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Difficulty: HardFractions, Decimals, and Percents Arithmetic

At the beginning of the year, a logistics company's fleet consisted of trucks, vans, and cargo planes. Exactly 0.400.40 of the total fleet were trucks, and 38\frac{3}{8} of the total fleet were vans, with the remaining vehicles being cargo planes. Over the course of the year, the number of trucks increased by 25%25\%, the number of vans decreased by 20%20\%, and the number of cargo planes increased by 50%50\%. By what percentage did the total number of vehicles in the company's fleet increase over the year?

  1. A
    11.25%11.25\%
  2. 13.75%13.75\%Answer
  3. C
    15.00%15.00\%
  4. D
    18.33%18.33\%
  5. E
    55.00%55.00\%

Answer

The total number of vehicles in the fleet increased by 13.75%13.75\%.
To find the net percentage increase of the entire fleet, we express each vehicle subgroup as a fraction or decimal of the initial total fleet TT. Trucks account for 0.40T0.40T, vans account for 38T=0.375T\frac{3}{8}T = 0.375T, and cargo planes account for the remaining 1−(0.40+0.375)=0.225T1 - (0.40 + 0.375) = 0.225T. Applying the respective percentage changes gives an increase of 0.25×0.40T=+0.10T0.25 \times 0.40T = +0.10T for trucks, a decrease of 0.20×0.375T=−0.075T0.20 \times 0.375T = -0.075T for vans, and an increase of 0.50×0.225T=+0.1125T0.50 \times 0.225T = +0.1125T for cargo planes. Summing these changes yields a net gain of 0.10T−0.075T+0.1125T=0.1375T0.10T - 0.075T + 0.1125T = 0.1375T, which corresponds to an overall increase of 13.75%13.75\%.

Step-by-Step Solution

1
Determine the initial proportions of each vehicle type in the fleet.
Trucks represent 0.400.40 of the fleet. Vans represent 38=0.375\frac{3}{8} = 0.375 of the fleet. Cargo planes represent 1−(0.40+0.375)=0.2251 - (0.40 + 0.375) = 0.225 of the fleet.
The sum of all component proportions must equal 1.01.0 (or 100%100\% of the initial fleet).
2
Calculate the weighted net change contributed by each vehicle type relative to the initial fleet size TT.
Truck change: +25% of 0.40T=0.25×0.40T=+0.100T+25\% \text{ of } 0.40T = 0.25 \times 0.40T = +0.100T.
Van change: −20% of 0.375T=−0.20×0.375T=−0.075T-20\% \text{ of } 0.375T = -0.20 \times 0.375T = -0.075T.
Cargo plane change: +50% of 0.225T=0.50×0.225T=+0.1125T+50\% \text{ of } 0.225T = 0.50 \times 0.225T = +0.1125T.
The net change contributed by a subgroup is its relative proportion multiplied by its specific percentage change.
3
Sum the component changes to find the total overall change.
Total net change =+0.100T−0.075T+0.1125T=+0.1375T= +0.100T - 0.075T + 0.1125T = +0.1375T.
Combining the individual net contributions yields the overall change in fleet size.
4
Convert the decimal net change to a percentage.
0.1375×100%=13.75%0.1375 \times 100\% = 13.75\%.
Multiplying the decimal fraction of total increase by 100%100\% gives the overall percentage increase.

Key Concept

Weighted Percentage Changes

Alternative Method

Assume a concrete total initial fleet size that works easily with the numbers, such as T=800T = 800 vehicles. Initial trucks =0.40×800=320= 0.40 \times 800 = 320. Initial vans =38×800=300= \frac{3}{8} \times 800 = 300. Initial planes =800−320−300=180= 800 - 320 - 300 = 180. After changes: new trucks =320×1.25=400= 320 \times 1.25 = 400, new vans =300×0.80=240= 300 \times 0.80 = 240, new planes =180×1.50=270= 180 \times 1.50 = 270. New total fleet =400+240+270=910= 400 + 240 + 270 = 910. Fleet increase =910−800=110= 910 - 800 = 110. Percentage increase =110800×100%=13.75%= \frac{110}{800} \times 100\% = 13.75\%.
Estimated Time:2m 0s
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Topics

Fractions, Decimals, and Percents Arithmetic

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