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Zorluk: ZorRatios, Rates, and Proportions

A renewable energy facility distributes stored electricity among three battery banks: Bank 1, Bank 2, and Bank 3. Initially, the ratio of the energy stored in Bank 1 to Bank 2 to Bank 3 is 3:4:53 : 4 : 5.

To balance the system, two sequential transfers of energy are performed without any energy loss:
1. Energy is transferred from Bank 3 to Bank 1 such that the ratio of the energy in Bank 1 to Bank 2 becomes 5:45 : 4.
2. Energy is then transferred from Bank 2 to Bank 3 such that the ratio of the energy in Bank 2 to Bank 3 becomes 1:41 : 4.

If Bank 3 contains 60 MWh60\text{ MWh} more energy after the second transfer than it did initially, what was the total amount of energy, in MWh, stored across all three battery banks?

  1. 1,200 MWh1,200\text{ MWh}Cevap
  2. B
    1,500 MWh1,500\text{ MWh}
  3. C
    900 MWh900\text{ MWh}
  4. D
    720 MWh720\text{ MWh}
  5. E
    500 MWh500\text{ MWh}

Cevap

The total energy stored across all three battery banks is 1,200 MWh1,200\text{ MWh}.
The correct answer is 1,200 MWh1,200\text{ MWh}. By setting up algebraic expressions for the energy in each bank relative to total energy TT, we track how each transfer modifies individual bank totals while conserving total energy. The intermediate transfer reduces Bank 3 from 2560T\frac{25}{60}T to 1560T\frac{15}{60}T, and the subsequent transfer increases it to 2860T\frac{28}{60}T. The resulting difference of 360T=120T\frac{3}{60}T = \frac{1}{20}T equals 60 MWh60\text{ MWh}, solving directly to T=1,200 MWhT = 1,200\text{ MWh}.

Adım Adım Çözüm

1
Express initial energy quantities in terms of total energy TT
Bank 1 = 312T\frac{3}{12}T, Bank 2 = 412T\frac{4}{12}T, Bank 3 = 512T=2560T\frac{5}{12}T = \frac{25}{60}T
The ratio 3:4:53 : 4 : 5 sums to 1212 total parts.
2
Calculate energy amounts after the first transfer (Bank 3 to Bank 1)
Bank 2 remains 412T\frac{4}{12}T; Bank 1 becomes 512T\frac{5}{12}T; Bank 3 becomes T512T412T=312TT - \frac{5}{12}T - \frac{4}{12}T = \frac{3}{12}T
The new ratio of Bank 1 to Bank 2 is 5:45 : 4, and Bank 2's energy did not change.
3
Calculate energy amounts after the second transfer (Bank 2 to Bank 3)
Combined energy in Banks 2 and 3 = 412T+312T=712T\frac{4}{12}T + \frac{3}{12}T = \frac{7}{12}T. Final Bank 3 = 45×712T=2860T\frac{4}{5} \times \frac{7}{12}T = \frac{28}{60}T
Bank 1 remains unchanged at 512T\frac{5}{12}T. The remaining energy is divided between Bank 2 and Bank 3 in a 1:41 : 4 ratio.
4
Determine the net change in Bank 3 and solve for total energy TT
Net change = 2860T2560T=360T=120T\frac{28}{60}T - \frac{25}{60}T = \frac{3}{60}T = \frac{1}{20}T. Since 120T=60 MWh\frac{1}{20}T = 60\text{ MWh}, T=1,200 MWhT = 1,200\text{ MWh}
Bank 3 ended with 60 MWh60\text{ MWh} more than its initial amount.

Anahtar Kavram

Multi-stage ratio rebalancing and conservation of total quantity
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