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5. 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.
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

1 answer

To solve for the adjusted production rates of the machines and find the ratio \( x : y : z \), we start by determining the original rates of production for the machines.

  1. Calculate Original Production Rates:

    • Machine A produces a batch in 3 hours, so its original rate: \[ R_A = \frac{1 \text{ batch}}{3 \text{ hours}} = \frac{1}{3} \text{ batches/hour} \]
    • Machine B produces a batch in 5 hours, so its original rate: \[ R_B = \frac{1 \text{ batch}}{5 \text{ hours}} = \frac{1}{5} \text{ batches/hour} \]
    • Machine C produces a batch in 8 hours, so its original rate: \[ R_C = \frac{1 \text{ batch}}{8 \text{ hours}} = \frac{1}{8} \text{ batches/hour} \]

  2. Inversely Proportional Adjusted Production Rates: The adjusted production rates are inversely proportional to the original production times:

    • Original times in hours:
      • Machine A (3 hours), Machine B (5 hours), Machine C (8 hours).
    • The relationship for adjusted rates \( x, y, z \) can be defined as follows: \[ x = k \cdot \frac{1}{3}, \quad y = k \cdot \frac{1}{5}, \quad z = k \cdot \frac{1}{8} \] where \( k \) is a constant of proportionality.

  3. Find the Common Factor: To find the ratio, we will express them without the constant \( k \):

    • Thus, we have: \[ x : y : z = \frac{1/3}{k} : \frac{1/5}{k} : \frac{1/8}{k} \]
    • We can multiply through by \( k \) (assuming \( k \) is positive) and simplify to find: \[ x : y : z = \frac{1}{3} : \frac{1}{5} : \frac{1}{8} \]

  4. Find a Common Denominator: Find the least common multiple (LCM) of the denominators 3, 5, and 8:

    • The LCM of 3, 5, and 8 is 120.

  5. Convert Each Ratio to Have the Same Denominator:

    • Convert to the common denominator: \[ x = \frac{1}{3} = \frac{40}{120}, \quad y = \frac{1}{5} = \frac{24}{120}, \quad z = \frac{1}{8} = \frac{15}{120} \]

  6. Form the Final Ratio:

    • Hence, we have: \[ x : y : z = 40 : 24 : 15 \]

  7. Simplify the Ratio:

    • We can simplify the ratio by dividing each term by their greatest common divisor (GCD). In this case, the numbers 40, 24, and 15 do not have a common factor greater than 1.
    • Therefore, the final simplified ratio remains: \[ x : y : z = 40 : 24 : 15 \]

Thus, the final ratio of the adjusted production rates \( x : y : z \) is:

\[ \boxed{40 : 24 : 15} \]