Question
In three hours, Company ABC can make X ball bearings. Company DEF can make X ball bearings in 4 hours. Company GHI can make X ball bearings in 6 hours. Working together, how long will it take all three companies to make X ball bearings?
Answers
GPT-4o mini
To find out how long it will take all three companies (ABC, DEF, and GHI) to make \(X\) ball bearings when they work together, we'll first determine the individual rates of production for each company.
1. **Company ABC:**
- Produces \(X\) ball bearings in 3 hours.
- Rate of production = \( \frac{X}{3} \) ball bearings per hour.
2. **Company DEF:**
- Produces \(X\) ball bearings in 4 hours.
- Rate of production = \( \frac{X}{4} \) ball bearings per hour.
3. **Company GHI:**
- Produces \(X\) ball bearings in 6 hours.
- Rate of production = \( \frac{X}{6} \) ball bearings per hour.
Next, we will sum their production rates to find the combined rate when they work together:
\[
\text{Combined rate} = \frac{X}{3} + \frac{X}{4} + \frac{X}{6}
\]
To add these fractions, we need a common denominator. The least common multiple of 3, 4, and 6 is 12.
Now we rewrite each rate with a denominator of 12:
\[
\frac{X}{3} = \frac{4X}{12}, \quad \frac{X}{4} = \frac{3X}{12}, \quad \frac{X}{6} = \frac{2X}{12}
\]
Adding these together:
\[
\text{Combined rate} = \frac{4X}{12} + \frac{3X}{12} + \frac{2X}{12} = \frac{(4 + 3 + 2)X}{12} = \frac{9X}{12} = \frac{3X}{4}
\]
This means together, they produce \( \frac{3X}{4} \) ball bearings per hour.
To find out how long it will take them to produce \(X\) ball bearings, we set up the equation where \(t\) is the time in hours:
\[
\text{Total production} = \text{Rate} \times \text{Time}
\]
So, we want to find \(t\) such that:
\[
X = \left(\frac{3X}{4}\right)t
\]
Dividing both sides by \(X\) (assuming \(X \neq 0\)):
\[
1 = \frac{3}{4}t
\]
Now, solve for \(t\):
\[
t = \frac{4}{3} \text{ hours}
\]
Therefore, the time it will take all three companies to make \(X\) ball bearings together is:
\[
\boxed{\frac{4}{3}} \text{ hours} \text{ or } 1 \text{ hour and } 20 \text{ minutes}.
\]
1. **Company ABC:**
- Produces \(X\) ball bearings in 3 hours.
- Rate of production = \( \frac{X}{3} \) ball bearings per hour.
2. **Company DEF:**
- Produces \(X\) ball bearings in 4 hours.
- Rate of production = \( \frac{X}{4} \) ball bearings per hour.
3. **Company GHI:**
- Produces \(X\) ball bearings in 6 hours.
- Rate of production = \( \frac{X}{6} \) ball bearings per hour.
Next, we will sum their production rates to find the combined rate when they work together:
\[
\text{Combined rate} = \frac{X}{3} + \frac{X}{4} + \frac{X}{6}
\]
To add these fractions, we need a common denominator. The least common multiple of 3, 4, and 6 is 12.
Now we rewrite each rate with a denominator of 12:
\[
\frac{X}{3} = \frac{4X}{12}, \quad \frac{X}{4} = \frac{3X}{12}, \quad \frac{X}{6} = \frac{2X}{12}
\]
Adding these together:
\[
\text{Combined rate} = \frac{4X}{12} + \frac{3X}{12} + \frac{2X}{12} = \frac{(4 + 3 + 2)X}{12} = \frac{9X}{12} = \frac{3X}{4}
\]
This means together, they produce \( \frac{3X}{4} \) ball bearings per hour.
To find out how long it will take them to produce \(X\) ball bearings, we set up the equation where \(t\) is the time in hours:
\[
\text{Total production} = \text{Rate} \times \text{Time}
\]
So, we want to find \(t\) such that:
\[
X = \left(\frac{3X}{4}\right)t
\]
Dividing both sides by \(X\) (assuming \(X \neq 0\)):
\[
1 = \frac{3}{4}t
\]
Now, solve for \(t\):
\[
t = \frac{4}{3} \text{ hours}
\]
Therefore, the time it will take all three companies to make \(X\) ball bearings together is:
\[
\boxed{\frac{4}{3}} \text{ hours} \text{ or } 1 \text{ hour and } 20 \text{ minutes}.
\]