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Zorluk: ZorAlgebraic Word Problems and Modeling

A technology company manages data storage across three servers: Server X, Server Y, and Server Z. Initially, the amount of data stored on Server Y is 25%25\% greater than the amount stored on Server X, and Server Z stores 4040 gigabytes less data than Server Y. During a system reorganization, the data on Server X increases by 20%20\%, the data on Server Y decreases by 20%20\%, and the data on Server Z increases by 6060 gigabytes. If the total amount of data stored across all three servers after the reorganization is 10%10\% greater than the total initial amount of data, what was the initial amount of data, in gigabytes, stored on Server X?

  1. A
    120
  2. B
    150
  3. 160Cevap
  4. D
    192
  5. E
    200

Cevap

160 gigabytes
Let xx be the initial storage on Server X in gigabytes. Then Server Y initially stores 1.25x1.25x gigabytes and Server Z stores 1.25x−401.25x - 40 gigabytes. The initial total storage is T1=x+1.25x+(1.25x−40)=3.5x−40T_1 = x + 1.25x + (1.25x - 40) = 3.5x - 40. After reorganization, Server X stores 1.20x1.20x, Server Y stores 0.80(1.25x)=1.00x0.80(1.25x) = 1.00x, and Server Z stores (1.25x−40)+60=1.25x+20(1.25x - 40) + 60 = 1.25x + 20. The new total storage is T2=1.20x+1.00x+1.25x+20=3.45x+20T_2 = 1.20x + 1.00x + 1.25x + 20 = 3.45x + 20. Since T2T_2 is 10%10\% greater than T1T_1, we have 3.45x+20=1.10(3.5x−40)3.45x + 20 = 1.10(3.5x - 40). Expanding the right side gives 3.45x+20=3.85x−443.45x + 20 = 3.85x - 44. Rearranging terms yields 0.40x=640.40x = 64, which simplifies to x=160x = 160 gigabytes.

Adım Adım Çözüm

1
Define variables for the initial storage on each server in terms of the initial storage on Server X, xx.
Server X = xx, Server Y = 1.25x1.25x, Server Z = 1.25x−401.25x - 40.
Server Y is 25% greater than Server X (1+0.25=1.251 + 0.25 = 1.25), and Server Z is 40 gigabytes less than Server Y.
2
Sum the initial storage values to find the total initial storage T1T_1.
T1=x+1.25x+(1.25x−40)=3.5x−40T_1 = x + 1.25x + (1.25x - 40) = 3.5x - 40.
Combining like terms gives the overall starting storage equation.
3
Express the storage on each server after the reorganization in terms of xx.
Server X' = 1.20x1.20x, Server Y' = 0.80(1.25x)=1.00x0.80(1.25x) = 1.00x, Server Z' = (1.25x−40)+60=1.25x+20(1.25x - 40) + 60 = 1.25x + 20.
Server X increases by 20%, Server Y decreases by 20% of its initial value, and Server Z gains 60 gigabytes.
4
Sum the updated storage values to find the new total storage T2T_2.
T2=1.20x+1.00x+(1.25x+20)=3.45x+20T_2 = 1.20x + 1.00x + (1.25x + 20) = 3.45x + 20.
Adding the three updated amounts gives the new total expression.
5
Set up and solve the equation representing the 10%10\% overall increase (T2=1.10T1T_2 = 1.10 T_1).
3.45x+20=1.10(3.5x−40)  ⟹  3.45x+20=3.85x−44  ⟹  64=0.40x  ⟹  x=1603.45x + 20 = 1.10(3.5x - 40) \implies 3.45x + 20 = 3.85x - 44 \implies 64 = 0.40x \implies x = 160.
Solving the linear algebraic model yields the value for initial storage on Server X.

Anahtar Kavram

Linear algebraic modeling of dynamic multi-variable systems involving percentage changes.
Tahmini Süre:2m 30s
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Algebraic Word Problems and Modeling

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