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Chapter 2: Problem 12

Solve each formula for the indicated letter. \(I=P r t,\) for \(t\)

Short Answer

Expert verified
t = \frac{I}{P r}

Step by step solution

01

- Understand the Problem

The given formula is the simple interest formula: \(I = P r t\). The goal is to solve this formula for the variable \(t\).
02

- Isolate the Variable

To isolate \(t\), we need to divide both sides of the equation by \(P r\). Start by writing: \[ t = \frac{I}{P r} \]
03

- Final Solution

The formula solved for \(t\) is: \[ t = \frac{I}{P r} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Interest Formula
The simple interest (SI) formula is essential in finance and mathematics. It's written as \(I = P r t\). Here:
  • \textbf{I} represents the interest earned
  • \textbf{P} stands for the principal amount or initial sum of money
  • \textbf{r} is the interest rate (expressed in decimal form)
  • \textbf{t} denotes the time period the money is invested or borrowed for
Our goal is to solve for \(t\).
That means we need to find out how much time passes given a specific amount of interest earned, a certain principal amount, and a fixed interest rate. Understanding this formula is crucial because it shows how changing one variable affects the others.
Isolating Variables
In mathematics, isolating a variable involves getting the variable by itself on one side of the equation. This is a fundamental skill when solving equations.
For example, to solve for \(t\) in the simple interest formula \(I = P r t\), we need \(t\) by itself on one side.
Here's how we isolate \(t\):
  • Start from the formula \(I = P r t\).
  • Divide both sides of the equation by \(P r\). This keeps the equation balanced.
  • We get \( \frac{I}{P r} = t \).
This means \( t = \frac{I}{P r} \).
Now, we have successfully isolated \(t\). It's now easy to substitute values and find the duration if we're given the interest, principal, and rate.
Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and simplifying equations using various algebraic rules.
Let's take a closer look at the steps involved in solving for \(t\) in the simple interest formula:
First, write the equation: \( I = P r t \).
To isolate \(t\), we need to clear the other variables and constants from its side of the equation. To do this:
  • We divide both sides by \(P r\), because \(P r\) is multiplied by \(t\): \[ \frac{I}{P r} = t \]
  • Now, \(t\) is isolated, and we obtain the final equation: \[ t = \frac{I}{P r} \]
This process involves understanding how to use inverse operations.
When \(P r\) multiplies \(t\), we use division to 'undo' it.
Understanding algebraic manipulation is key to solving more complex equations.
It provides a systematic approach to finding unknown values.

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