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Zorluk: ZorFractions, Decimals, and Percents Arithmetic

In 2025, an architectural design firm divided its working hours among commercial, residential, and municipal projects. Commercial projects accounted for 0.400.40 of the total hours. Of the remaining hours, 512\frac{5}{12} were spent on residential projects, and the rest were spent on municipal projects.

In 2026, the firm's total working hours increased by 25%25\%. Commercial project hours decreased by 15%15\%, while residential project hours increased by 40%40\%. By what percent did the number of hours spent on municipal projects increase from 2025 to 2026?

Cevap: 60 %

Cevap

The number of hours spent on municipal projects increased by 60%60\%.
By setting the total initial hours to a variable TT (or a convenient constant like 100100), we find that in 2025, commercial projects comprised 0.40T0.40T, residential comprised 512×0.60T=0.25T\frac{5}{12} \times 0.60T = 0.25T, and municipal comprised 0.35T0.35T. In 2026, total hours rose to 1.25T1.25T, commercial dropped to 0.34T0.34T, and residential rose to 0.35T0.35T, leaving 0.56T0.56T for municipal. The percent increase in municipal hours is 0.56T−0.35T0.35T=0.210.35=60%\frac{0.56T - 0.35T}{0.35T} = \frac{0.21}{0.35} = 60\%.

Adım Adım Çözüm

1
Determine the baseline breakdown of working hours for 2025 in terms of total hours TT.
Commercial hours = 0.40T0.40T, Residential hours = 0.25T0.25T, Municipal hours = 0.35T0.35T.
Commercial is given as 0.40T0.40T. The remaining fraction 0.60T0.60T is split such that 512×0.60T=0.25T\frac{5}{12} \times 0.60T = 0.25T goes to residential, leaving 0.60T−0.25T=0.35T0.60T - 0.25T = 0.35T for municipal.
2
Calculate the updated working hours for each category in 2026.
Total hours = 1.25T1.25T, Commercial hours = 0.34T0.34T, Residential hours = 0.35T0.35T.
A 25%25\% total increase yields 1.25T1.25T. A 15%15\% decrease in commercial hours yields 0.40T×0.85=0.34T0.40T \times 0.85 = 0.34T. A 40%40\% increase in residential hours yields 0.25T×1.40=0.35T0.25T \times 1.40 = 0.35T.
3
Find the municipal project hours for 2026 by subtracting commercial and residential hours from total 2026 hours.
Municipal hours (2026) = 0.56T0.56T.
Municipal hours in 2026 equal 1.25T−0.34T−0.35T=0.56T1.25T - 0.34T - 0.35T = 0.56T.
4
Compute the percent change in municipal project hours from 2025 to 2026.
Percent Increase = 60%60\%.
Percent increase is Municipal2026−Municipal2025Municipal2025×100%=0.56T−0.35T0.35T×100%=0.210.35×100%=60%\frac{\text{Municipal}_{2026} - \text{Municipal}_{2025}}{\text{Municipal}_{2025}} \times 100\% = \frac{0.56T - 0.35T}{0.35T} \times 100\% = \frac{0.21}{0.35} \times 100\% = 60\%.

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Multi-step percentage change and fraction-decimal conversions
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Fractions, Decimals, and Percents Arithmetic

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