You're solving a classic
work rate problem. The goal is to find how long it takes Kent and Kendra to paint the room
together, given their individual times.
---
Step 1: Understand what “rate” means
- If Kent can paint the whole room in
6 hours, then in
1 hour, he paints
1/6 of the room.
- If Kendra can paint the whole room in
4 hours, then in
1 hour, she paints
1/4 of the room.
---
Step 2: Add their rates together
When they work together, their rates add up:
>
Combined rate = Kent’s rate + Kendra’s rate
> $$
\frac{1}{6} + \frac{1}{4}
$$
To add these fractions, find a common denominator — the least common multiple of 6 and 4 is
12.
> $$
\frac{1}{6} = \frac{2}{12}, \quad \frac{1}{4} = \frac{3}{12}
$$
> $$
\frac{2}{12} + \frac{3}{12} = \frac{5}{12}
$$
✔ So together, they paint
5/12 of the room per hour.
---
Step 3: Find total time to paint the whole room
If they paint
5/12 of the room in 1 hour, then to paint
1 whole room, it will take:
> $$
\text{Time} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4 \text{ hours}
$$
---
Step 4: Convert to hours and minutes (optional)
0.4 hours = 0.4 × 60 minutes =
24 minutes
✔ So,
2 hours and 24 minutes.
---
Final Answer:
>
It takes them 2.4 hours (or 2 hours and 24 minutes) to paint the room together.
---
Why the equation in the image works:
The image sets up:
> $$
\frac{1}{6} + \frac{1}{4} = \frac{1}{x}
$$
This is correct! Because:
- Left side: combined work rate (fraction of job per hour)
- Right side: if it takes `x` hours to do the whole job together, then the rate is `1/x`
So solving:
> $$
\frac{1}{6} + \frac{1}{4} = \frac{1}{x}
\Rightarrow \frac{5}{12} = \frac{1}{x}
\Rightarrow x = \frac{12}{5} = 2.4
$$
✔️ Perfect!
---
✔ Answer: 2.4 hours (or 2 hours and 24 minutes)
Parent Tip: Review the logic above to help your child master the concept of rational equation word problems worksheet.
