Problem 3 - Algebra for College Students

Question

3\. Solve \(i=P r t\) for \(t\), given that \(P=\$ 400, r=11 \%\), and \(i=\$ 132\).

Step-by-Step Solution

Answer

The value of \(t\) is 3.

Step 1: Understand the Problem

The problem provides the formula for interest, which is \(i = P r t\). You need to solve for \(t\) given the values for \(i\), \(P\), and \(r\).

Step 2: Input the Given Values

Substitute the given values into the equation: \(132 = 400 \times 0.11 \times t\). (Note that the interest rate \(r\) is given as a percentage and needs to be converted to a decimal by dividing by 100.)

Step 3: Simplify the Equation

Multiply \(400\) and \(0.11\): \(400 \times 0.11 = 44\). Thus, the equation becomes \(132 = 44t\).

Step 4: Solve for \(t\)

To isolate \(t\), divide both sides of the equation by \(44\): \(t = \frac{132}{44}\).

Step 5: Calculate \(t\)

Calculate \(t\) by dividing: \(t = 3\).

Key Concepts

Interest Calculation

Interest calculation is an essential concept in finance and mathematics for determining how much interest will accumulate over a given period. When dealing with simple interest, like in this exercise, the formula used is \( i = Prt \), where:

• \(i\) is the interest
• \(P\) is the principal amount
• \(r\) is the rate of interest per period, and
• \(t\) is the time the money is invested for.

Simple interest assumes that the interest does not compound over time—interest earned is not reinvested. To apply this, you need to know three of the four variables to find the unknown one. In our problem, we know the principal \(P\), rate \(r\), and interest \(i\), so we solve for \(t\), the time period.
Understanding this formula is crucial as it can be applied to various real-life situations like loans, savings, and investments.

Algebraic Manipulation

Algebraic manipulation is a strategy used to rearrange and simplify equations to isolate a specific variable. In the context of the problem, we aim to solve for \(t\) in the interest equation \(i = Prt\). Here are some steps:

• The first step is to substitute known values into the equation. You replace \(i\) with \(132\), \(P\) with \(400\), and \(r\) with \(0.11\) (note: the rate is given as 11% and needs to be converted to decimal by dividing by 100).
• Once the equation is \(132 = 400 \times 0.11 \times t\), simplify the equation by performing multiplication on the given values which results in \(132 = 44t\).

These steps show how algebraic manipulation helps to rearrange and solve equations efficiently. This skill is essential for solving many algebraic problems and forms the foundation for advanced algebraic concepts.

Equation Simplification

Equation simplification involves reducing an equation to its simplest form to make the problem easier to solve. In the exercise, simplifying involves two main actions:

• First, you multiply the constants together, resulting in the equation \(132 = 44t\).
• Second, isolate \(t\) by dividing both sides of the equation by \(44\). This is a straightforward division operation that leaves you with \(t = \frac{132}{44}\).

Once you've simplified the equation, the final calculation involves finding \(t\) by performing the division, which gives \(t = 3\). This process exemplifies the power of simplification in breaking down complex problems and finding solutions quickly. Being able to simplify equations effectively is fundamental to mastering algebra.