Twin​ brothers, Billy and​ Bobby, can mow their​ grandparent's lawn together in 88 minutes. Billy  could mow the lawn by himself in 20 minutes less time that it would take Bobby. How long would it take Bobby to mow the lawn by​ himself?

Let's denote the time it would take Bobby to mow the lawn by himself as \( t \) minutes.

### Step 1: Express Billy's time in terms of Bobby's time
Since Billy can mow the lawn 20 minutes faster than Bobby, Billy's time would be \( t - 20 \) minutes.

### Step 2: Work rates
The work rate is the amount of work done per minute. If Bobby can mow the lawn in \( t \) minutes, his work rate is:

\[
\frac{1}{t} \text{ lawn per minute}
\]

If Billy can mow the lawn in \( t - 20 \) minutes, his work rate is:

\[
\frac{1}{t - 20} \text{ lawn per minute}
\]

Together, their combined work rate is the sum of their individual work rates. So, the combined work rate is:

\[
\frac{1}{t} + \frac{1}{t - 20}
\]

### Step 3: Use the information that they mow the lawn together in 88 minutes
When working together, they mow the lawn in 88 minutes, meaning their combined work rate is:

\[
\frac{1}{88} \text{ lawn per minute}
\]

Thus, the equation becomes:

\[
\frac{1}{t} + \frac{1}{t - 20} = \frac{1}{88}
\]

### Step 4: Solve the equation
To solve this, we first find a common denominator for the left-hand side:

\[
\frac{1}{t} + \frac{1}{t - 20} = \frac{(t - 20) + t}{t(t - 20)} = \frac{2t - 20}{t(t - 20)}
\]

Now, set this equal to \( \frac{1}{88} \):

\[
\frac{2t - 20}{t(t - 20)} = \frac{1}{88}
\]

### Step 5: Cross multiply to solve for \( t \)
Now cross-multiply:

\[
88(2t - 20) = t(t - 20)
\]

Expand both sides:

\[
176t - 1760 = t^2 - 20t
\]

Move all terms to one side:

\[
t^2 - 20t - 176t + 1760 = 0
\]

Simplify:

\[
t^2 - 196t + 1760 = 0
\]

### Step 6: Solve the quadratic equation
We now solve the quadratic equation \( t^2 - 196t + 1760 = 0 \) using the quadratic formula:

\[
t = \frac{-(-196) \pm \sqrt{(-196)^2 - 4(1)(1760)}}{2(1)}
\]

Simplifying:

\[
t = \frac{196 \pm \sqrt{38416 - 7040}}{2} = \frac{196 \pm \sqrt{31376}}{2} = \frac{196 \pm 177.11}{2}
\]

Thus:

\[
t = \frac{196 + 177.11}{2} \quad \text{or} \quad t = \frac{196 - 177.11}{2}
\]

Calculating these:

\[
t = \frac{373.11}{2} \approx 186.56 \quad \text{or} \quad t = \frac{18.89}{2} \approx 9.45
\]

Since \( t \) represents Bobby's time to mow the lawn, the smaller value \( t = 9.45 \) doesn't make sense in this context (it would imply Billy is much faster than reasonable), so we take:

\[
t \approx 186.56 \text{ minutes}
\]

Thus, it would take Bobby approximately **186.56 minutes** to mow the lawn by himself.

### Do you want more details or have any questions?

Here are five related questions for practice:
1. How long would it take Billy to mow the lawn by himself?
2. What is Billy's work rate in lawns per minute?
3. If Bobby could mow the lawn 30 minutes faster than Billy, what would their combined time be?
4. How would the combined time change if Bobby improved his speed by 10%?
5. What if a third person helped them, and together they could finish the lawn in 60 minutes?

**Tip:** Work rate problems can be solved by first defining the rates for each worker and then setting up an equation for their combined rate.

Twin​ brothers, Billy and​ Bobby, can mow their​ grandparent's lawn together in 88 minutes. Billy could mow the lawn by himself in 20 minutes less time than it would take Bobby. How long would it take Bobby to mow the lawn by​ himself?

Explore how to solve a work rate problem where Billy and Bobby mow their grandparent's lawn together in 88 minutes. Learn how to set up and solve the quadratic equation to find how long it would take Bobby to mow the lawn alone. Suitable for students in Grades 9-11.

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