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Question

Difficulty: MediumPercents and Percent Change

A city's annual budget is divided into education, public safety, and other expenses. In 2025, the budget allocated for education was 1212 million dollars. For 2026, the education budget is increased by 15%15\%, and the public safety budget is decreased by 8%8\% from its 2025 value. If the public safety budget in 2026 is equal to the education budget in 2026, what was the public safety budget, in millions of dollars, in 2025?

Answer: 15 million dollars

Answer

The public safety budget in 2025 was 1515 million dollars.
To find the 2025 public safety budget, we first determine the 2026 education budget. A 15%15\% increase on the 2025 education budget of 1212 million yields 12×1.15=13.812 \times 1.15 = 13.8 million. Since the 2026 public safety budget is equal to this amount, it is also 13.813.8 million. Given that the 2026 public safety budget is an 8%8\% decrease from its 2025 value (PP), we set up the equation P×(1−0.08)=13.8P \times (1 - 0.08) = 13.8. Solving for PP gives P=13.80.92=15P = \frac{13.8}{0.92} = 15 million.

Step-by-Step Solution

1
Calculate the education budget for 2026.
The education budget in 2026 is 13.813.8 million dollars.
The education budget in 2026 is a 15%15\% increase from the 2025 education budget of 1212 million dollars: 12×(1+0.15)=12×1.15=13.812 \times (1 + 0.15) = 12 \times 1.15 = 13.8 million dollars.
2
Express the relationship between the 2025 and 2026 public safety budgets.
0.92×P=13.80.92 \times P = 13.8, where PP is the 2025 public safety budget.
The public safety budget in 2026 is equal to the education budget in 2026 (13.813.8 million dollars) and represents an 8%8\% decrease from the 2025 public safety budget PP: P×(1−0.08)=13.8P \times (1 - 0.08) = 13.8.
3
Solve for the 2025 public safety budget PP.
P=15P = 15 million dollars.
Dividing both sides of the equation by 0.920.92 yields P=13.80.92=15P = \frac{13.8}{0.92} = 15 million dollars.

Key Concept

Calculating initial values after a percentage change

Alternative Method

Instead of converting to decimals immediately, you can use fractions: 12×115100=12×2320=695=13.812 \times \frac{115}{100} = 12 \times \frac{23}{20} = \frac{69}{5} = 13.8. Let PP be the 2025 public safety budget. Then 92100P=13.8\frac{92}{100} P = 13.8, so P=13.8×10092=138092=15P = 13.8 \times \frac{100}{92} = \frac{1380}{92} = 15.
Estimated Time:1m 30s
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Topics

Percents and Percent Change

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