A highway construction company is contracted to pave a new road. In the first month, they … — State PSC Exam

A highway construction company is contracted to pave a new road. In the first month, they complete exactly $0.\overline{3}$ of the total project length. In the second month, they pave $0.625$ of the *remaining* length. If there are exactly $21$ kilometers left to pave in the third month to finish the project, what is the total length of the highway in kilometers?

84Answer
B
80
C
63
D
504

Answer

The total length of the highway is 84 kilometers.

The total length of the highway is 84 km. In the first month,

1/3

is completed, leaving

2/3

. In the second month,

5/8

of this remaining

2/3

is completed, which means

(1 - 5/8) = 3/8

of the remainder is left for the third month. Calculating this relative to the total:

3/8 \times 2/3 = 1/4

. Since

1/4

of the total project equals 21 km, the total length must be

21 \times 4 = 84

km.

Step-by-Step Solution

Convert the decimal and recurring decimal into fractions for exact calculation.

0.\overline{3} = \frac{1}{3}

and

0.625 = \frac{625}{1000} = \frac{5}{8}

Fractional calculations prevent rounding errors and simplify multi-step multiplication.

Determine the fraction of the highway remaining after the first month.

1 - \frac{1}{3} = \frac{2}{3}

of the total length remains.

The second month's progress is based on this remaining amount, not the total.

Calculate the fraction of the total project paved in the second month.

\frac{5}{8} \times \frac{2}{3} = \frac{10}{24} = \frac{5}{12}

of the total length.

The second month's work is

0.625

of the remaining

\frac{2}{3}

Find the final remaining fraction for the third month.

\frac{2}{3} - \frac{5}{12} = \frac{8}{12} - \frac{5}{12} = \frac{3}{12} = \frac{1}{4}

of the total project.

Subtracting the second month's progress from what was left after the first month gives the final remainder.

Set the remaining fraction equal to the given remaining distance and solve for the total.

\frac{1}{4} \times \text{Total} = 21 \Rightarrow \text{Total} = 21 \times 4 = 84

km.

This establishes the relationship between the fractional parts and the actual physical distance.

Key Concept

Converting recurring and terminating decimals to fractions and computing successive remainders.

Estimated Time:1m 30s

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