Four investment options, each with a different starting value (expressed as a fraction, de… — ACT

Four investment options, each with a different starting value (expressed as a fraction, decimal, or percentage of a standard initial budget $B$ ) and a specified first-year growth rate, are described below. Arrange the options in order from the lowest final value to the highest final value after the first year.

1Option Y: A starting value of $90\%$ of $B$ that increases by $15\%$ .
2Option Z: A starting value of $\frac{7}{8}$ of $B$ that increases by $18.4\%$ .
3Option X: A starting value of $0.85$ of $B$ that increases by $22\%$ .
4Option W: A starting value of $\frac{4}{5}$ of $B$ that increases by $30\%$ .

Answer

Option Y, Option Z, Option X, Option W

The correct order is Option Y, Option Z, Option X, Option W. Converting each initial value to a decimal fraction of

B

and applying the respective growth multipliers yields: Option Y is

0.90 \times 1.15 = 1.035B

, Option Z is

0.875 \times 1.184 = 1.036B

, Option X is

0.85 \times 1.22 = 1.037B

, and Option W is

0.80 \times 1.30 = 1.040B

. Comparing these decimal values gives

1.035 < 1.036 < 1.037 < 1.040

Step-by-Step Solution

Calculate the final value of Option W relative to

B

0.80 \times 1.30 \times B = 1.040B

Convert the starting fraction

\frac{4}{5}

to the decimal

0.80

. A

30\%

increase corresponds to a multiplier of

1 + 0.30 = 1.30

. Multiply these values together:

0.80 \times 1.30 = 1.040

Calculate the final value of Option X relative to

B

0.85 \times 1.22 \times B = 1.037B

The starting decimal is

0.85

. A

22\%

increase corresponds to a multiplier of

1 + 0.22 = 1.22

. Multiply these values together:

0.85 \times 1.22 = 1.037

Calculate the final value of Option Y relative to

B

0.90 \times 1.15 \times B = 1.035B

Convert the starting percentage

90\%

to the decimal

0.90

. A

15\%

increase corresponds to a multiplier of

1 + 0.15 = 1.15

. Multiply these values together:

0.90 \times 1.15 = 1.035

Calculate the final value of Option Z relative to

B

0.875 \times 1.184 \times B = 1.036B

Convert the starting fraction

\frac{7}{8}

to the decimal

0.875

. An

18.4\%

increase corresponds to a multiplier of

1 + 0.184 = 1.184

. Multiply these values together:

0.875 \times 1.184 = 1.036

Compare the resulting multipliers to order the final values from lowest to highest.

1.035 < 1.036 < 1.037 < 1.040

, which translates to Option Y < Option Z < Option X < Option W.

By arranging the computed multipliers in ascending order, we find the correct sequence.

Key Concept

Converting fractions, decimals, and percentages to common decimal representations to compute and compare multi-step percentage increases.

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