Question

alone it takes Khalil 10 hours to complete a one-hour long presentation. it takes Teddy 8 hours to complete a one hour long presentation using a rational equation calculate how long it would take them to complete a one hour presentation if they work together

you may solve algebraically or graphically

Answer 1

To determine how long it would take Khalil and Teddy to complete a one-hour presentation when they work together, we can use the concept of their work rates.

1. **Find the work rates of Khalil and Teddy:**
- Khalil takes 10 hours to complete a one-hour presentation, so his work rate is \( \frac{1 \text{ hour}}{10 \text{ hours}} = \frac{1}{10} \) of a presentation per hour.
- Teddy takes 8 hours to complete a one-hour presentation, so his work rate is \( \frac{1 \text{ hour}}{8 \text{ hours}} = \frac{1}{8} \) of a presentation per hour.

2. **Combine their work rates:**
When they work together, their combined work rate is the sum of their individual work rates:
\[
\text{Combined work rate} = \frac{1}{10} + \frac{1}{8}
\]

To add these fractions, we need a common denominator. The least common multiple of 10 and 8 is 40.

- Convert \( \frac{1}{10} \) to have a denominator of 40:
\[
\frac{1}{10} = \frac{4}{40}
\]
- Convert \( \frac{1}{8} \) to have a denominator of 40:
\[
\frac{1}{8} = \frac{5}{40}
\]

Now, add the two fractions:
\[
\text{Combined work rate} = \frac{4}{40} + \frac{5}{40} = \frac{9}{40}
\]

3. **Calculate the time to complete the presentation together:**
Let \( t \) be the time (in hours) it takes for them to complete the one-hour presentation together. The equation will be:
\[
\left(\frac{9}{40}\right) t = 1 \quad \text{(since they need to complete 1 full presentation)}
\]

To find \( t \), rearrange the equation:
\[
t = \frac{1}{\frac{9}{40}} = \frac{40}{9}
\]

Hence, \( t \approx 4.44 \), which can also be expressed as:
\[
t \approx 4 \text{ hours and } 26.67 \text{ minutes}
\]

Therefore, Khalil and Teddy can complete a one-hour presentation together in approximately **4 hours and 27 minutes**.