Let's solve the problem using the given rates and times.
We are given:
- Bethany can mow the lawn in 4 hours, so her rate is [tex]\( \frac{1}{4} \)[/tex] lawns per hour.
- Colin can mow the lawn in 3 hours, so his rate is [tex]\( \frac{1}{3} \)[/tex] lawns per hour.
### Step-by-Step Solution
1. Determine the combined work rate:
- Bethany's rate: [tex]\( \frac{1}{4} \)[/tex] lawns per hour.
- Colin's rate: [tex]\( \frac{1}{3} \)[/tex] lawns per hour.
Together, their combined rate is the sum of their individual rates:
[tex]\[
\text{Combined rate} = \frac{1}{4} + \frac{1}{3}
\][/tex]
2. Find a common denominator and add the fractions:
[tex]\[
\frac{1}{4} + \frac{1}{3}
\][/tex]
The least common multiple of 4 and 3 is 12. So, convert each rate to have a denominator of 12:
[tex]\[
\frac{1}{4} = \frac{3}{12} \quad \text{and} \quad \frac{1}{3} = \frac{4}{12}
\][/tex]
Now add these fractions:
[tex]\[
\frac{3}{12} + \frac{4}{12} = \frac{7}{12}
\][/tex]
Thus, their combined rate is [tex]\( \frac{7}{12} \)[/tex] lawns per hour.
3. Set up the equation for the combined work:
If [tex]\( x \)[/tex] is the number of hours needed for them to mow the lawn together, the equation becomes:
[tex]\[
\text{(Combined rate)} \times x = 1 \quad \text{(1 full lawn mowed)}
\][/tex]
Substituting the combined rate [tex]\( \frac{7}{12} \)[/tex]:
[tex]\[
\frac{7}{12} x = 1
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{1}{\frac{7}{12}} = \frac{12}{7}
\][/tex]
Simplify the fraction:
[tex]\[
x = \frac{12}{7} \approx 1.71428571428571
\][/tex]
Therefore, it would take approximately [tex]\( 1.714 \)[/tex] hours for Bethany and Colin to mow the lawn together.
