At a pharmaceutical manufacturing facility, a liquid solution undergoes a three-stage puri… — GRE General Test

At a pharmaceutical manufacturing facility, a liquid solution undergoes a three-stage purification process to remove a specific chemical compound. Stage I removes $20\%$ of the compound present at the start of Stage I. Stage II removes $25\%$ of the compound present at the start of Stage II. Stage III removes $30\%$ of the compound present at the start of Stage III. If $16.8\text{ grams}$ of the compound remain in the solution after Stage III, how many grams of the compound were present in the solution immediately before Stage I?

A
$28.0\text{ grams}$
B
$33.6\text{ grams}$
$40.0\text{ grams}$ Answer
D
$56.0\text{ grams}$
E
$67.2\text{ grams}$

Answer

40.0 grams

To find the initial quantity, we determine the compound fraction remaining after each stage. Stage I leaves

80\%

, Stage II leaves

75\%

, and Stage III leaves

70\%

. The overall fraction remaining after all three stages is

0.80 \times 0.75 \times 0.70 = 0.42

. Given that

0.42

of the original amount equals

16.8\text{ grams}

, dividing

16.8

0.42

gives the initial mass of

40.0\text{ grams}

Step-by-Step Solution

Determine the remaining fraction of the compound after each individual stage.

Stage I leaves

1 - 0.20 = 0.80

of its input; Stage II leaves

1 - 0.25 = 0.75

of its input; Stage III leaves

1 - 0.30 = 0.70

of its input.

Percentage decreases must be represented as multiplicative factors of the base amount entering each stage.

Calculate the combined multi-stage remaining multiplier.

Combined multiplier

= 0.80 \times 0.75 \times 0.70 = 0.42

(or

42\%

Successive percentage reductions compound multiplicatively, not additively.

Set up an equation relating the initial mass

X

to the final remaining mass.

0.42 X = 16.8\text{ grams}

The final mass is equal to the initial mass multiplied by the overall remaining decimal fraction.

Solve for the initial mass

X

X = \frac{16.8}{0.42} = 40.0\text{ grams}

Dividing the final amount by

0.42

yields the exact starting quantity.

Key Concept

Successive Percent Reductions and Base Shifts

Alternative Method

Work backwards from the final amount stage by stage: Before Stage III, mass

= 16.8 \div 0.70 = 24.0\text{ g}

. Before Stage II, mass

= 24.0 \div 0.75 = 32.0\text{ g}

. Before Stage I, mass

= 32.0 \div 0.80 = 40.0\text{ g}

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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