Question

A typist has already typed 3,500 words of a document. How many total words, \(W(t)\), of the document will he have typed if he can type 70 words per minute and types for an additional \(t\) minutes? (A) \(W(t)=3,500 t\) (B) \(W(t)=70 \mathrm{t}-3,500\) (C) \(W(t)=3,500 \mathrm{t}+70\) (D) \(W(t)=3,500+70 t\)

Short answer

The correct answer is (D) \( W(t)=3,500+70t \).

Step-by-step solution

Understand the Problem

The typist has already typed 3,500 words. He types at a rate of 70 words per minute for an additional \( t \) minutes. We need to find the total number of words \( W(t) \) he will have typed after \( t \) more minutes of typing.

Set Up the Equation

The typist adds new words to the already typed 3,500 words. He types \( 70 \) words each minute, and for \( t \) minutes, the additional words are calculated as \( 70t \). So, the total words typed after \( t \) minutes is the sum of the already typed words and the additional words: \[ W(t) = 3,500 + 70t \]

Match with Given Options

Compare our equation \( W(t) = 3,500 + 70t \) to the provided answer choices. Our equation matches option (D): \( W(t) = 3,500 + 70t \).

Key concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations. They allow us to represent real-world problems in a mathematical form, helping us solve these problems more easily. In this scenario, the algebraic expression helps us calculate the total number of words a typist has typed.

To set up an algebraic expression, identify the constant values and variables in the problem. Here, 3,500 is the constant number of words already typed, and 70 is the rate of typing words per minute. The variable is the time in minutes, denoted as \( t \). These components combine to form the expression \( W(t) = 3,500 + 70t \), which tells us how many words have been typed after \( t \) additional minutes.

• First, recognize the initial value: 3,500 words are already typed.
• Next, understand the rate: 70 words per minute.
• Finally, use the variable \( t \) to represent time in minutes.

By putting these elements together, you create an expression that can dynamically calculate total words typed for varying amounts of time. This process exemplifies how algebra simplifies complex problems into solvable equations.

Rate of Work Problems

Rate of work problems involve calculating how much work is done over a period of time. They often involve using rates to find the quantity of work completed. In the typist problem, the typist's work rate is 70 words per minute. Rate of work problems follow a common pattern and formula:

\[ \text{Work Done} = \text{Rate} \times \text{Time} \]
In our case, the work done is the additional number of words typed, the rate is 70 words per minute, and the time is \( t \) minutes. The additional words typed while the typist continues to type for \( t \) minutes are calculated as \( 70t \).

• Identify the rate: How fast is the work being done? Here, it's 70 words per minute.
• Determine the time: How long will the work continue? In this problem, it's represented by \( t \).
• Compute the work done: Multiply the rate by the time to find out additional words typed. Hence, \( 70t \).

Add this to the already completed 3,500 words for the total typed. Understanding these problems can greatly assist in a variety of real-world tasks, from calculating labor work to determining project timelines.

PSAT Math Questions

The PSAT, or Preliminary SAT, includes math questions designed to test a range of skills, including algebra, arithmetic, geometry, and problem-solving. By understanding how to approach these questions, you're better prepared for both the PSAT and other standardized tests.
To solve PSAT math questions effectively, it's important to follow a systematic approach. First, carefully read the problem to understand what is being asked. For the typist's problem, we determine the working rate and duration to calculate total word output. Make sure to extract key pieces of data that translate into mathematical operations, such as the rate (70 words/minute) and initial words typed (3,500).

• Focus on understanding what is being asked in the question. Look for keywords.
• Translate the problem into a mathematical equation using known quantities and variables.
• Cross-check your equation with given options to ensure accuracy.

Developing these skills in addressing PSAT math questions, prepares you not only for exams but also for mathematical problems you might encounter in everyday life. Practice and familiarity with such structured approaches will enable you to tackle problems confidently and with ease.