if we test the options:
- Option A: \( j = 7 \), Kevin's time \( 7+21 = 28 \), \( \frac{1}{7}+\frac{1}{28}=\frac{4 + 1}{28}=\frac{5}{28} eq\frac{1}{14} \)
- Option B: \( j = 21 \), Kevin's time \( 21 + 21=42 \), \( \frac{1}{21}+\frac{1}{42}=\frac{2+1}{42}=\frac{3}{42}=\frac{1}{14} \), which works. But wait, the option D is 42. Wait, maybe I misread the problem. Wait, the problem says "Kevin 21 minutes longer to clear the driveway of snow than it would take Jay working alone". So Jay's time is \( j \), Kevin's is \( j + 21 \). When we solved, \( j = 21 \), but let's check the equation again. Wait, the equation is \( \frac{1}{j}+\frac{1}{j + 21}=\frac{1}{14} \)
Wait, if \( j = 21 \), then \( \frac{1}{21}+\frac{1}{42}=\frac{2 + 1}{42}=\frac{3}{42}=\frac{1}{14} \), which is correct. But the option B is 21 minutes. But wait, maybe I made a mistake in the quadratic solution. Wait, no, the quadratic solution gives \( j = 21 \) (since \( j=-14 \) is invalid). So Jay's time is 21 minutes? But let's check the answer options again. Wait, the option D is 42. Wait, maybe I mixed up Jay and Kevin. Wait, no, the problem says "How long would it take Jay to clear the driveway of snow working alone?". So if \( j = 21 \), that's Jay's time. But let's check the equation again. The combined rate is \( \frac{1}{14} \) (driveway per minute). Jay's rate is \( \frac{1}{j} \), Kevin's rate is \( \frac{1}{j + 21} \). So when \( j = 21 \), Jay's rate is \( \frac{1}{21} \), Kevin's rate is \( \frac{1}{42} \), and \( \frac{1}{21}+\frac{1}{42}=\frac{2 + 1}{42}=\frac{3}{42}=\frac{1}{14} \), which is correct. So the answer should be 21 minutes? But wait, the option B is 21 minutes. But let's check the problem statement again. Wait, the user's image shows option D as 42 minutes highlighted. Wait, maybe I misread the equation. Wait, the equation is \( \frac{1}{j}+\frac{1}{j + 21}=\frac{1}{14} \). Wait, if Jay's time is 42, then Kevin's time is \( 42+21 = 63 \). Then \( \frac{1}{42}+\frac{1}{63}=\frac{3+2}{126}=\frac{5}{126} eq\frac{1}{14} \) (since \( \frac{1}{14}=\frac{9}{126} \)). So that's not correct. Wait, when \( j = 21 \), Kevin's time is 42, and \( \frac{1}{21}+\frac{1}{42}=\frac{1}{14} \), which is correct. So Jay's time is 21 minutes, which is option B. But the highlighted option is D. Maybe there is a misprint. But according to the equation, Jay's time is 21 minutes.

Answer:

B. 21 minutes