Question

Working alone at its constant rate, machine A produces kkk liters of a chemical in 10 minutes. Working alone at its constant rate, machine B produces kkk liters of the chemical in 15 minutes. How many minutes does it take machines A and B, working simultaneously at their respective constant rates, to produce kkk liters of the chemical?

Hint:

For problems where two entities work together to complete one job, you can use the formula \( \frac{1}{T_{total}} = \frac{1}{T_1} + \frac{1}{T_2} \). Here, \( \frac{1}{T} = \frac{1}{10} + \frac{1}{15} = \frac{3+2}{30} = \frac{5}{30} = \frac{1}{6} \). Therefore, \( T = 6 \) minutes.

Answer 1

Step 1: Understanding the Concept:
This is a classic work-rate problem. The key is to find the rate at which each machine works, then add their rates to find their combined rate. Let 'k' represent the quantity 'kkk' liters.
Step 2: Key Formula or Approach:
Work = Rate × Time. Therefore, Rate = Work / Time.
The combined rate of two machines working together is the sum of their individual rates:
\[ R_{\text{combined}} = R_A + R_B \] The time to complete a job together is:
\[ T_{\text{combined}} = \frac{\text{Work}}{R_{\text{combined}}} \] A shortcut formula for two workers is \( T_{\text{combined}} = \frac{T_A \times T_B}{T_A + T_B} \).
Step 3: Detailed Explanation:
Step 3a: Find the individual rates.
The work is to produce k liters of the chemical.
Rate of Machine A (\( R_A \)):
\[ R_A = \frac{\text{Work}}{\text{Time}} = \frac{k \text{ liters}}{10 \text{ minutes}} \] Rate of Machine B (\( R_B \)):
\[ R_B = \frac{\text{Work}}{\text{Time}} = \frac{k \text{ liters}}{15 \text{ minutes}} \] Step 3b: Find the combined rate.
\[ R_{\text{combined}} = R_A + R_B = \frac{k}{10} + \frac{k}{15} \] To add these fractions, find a common denominator, which is 30.
\[ R_{\text{combined}} = \frac{3k}{30} + \frac{2k}{30} = \frac{5k}{30} = \frac{k}{6} \text{ liters per minute} \] Step 3c: Calculate the time taken together.
The work is to produce k liters. The combined rate is \( \frac{k}{6} \) liters per minute.
\[ T_{\text{combined}} = \frac{\text{Work}}{R_{\text{combined}}} = \frac{k}{k/6} = k \times \frac{6}{k} = 6 \text{ minutes} \] Step 4: Final Answer:
It takes machines A and B 6 minutes to produce kkk liters of the chemical when working simultaneously.