If four control devices each have an efficiency of 90%, what would be the overall efficiency of the combined system?

Understand the Problem

The question is asking us to calculate the overall efficiency of a system comprised of four control devices, each with an efficiency of 90%. The overall efficiency can be derived by multiplying the efficiencies of the individual devices together, considering that efficiencies are represented as decimals in multiplication. The approach will involve converting percentages to decimals, multiplying them, and then converting the final result back to a percentage.

Answer

$65.61\%$

Steps to Solve

Convert Percentages to Decimals

Each control device has an efficiency of 90%. To use this in our calculations, we need to convert it from a percentage to a decimal.

The decimal representation of 90% is:

$$ 0.90 $$

Determine Total Efficiency

Since there are four control devices, and we assume that they work in series, the overall efficiency of the system is calculated by multiplying the efficiency of each device together:

$$ \text{Overall Efficiency} = 0.90 \times 0.90 \times 0.90 \times 0.90 $$

Calculate the Overall Efficiency

Now, we perform the multiplication:

$$ \text{Overall Efficiency} = 0.90^4 $$

Calculating this gives:

$$ 0.90^4 = 0.6561 $$

Convert Decimal Back to Percentage

To express the overall efficiency as a percentage, multiply the result by 100:

$$ \text{Overall Efficiency as Percentage} = 0.6561 \times 100 = 65.61% $$

The overall efficiency of the system is $65.61%$.

More Information

This means that when four control devices, each with an efficiency of 90%, are used together, the effectiveness of the whole system drops to approximately 65.61%. This reduction occurs because each device reduces the overall efficiency through cumulative effects.

Tips

Forgetting to convert the percentages to decimals before multiplying them.
Failing to multiply all percentages together, or mistakenly adding them instead.
Miscalculating the final percentage after converting back from decimal form.

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