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Difficulty: Very hardAlgebraic Word Problems and Equation Modeling

Two fulfillment centers, Center X and Center Y, process customer orders at constant individual rates of xx orders per hour and yy orders per hour, respectively, where x>y>0x > y > 0. Working together simultaneously, Center X and Center Y complete a total batch of NN orders in TT hours. In a separate shift, Center X works alone for hh hours (where 0<h<T0 < h < T) to process a portion of the NN orders, after which Center Y works alone for kk hours to finish all remaining orders in the batch.

Which of the following algebraic statements MUST be true? Select all that apply.

  1. The ratio of the processing rates satisfies xy=k−TT−h\frac{x}{y} = \frac{k - T}{T - h}.Answer
  2. The processing rate of Center Y is expressed as y=N(T−h)T(k−h)y = \frac{N(T - h)}{T(k - h)}.Answer
  3. C
    The total sequential processing time satisfies h+k=2Th + k = 2T.
  4. D
    The ratio of the processing rates simplifies directly to xy=kh\frac{x}{y} = \frac{k}{h}.
  5. E
    The processing rate of Center X is expressed as x=N(k−T)T(k+h)x = \frac{N(k - T)}{T(k + h)}.

Answer

The valid algebraic statements are the rate ratio xy=k−TT−h\frac{x}{y} = \frac{k - T}{T - h} and the expression for Center Y's rate y=N(T−h)T(k−h)y = \frac{N(T - h)}{T(k - h)}.
Equating the total work from combined operation N=(x+y)TN = (x + y)T and sequential operation N=xh+ykN = xh + yk yields x(T−h)=y(k−T)x(T - h) = y(k - T), which proves that the ratio xy=k−TT−h\frac{x}{y} = \frac{k - T}{T - h} must be true. Furthermore, substituting x=NT−yx = \frac{N}{T} - y into xh+yk=Nxh + yk = N gives NhT+y(k−h)=N\frac{Nh}{T} + y(k - h) = N, which simplifies to y=N(T−h)T(k−h)y = \frac{N(T - h)}{T(k - h)}. Therefore, both of these algebraic formulations are correct.

Step-by-Step Solution

1
Formulate the total work equations for both operational scenarios.
From simultaneous operation: N=(x+y)T=xT+yTN = (x + y)T = xT + yT. From sequential operation: N=xh+ykN = xh + yk.
Total work completed in both scenarios equals the batch size NN.
2
Equate the two expressions for NN to derive the relationship between rates and times.
xT+yT=xh+yk  ⟹  xT−xh=yk−yT  ⟹  x(T−h)=y(k−T)xT + yT = xh + yk \implies xT - xh = yk - yT \implies x(T - h) = y(k - T).
Grouping like rate terms enables solving for the rate ratio xy\frac{x}{y}.
3
Divide by y(T−h)y(T - h) to find the rate ratio xy\frac{x}{y}.
\frac{x}{y} = \frac{k - T}{T - h}.
Since 0<h<T0 < h < T, T−h>0T - h > 0, so the division is valid and confirms the rate ratio relation.
4
Express xx in terms of y,N,Ty, N, T and substitute into the sequential work equation.
x=NT−y  ⟹  (NT−y)h+yk=N  ⟹  NhT+y(k−h)=Nx = \frac{N}{T} - y \implies \left(\frac{N}{T} - y\right)h + yk = N \implies \frac{Nh}{T} + y(k - h) = N.
Eliminating xx isolates yy in terms of given parameters N,T,h,kN, T, h, k.
5
Solve for yy.
y(k−h)=N−NhT=N(1−hT)=N(T−h)T  ⟹  y=N(T−h)T(k−h)y(k - h) = N - \frac{Nh}{T} = N\left(1 - \frac{h}{T}\right) = \frac{N(T - h)}{T} \implies y = \frac{N(T - h)}{T(k - h)}.
This establishes the exact formula for yy.

Key Concept

Multi-variable system modeling of rate, time, and work constraints.
Estimated Time:2m 30s
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Topics

Algebraic Word Problems and Equation Modeling

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