At the beginning of a fiscal year, a municipal transit authority allocated its capital exp… — GMAT

At the beginning of a fiscal year, a municipal transit authority allocated its capital expenditure budget among three projects: Bus Rapid Transit, Rail Modernization, and Station Upgrades. Exactly $0.30$ of the total budget was allocated to Bus Rapid Transit. Of the remaining budget, exactly $\frac{3}{7}$ was allocated to Rail Modernization, and the rest was allocated to Station Upgrades. By the end of the year, expenditures on Bus Rapid Transit exceeded its initial allocation by $25\%$ , expenditures on Rail Modernization were $15\%$ below its initial allocation, and expenditures on Station Upgrades exceeded its initial allocation by $10\%$ . By what percent did the transit authority's total expenditures across all three projects exceed its initial total budget?

Answer: 7 percent

Answer

The total expenditures across all three projects exceeded the initial total budget by

7\%

To solve this problem, represent the total initial budget as

B

. The Bus Rapid Transit allocation is

0.30B

, leaving

0.70B

. Rail Modernization receives

\frac{3}{7}

0.70B

, which equals

0.30B

. The remaining portion for Station Upgrades is

0.70B - 0.30B = 0.40B

. End-of-year expenditures are calculated by multiplying each allocation by its respective growth multiplier: Bus Rapid Transit is

0.30B \times 1.25 = 0.375B

, Rail Modernization is

0.30B \times 0.85 = 0.255B

, and Station Upgrades is

0.40B \times 1.10 = 0.44B

. Summing these expenditures gives

0.375B + 0.255B + 0.44B = 1.07B

. Comparing

1.07B

to the initial

1.00B

reveals an overall increase of

0.07B

, or

7\%

Step-by-Step Solution

Express the initial allocations for each project as fractions of the total budget B

Bus Rapid Transit =

0.30B

, Rail Modernization =

0.30B

, Station Upgrades =

0.40B

Bus Rapid Transit is explicitly

0.30B

. The remaining

0.70B

is split such that Rail Modernization receives

\frac{3}{7} \times 0.70B = 0.30B

, leaving

0.70B - 0.30B = 0.40B

for Station Upgrades.

Apply the individual percentage changes to determine end-of-year expenditures

Bus Rapid Transit =

0.375B

, Rail Modernization =

0.255B

, Station Upgrades =

0.44B

25\%

increase corresponds to a multiplier of

1.25

, a

15\%

decrease corresponds to a multiplier of

0.85

, and a

10\%

increase corresponds to a multiplier of

1.10

Sum the project expenditures and calculate the net percent change relative to B

Total expenditure =

1.07B

, corresponding to a

7\%

net increase

Adding

0.375B + 0.255B + 0.44B

yields

1.07B

. Subtracting the original budget

1.00B

gives

0.07B

, which is

7\%

B

Key Concept

Weighted Percentage Changes and Sequential Fraction-Decimal Operations

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