Problem 3 - Intermediate Algebra

Question

Use the formula to solve for the given variable. Solve \(i=\) Prt for \(t\), given that P= 4000 dollars, r=5 %, and \(i\)= 600 dollars.

Step-by-Step Solution

Answer

The time \(t\) is 3 years.

Step 1: Understand the given formula

The formula provided is for simple interest, which is \( i = P \times r \times t \). This formula relates interest (\(i\)), principal amount (\(P\)), interest rate (\(r\)), and time (\(t\)). Our task is to solve for the variable \(t\).

Step 2: Rearrange the formula to solve for \(t\)

We need to isolate \(t\) on one side of the equation. Starting with the formula \( i = P \times r \times t \), divide both sides by \(P \times r\) to get: \( t = \frac{i}{P \times r} \).

Step 3: Substitute the given values into the formula

Substitute the given values: \(P = 4000\), \(r = 0.05\) (note that 5% is the same as 0.05), and \(i = 600\) dollars into the formula we found in Step 2:\[ t = \frac{600}{4000 \times 0.05} \].

Step 4: Perform the calculation

Calculate the denominator first: \(4000 \times 0.05 = 200\).Now substitute back into the equation for \(t\):\[ t = \frac{600}{200} \].Perform the division:\( t = 3 \).

Key Concepts

Understanding Simple Interest

Simple interest is a straightforward way to calculate the interest you earn or owe on a sum of money. It's based on the principal amount — the initial amount of money — a fixed interest rate, and the time period the money is borrowed or invested. The formula for simple interest is:

• $i\; =\; P\; \backslash times\; r\; \backslash times\; t$

Where:- \(i\) is the interest generated.- \(P\) is the principal amount or original sum.- \(r\) is the rate of interest per time period.- \(t\) is the time period for which the interest is calculated.One of the key characteristics of simple interest is that it does not compound. This means the interest earned is only on the principal amount, not on any accumulated interest. This type of interest calculation is common in short-term loans or savings accounts.

Solving Equations in Context

Solving equations involves finding the value of an unknown variable. In algebra, you'll often encounter formulas where you need to solve for a specific variable. Let's walk through how this process works using simple interest as an example.First, you're given the formula: \(i = P \times r \times t\). If the problem asks you to solve for \(t\), your primary goal is to isolate \(t\) on one side of the equation. This process involves doing the same operation to both sides of the equation to keep it balanced.Steps to rearrange the equation:1. Start with the equation: \(i = P \times r \times t\).2. To solve for \(t\), divide both sides by \(P \times r\):

• \(t = \frac{i}{P \times r}\)

This gives you a new equation where \(t\) is isolated, making it simple to plug in your known values and solve for \(t\). Rearranging formulas and solving them is a fundamental skill in algebra, often used in finance, physics, and engineering.

Understanding Interest Rate Calculations

Interest rate calculations are crucial when working with financial math. The interest rate is typically expressed as a percentage and determines how much the principal amount will earn or owe over a particular time period.When calculating simple interest, convert the percentage to a decimal by dividing by 100. For instance, a 5% interest rate becomes \(0.05\) in decimal form. This conversion is necessary for accurate calculation in the simple interest formula.Here's how you substitute and calculate with a given interest rate:- Given rate \(r = 5\%\)- Convert to decimal: \(r = 0.05\)By substituting this rate into the calculation, it helps you determine how much interest is generated for each dollar of the principal per time period.Accurate interest rate calculations affect the outcomes of financial decisions, savings, and investments. It helps in predicting how costs or earnings will evolve over time, assisting with planning and budgeting.