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Difficulty: Very hardFractions, Decimals, and Percentages

At the start of the fiscal year, a city's municipal budget is divided among three departments: Education, Healthcare, and Infrastructure. Education receives 38\frac{3}{8} of the total budget, Healthcare receives 40%40\% of the total budget, and Infrastructure receives the remainder. Midyear, the total municipal budget is increased by 25%25\%. The budget allocated to Education is increased by 20%20\% of its original amount, and the budget allocated to Healthcare is increased by 15%15\% of its original amount. What percent of the new total municipal budget is allocated to Infrastructure?

Answer: 27.2 %

Answer

27.2
The correct answer is 27.2%27.2\%. By representing the initial total budget as 11, we find the initial Infrastructure allocation is 1−(0.375+0.40)=0.2251 - (0.375 + 0.40) = 0.225. After the budget increase of 25%25\%, the new total budget is 1.251.25. The new Education allocation is 0.375×1.20=0.450.375 \times 1.20 = 0.45, and the new Healthcare allocation is 0.40×1.15=0.460.40 \times 1.15 = 0.46. The remaining amount for Infrastructure is 1.25−0.45−0.46=0.341.25 - 0.45 - 0.46 = 0.34. Expressing 0.340.34 as a percentage of the new total budget 1.251.25 gives 0.341.25×100%=27.2%\frac{0.34}{1.25} \times 100\% = 27.2\%.

Step-by-Step Solution

1
Convert the initial budget allocations to decimal shares of the original total budget.
Education share is 0.3750.375, Healthcare share is 0.400.40, and Infrastructure share is 0.2250.225.
Expressing all initial shares as decimals relative to a total budget of 1.01.0 makes successive calculations straightforward.
2
Calculate the updated budget amounts for the entire city, Education, and Healthcare.
The new total budget is 1.251.25, the new Education allocation is 0.375×1.20=0.450.375 \times 1.20 = 0.45, and the new Healthcare allocation is 0.40×1.15=0.460.40 \times 1.15 = 0.46.
Adjust each allocation by its respective percentage change to find its share relative to the initial total.
3
Compute the remaining budget share left for Infrastructure.
New Infrastructure share is 1.25−(0.45+0.46)=0.341.25 - (0.45 + 0.46) = 0.34 of the initial budget.
The sum of the three department budgets must equal the new total budget of 1.251.25.
4
Find the percentage of the new total budget represented by the new Infrastructure allocation.
0.341.25×100%=27.2%\frac{0.34}{1.25} \times 100\% = 27.2\%
Divide the new Infrastructure share by the new total budget to find the new ratio, then convert to a percentage.

Key Concept

Converting between fractions, decimals, and percentages, and tracking changes across multiple base values.

Alternative Method

Instead of using 1.01.0 as the base, you can assume an initial total budget of 800800 dollars (since 800800 is a multiple of 88, which simplifies 38\frac{3}{8}). Initial Education = 300300, Healthcare = 320320, Infrastructure = 180180. The new total budget is 800×1.25=1000800 \times 1.25 = 1000. New Education = 300×1.20=360300 \times 1.20 = 360. New Healthcare = 320×1.15=368320 \times 1.15 = 368. New Infrastructure = 1000−(360+368)=2721000 - (360 + 368) = 272. Percentage = 2721000×100%=27.2%\frac{272}{1000} \times 100\% = 27.2\%.
Estimated Time:3m 0s
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Topics

Fractions, Decimals, and Percentages

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