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Zorluk: OrtaLinear Equations in One Variable

Two water reservoirs, Reservoir A and Reservoir B, contain a combined total of 360360 liters of water. After 3030 liters of water are transferred from Reservoir A to Reservoir B, the volume of water in Reservoir B is equal to 23\frac{2}{3} of the volume of water remaining in Reservoir A. How many liters of water were originally in Reservoir A?

  1. A
    114114
  2. B
    210210
  3. C
    216216
  4. 246246Cevap
  5. E
    228228

Cevap

246 liters were originally in Reservoir A.
Let xx represent the original volume of Reservoir A. Since the combined volume is 360360 liters, Reservoir B initially contains 360−x360 - x liters. After transferring 3030 liters from A to B, Reservoir A contains x−30x - 30 liters and Reservoir B contains (360−x)+30=390−x(360 - x) + 30 = 390 - x liters. Setting up the equation 390−x=23(x−30)390 - x = \frac{2}{3}(x - 30) and multiplying by 3 gives 1170−3x=2x−601170 - 3x = 2x - 60. Combining like terms yields 5x=12305x = 1230, so x=246x = 246 liters. Thus, 246246 is the correct original volume.

Adım Adım Çözüm

1
Define the variable for the unknown quantity.
Let xx be the original volume of water in Reservoir A (in liters).
Choosing a single variable simplifies setting up a linear equation.
2
Express the original volume of Reservoir B in terms of xx.
Original volume in Reservoir B is 360−x360 - x.
The total volume across both reservoirs is given as 360360 liters.
3
Write expressions for the volumes in each reservoir after the transfer of 30 liters.
Volume in Reservoir A after transfer: x−30x - 30.
Volume in Reservoir B after transfer: (360−x)+30=390−x(360 - x) + 30 = 390 - x.
Transferring 30 liters removes 30 liters from A and adds 30 liters to B.
4
Set up the linear equation based on the given relationship.
390−x=23(x−30)390 - x = \frac{2}{3}(x - 30)
The problem specifies that the new volume in Reservoir B is 23\frac{2}{3} of the new volume in Reservoir A.
5
Solve the equation for xx.
3(390−x)=2(x−30)  ⟹  1170−3x=2x−60  ⟹  1230=5x  ⟹  x=2463(390 - x) = 2(x - 30) \implies 1170 - 3x = 2x - 60 \implies 1230 = 5x \implies x = 246.
Multiply both sides by 3 to eliminate the fraction, then collect like terms.

Anahtar Kavram

Formulating and solving a linear equation in one variable from a real-world conservation/transfer scenario.
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Konular

Linear Equations in One Variable

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