Problem 48 - Prealgebra

Question

Mary and Tabitha ran in a race. Mary's time was 12 minutes, which was \(\frac{3}{4}\) of Tabitha's time. Using \(t\) for Tabitha's time, which equation represents the situation? \(\begin{array}{ll}\text { A } \frac{3}{4} t=12 & \text { C } t-\frac{3}{4}=12\end{array}\) B \(t+\frac{3}{4}=12\) D \(12 t=\frac{3}{4}\)

Step-by-Step Solution

Answer

The correct equation is \(\frac{3}{4} t = 12\) (Option A).

Step 1: Understand the Relationship

Mary's time is given as 12 minutes, and it is stated to be \(\frac{3}{4}\) of Tabitha's time. This indicates that Tabitha's time is longer since taking \(\frac{3}{4}\) of it gives Mary's time.

Step 2: Set Up the Proportion

Since Mary's time (12 minutes) is \(\frac{3}{4}\) of Tabitha's time \(t\), we translate this statement into the equation \(\frac{3}{4} \times t = 12\).

Step 3: Identify the Correct Option

Compare the set up equation \(\frac{3}{4} \times t = 12\) with the given options. This matches option A: \(\frac{3}{4} t = 12\).

Key Concepts

Understanding Proportions

Proportions are mathematical expressions that show how two different quantities relate to each other. They are often found in equations where one quantity is a fraction or multiple of another. In the problem with Mary and Tabitha, Mary's time is expressed as a fraction of Tabitha's time. This is an example of a proportion. To understand this concept better, consider the formula used in the solution:

• The proportion being represented is "Mary's time is \(\frac{3}{4}\) of Tabitha's time".
• In mathematical terms, this means multiplying Tabitha's time by \(\frac{3}{4}\) equals Mary's time.
• Therefore, the equation \(\frac{3}{4} \times t = 12\) shows this proportional relationship succinctly.

Proportions are very useful for comparing quantities and solving problems in real-world scenarios, such as this race. By understanding how proportions work, you can solve many similar problems by setting up equations that represent real-life situations accurately.

Working with Equations

Equations are mathematical statements that assert the equality of two expressions. In solving word problems, setting up an equation is often a critical step in finding the solution. In this exercise, the main task is to create an equation based on the word problem description.

• Start by identifying what each part of the sentence means mathematically. "Mary's time was \(\frac{3}{4}\) of Tabitha's time" is translated into the equation \(\frac{3}{4} \times t = 12\).
• On the left side of the equation, you have \(\frac{3}{4} \times t\), which represents the operation you need to perform on \( t \) to get Mary's time.
• On the right side, 12 is Mary's time in minutes.

Solving the equation involves isolating the variable \(t\) to find out the time Tabitha took. This is accomplished by performing the inverse operation. Equations like this often play a role in finding unknown values when certain conditions are known.

Exploring Variables

Variables are symbols used in mathematics to represent unknown or changeable values. They allow us to formulate equations and solve problems where not all quantities are known. In the present problem, the variable \(t\) is used to represent the time Tabitha took to finish the race.

• Choosing \(t\) as a variable helps in setting up a general equation where you can input any specific values to find unknowns.
• Variables are placeholders that must be solved for by using the information given and applying mathematical operations.
• In this case, "\(t\)" helps you work out the equation \(\frac{3}{4} \times t = 12\) and ultimately find Tabitha's race time.

Variables provide flexibility and allow mathematicians and students alike to explore multiple scenarios just by replacing numbers into equations. They are fundamental in learning algebra and prealgebra as they bridge the gap between conceptual understanding and computational skills.