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Difficulty: Very hardBasic Algebraic Expressions and One-Step Equations

A researcher is cataloging a collection of TT historic documents. She catalogs the documents at a constant rate of rr documents per day. After dd days of cataloging, the remaining documents represent a fraction, ff, of the entire collection. Which of the following equations correctly expresses the total number of documents in the collection, TT, in terms of rr, dd, and ff?

  1. A
    T=rdfT = \frac{rd}{f}
  2. B
    T=rd(1−f)T = rd(1-f)
  3. T=rd1−fT = \frac{rd}{1-f}Answer
  4. D
    T=r(1−f)dT = \frac{r(1-f)}{d}
  5. E
    T=rdfT = rdf

Answer

The correct equation is T=rd1−fT = \frac{rd}{1-f}.
The correct equation is found by identifying that the total number of documents cataloged is the rate rr multiplied by the number of days dd, which equals rdrd. Because the remaining fraction of the collection is ff, the completed fraction of the collection is 1−f1-f. The number of completed documents, rdrd, must equal the completed fraction of the total collection, (1−f)T(1-f)T. Setting these equal gives (1−f)T=rd(1-f)T = rd. Dividing both sides by the group (1−f)(1-f) isolates the total collection, resulting in T=rd1−fT = \frac{rd}{1-f}.

Step-by-Step Solution

1
Calculate the total number of documents cataloged.
The number of cataloged documents is rdrd.
The total quantity completed is the constant rate rr multiplied by the number of days dd.
2
Determine the fraction of the collection that has been cataloged.
The cataloged fraction of the collection is 1−f1-f.
Since the remaining fraction is ff, the completed fraction must be 1−f1-f because the sum of the completed and remaining fractions must equal 1.
3
Set up an equation relating the cataloged documents to the total collection.
(1−f)T=rd(1-f)T = rd
The fraction of the total collection that is cataloged, (1−f)T(1-f)T, must equal the actual number of cataloged documents, rdrd.
4
Solve for the total number of documents, TT.
T=rd1−fT = \frac{rd}{1-f}
Dividing both sides of the equation by the coefficient (1−f)(1-f) isolates TT.

Key Concept

Translating a real-world scenario with rates and fractional parts into a solvable one-step equation.
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Topics

Basic Algebraic Expressions and One-Step Equations

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