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Zorluk: ZorRatios, Rates, and Proportions

A renewable energy facility operates three solar arrays: Alpha, Beta, and Gamma. The ratio of energy generated by Array Alpha to Array Beta is 4:54:5, and the ratio of energy generated by Array Beta to Array Gamma is 3:23:2. On a given day, Array Gamma generated 120120 kilowatt-hours (kWh) less energy than Array Alpha. What is the total energy, in kWh, generated by all three arrays combined on that day?

  1. A
    660660
  2. B
    900900
  3. C
    1,3201,320
  4. 2,2202,220Cevap
  5. E
    4,4404,440

Cevap

The total energy generated by all three arrays combined on that day is 2,2202,220 kWh.
To solve this problem, first express the energy outputs as a unified ratio Alpha : Beta : Gamma. Since Alpha : Beta = 4:54:5 and Beta : Gamma = 3:23:2, convert both so that Beta has a common ratio value of 1515. This gives Alpha : Beta : Gamma = 12:15:1012 : 15 : 10. Define the outputs in terms of a multiplier xx: Alpha = 12x12x, Beta = 15x15x, and Gamma = 10x10x. The problem states that Gamma generates 120120 kWh less than Alpha, so 12x−10x=12012x - 10x = 120, which simplifies to 2x=1202x = 120 or x=60x = 60. The total output for all three arrays combined is 12x+15x+10x=37x12x + 15x + 10x = 37x. Multiplying 37×6037 \times 60 yields 2,2202,220 kWh.

Adım Adım Çözüm

1
Unify the two separate ratios into a single continuous ratio for Alpha : Beta : Gamma.
The ratio Alpha : Beta is 4:54:5 and Beta : Gamma is 3:23:2. The common term is Beta. Multiply 4:54:5 by 33 to get 12:1512:15, and multiply 3:23:2 by 55 to get 15:1015:10. The combined ratio is Alpha : Beta : Gamma = 12:15:1012 : 15 : 10.
Array Beta must have the same relative value in both ratios to enable direct comparison across all three arrays.
2
Set up an equation for the given difference between Array Alpha and Array Gamma.
Let xx be the common multiplier. Array Alpha produces 12x12x kWh and Array Gamma produces 10x10x kWh. The difference is 12x−10x=2x12x - 10x = 2x. We are given 2x=1202x = 120, so x=60x = 60.
Using a single variable xx allows conversion from ratio units to actual kilowatt-hours.
3
Calculate the total energy generated by summing the parts of all three arrays.
Total energy = 12x+15x+10x=37x12x + 15x + 10x = 37x. Substituting x=60x = 60: 37×60=2,22037 \times 60 = 2,220 kWh.
The question asks for the total energy produced by all three arrays combined.

Anahtar Kavram

Combining Three-Part Ratios and Proportions
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Konular

Ratios, Rates, and Proportions

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