A civil contractor is tasked with upgrading a rural highway. In the first month, the crew … — State PSC Exam

A civil contractor is tasked with upgrading a rural highway. In the first month, the crew successfully upgrades $0.275$ of the total highway length. During the second month, they upgrade $\frac{5}{12}$ of the remaining length. If the crew has exactly $10.15$ kilometers left to upgrade in the third month to complete the project, what is the total length of the highway in kilometers?

Cevap: 24 km

Cevap

The total length of the highway is 24 kilometers.

The correct total length is calculated by accurately determining the remaining fraction of the highway step-by-step. Converting the decimal 0.275 to a fraction (11/40) allows for clean, exact operations without repeating decimal rounding errors. By tracking the remainder accurately, we establish that the final 10.15 km represents exactly 203/480 of the total length, resulting in a total of 24 km.

Adım Adım Çözüm

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

\frac{29}{40}

of the total length remains.

The crew upgraded

0.275

of the total. Converting

0.275

to a fraction gives

\frac{275}{1000}

, which simplifies to

\frac{11}{40}

. Subtracting this from the whole gives

1 - \frac{11}{40} = \frac{29}{40}

Calculate the fraction of the total highway upgraded in the second month.

\frac{29}{96}

of the total length.

The crew upgraded

\frac{5}{12}

of the remaining length. Therefore, multiply the two fractions:

\frac{5}{12} \times \frac{29}{40} = \frac{1 \times 29}{12 \times 8} = \frac{29}{96}

Calculate the total fraction of the highway remaining for the third month.

\frac{203}{480}

of the total length.

Subtract the second month's progress from the remainder after the first month:

\frac{29}{40} - \frac{29}{96}

. The least common multiple of

40

and

96

480

. Converting to common denominators gives

\frac{348}{480} - \frac{145}{480} = \frac{203}{480}

Set up the final equation and solve for the total highway length.

24

The remaining fraction equals the given physical distance of

10.15

km. So,

\frac{203}{480} \times \text{Total} = 10.15

. Solving for Total yields

\frac{10.15 \times 480}{203}

. Since

10.15 \div 203 = 0.05

, the Total is

0.05 \times 480 = 24

km.

Anahtar Kavram

Solving sequential parts-of-a-whole word problems by effectively converting between decimals and fractions to find a remaining proportion.

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