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: Multiply all terms by \( j(j + 21)14 \) to eliminate denominators.

\( 14(j + 21) + 14j = j(j + 21) \)

Step2: Expand and simplify the equation.

\( 14j + 294 + 14j = j^2 + 21j \)
\( 28j + 294 = j^2 + 21j \)
\( j^2 - 7j - 294 = 0 \)

Step3: Factor the quadratic equation.

\( (j - 21)(j + 14) = 0 \)

Step4: Solve for \( j \).

\( j = 21 \) or \( j = -14 \) (time can't be negative, so discard \( j = -14 \))
Wait, but let's check the options. Wait, maybe I made a mistake. Wait, let's re - check the equation. The original equation is \( \frac{1}{j}+\frac{1}{j + 21}=\frac{1}{14} \). Let's test the options.
Option A: \( j = 7 \). Then \( \frac{1}{7}+\frac{1}{7 + 21}=\frac{1}{7}+\frac{1}{28}=\frac{4 + 1}{28}=\frac{5}{28}
eq\frac{1}{14} \)
Option B: \( j = 21 \). Then \( \frac{1}{21}+\frac{1}{21+21}=\frac{1}{21}+\frac{1}{42}=\frac{2 + 1}{42}=\frac{3}{42}=\frac{1}{14} \). Oh! Wait, I factored wrong earlier. Wait, when we have \( j^2-7j - 294 = 0 \), discriminant \( D=49+4\times294 = 49 + 1176=1225 = 35^2 \). Then \( j=\frac{7\pm35}{2} \). So \( j=\frac{7 + 35}{2}=21 \) or \( j=\frac{7-35}{2}=-14 \). So the correct solution is \( j = 21 \)? Wait, but let's check the option B is 21 minutes. Wait, but let's re - check the problem statement. Wait, the problem says "it would take Kevin 21 minutes longer to clear the driveway of snow than it would take Jay working alone". If Jay takes 21 minutes, Kevin takes 42 minutes. Then together, their rate is \( \frac{1}{21}+\frac{1}{42}=\frac{2 + 1}{42}=\frac{3}{42}=\frac{1}{14} \), which matches the combined time of 14 minutes. So the correct answer is 21 minutes.

Answer:

B. 21 minutes