In a factory, there are three machines, Machine A, Machine B, and Machine C, that are used to produce widgets. Machine A can produce a batch of widgets in 3 hours, Machine B can produce the same batch in 5 hours, and Machine C can produce the batch in 8 hours. The factory manager decides to adjust the production rates of the machines so that the time taken by each machine to produce the batch is inversely proportional to their original production times.

Question

If the adjusted production rate of Machine A is x widgets per hour, the adjusted production rate of Machine B is y widgets per hour, and the adjusted production rate of Machine C is z widgets per hour, find the ratio of x to y to z.

Answer 1

To find the adjusted production rates of machines A, B, and C and express them in a ratio, we will first determine the original production rates of each machine, and then apply the principle of inverse proportionality based on their original production times.

1. Original Production Rates:

   • For Machine A, the production time is 3 hours for 1 batch of widgets. Therefore, the production rate $R_A$ is: $$R_A=\frac{1\text{ batch}}{3\text{ hours}}=\frac{1}{3}\text{ batches per hour}$$
   • For Machine B, the production time is 5 hours. Thus, the production rate $R_B$ is: $$R_B=\frac{1\text{ batch}}{5\text{ hours}}=\frac{1}{5}\text{ batches per hour}$$
   • For Machine C, the production time is 8 hours. Thus, the production rate $R_C$ is: $$R_C=\frac{1\text{ batch}}{8\text{ hours}}=\frac{1}{8}\text{ batches per hour}$$

2. Finding the Adjusted Production Rates: According to the problem, the adjusted production rates $x$, $y$, and $z$ are inversely proportional to their original production times. The proportionality can be expressed as: $$x\propto \frac{1}{3},y\propto \frac{1}{5},z\propto \frac{1}{8}$$

3. Setting the Ratios: We can express this as: $$x=k\cdot \frac{1}{3},y=k\cdot \frac{1}{5},z=k\cdot \frac{1}{8}$$ for some constant $k$.

4. Finding the Common Denominator: To express $x$, $y$, and $z$ as a ratio, we need to get a common denominator for $\frac{1}{3},\frac{1}{5},\frac{1}{8}$. The least common multiple of the denominators $3,5$, and $8$ is $120$.

   Thus:

   • $x=k\cdot \frac{1}{3}=k\cdot \frac{40}{120}=\frac{40k}{120}$
   • $y=k\cdot \frac{1}{5}=k\cdot \frac{24}{120}=\frac{24k}{120}$
   • $z=k\cdot \frac{1}{8}=k\cdot \frac{15}{120}=\frac{15k}{120}$

5. Expressing the Ratio: Now, we can write the ratios of their adjusted production rates as: $$x:y:z=40k:24k:15k$$

6. Eliminating $k$: Since $k$ is a common factor, we can simplify this to: $$40:24:15$$

7. Simplifying Further: To further simplify the ratio, we divide each term by their greatest common divisor (GCD). The GCD of $40$, $24$, and $15$ is $1$ (as they have no common factors). Thus, the ratio remains: $$40:24:15$$

Therefore, the final ratio of $x$ to $y$ to $z$ is: $$\boxed{{\displaystyle 40:24:15}}$$