A manufacturer of solar panels inspects their production in three stages. In the first sta… — ACT

A manufacturer of solar panels inspects their production in three stages. In the first stage, $\frac{1}{12}$ of the total panels produced are identified as defective and recycled. In the second stage, $20\%$ of the remaining non-defective panels are found to have cosmetic flaws and are sold at a discount. In the third stage, $\frac{1}{5}$ of the panels that passed the first two stages without any defects or flaws are selected for quality testing. If $1,760$ panels passed the first two stages but were NOT selected for quality testing, what was the total number of panels originally produced?

Answer: 3000 panels

Answer

The total number of panels originally produced was

3,000

The correct answer of

3,000

is determined by calculating the cumulative fraction of panels that are not defective, do not have cosmetic flaws, and are not selected for testing. In Stage 1, subtracting

\frac{1}{12}

leaves

\frac{11}{12}

of the original panels. In Stage 2, subtracting

20\%

of those leaves

80\%

(or

\frac{4}{5}

), resulting in

\frac{11}{15}

of the original panels. In Stage 3, subtracting

\frac{1}{5}

of these leaves

\frac{4}{5}

, resulting in

\frac{44}{75}

of the original panels. Setting

\frac{44}{75}

of the original panels equal to

1,760

and solving for the total original panels yields

3,000

Step-by-Step Solution

Calculate the fraction of panels remaining after the first stage of inspection.

\frac{11}{12}

of the original panels

Since

\frac{1}{12}

of the total panels are defective and recycled, the remaining portion is

1 - \frac{1}{12} = \frac{11}{12}

Calculate the fraction of the original panels that pass the second stage without defects or flaws.

\frac{11}{15}

of the original panels

Of the remaining

\frac{11}{12}

of the panels,

20\%

have cosmetic flaws, meaning

100\% - 20\% = 80\%

(or

\frac{4}{5}

) do not have flaws. Multiplying these fractions gives

\frac{4}{5} \times \frac{11}{12} = \frac{11}{15}

Calculate the fraction of original panels that pass the first two stages but are not selected for testing.

\frac{44}{75}

of the original panels

Since

\frac{1}{5}

of the flawless panels are selected for testing, the remaining

1 - \frac{1}{5} = \frac{4}{5}

are not selected. Multiplying this by the fraction from Step 2 gives

\frac{4}{5} \times \frac{11}{15} = \frac{44}{75}

Solve for the original number of panels,

P

, using the given value of

1,760

P = 3,000

panels

Set up the equation

\frac{44}{75} P = 1,760

. Solving for

P

gives

P = 1,760 \times \frac{75}{44} = 40 \times 75 = 3,000

Key Concept

Applying successive fractions and percentages to resolve a multi-stage word problem by working backward to the initial value.

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