Question

A sum of money \(a\) when invested for \(t\) years at an interest rate \(n\) will accumulate to an amount \(y,\) where \(y=a+a n t .\) Factor the right side of this equation.

Short answer

The factored form of the equation is \(y = a(1 + n t)\).

Step-by-step solution

Understand the Given Equation

The given equation is a formula for calculating the accumulated amount of a sum of money with simple interest: \(y = a + a n t\). In this formula, \(a\) is the initial amount, \(n\) is the interest rate, and \(t\) is the time in years. The goal is to factor the right side of the equation.

Factor Out the Common Factor

The common factor on the right side of the equation is \(a\). We can factor \(a\) out from both terms to simplify the equation: \(y = a(1 + n t)\). By factoring \(a\) out, we're applying the distributive property in reverse.

Write the Final Factored Equation

With the common factor factored out, the final factored form of the equation is: \(y = a(1 + n t)\). This equation shows the relationship between the initial sum, the interest rate, the time, and the final accumulated amount.

Key concepts

Simple Interest Formula

Understanding the simple interest formula is crucial for anyone dealing with loans, investments, or any financial activities where interest is involved. It helps you calculate the interest earned or paid on a given principal amount over a specific period.

The formula for simple interest is usually expressed as:
I = Prt,
where I is the interest accrued, P represents the principal amount (initial sum of money), r denotes the interest rate, and t signifies the time period in years. Interest rate is typically provided as a decimal, so an interest rate of 5% would be 0.05 in the formula.

The beauty of the simple interest formula lies in its directness—interest is calculated only on the original principal, which means that it doesn't compound over time. This makes it a great introductory concept for students beginning their journey into understanding interest and finance.

In the context of our original exercise, we can spot the simple interest formula within it: y = a + ant. Here, the term ant represents the interest (I) calculated by the simple interest formula, while a reminds us of the principal amount.

Distributive Property

The distributive property is an essential algebraic rule that allows us to multiply a single term by two or more terms inside a set of parentheses. In other words, it distributes the multiplication across the terms inside the parentheses.

Formally, it can be stated as:
a(b + c) = ab + ac,
where a, b, and c are algebraic expressions. This property is particularly useful when simplifying algebraic expressions or solving equations.

Let’s see an example using numbers:

• If you have an expression like 3(4 + 5), using the distributive property, you can simplify this to 3*4 + 3*5, which equals 12 + 15 = 27.

In our original exercise with the simple interest formula, when factoring out a from a + ant, we reversed the distributive property. We took the sum of two terms, both sharing a common factor, and we factored out that common element, which is a key step in simplifying expressions.

Mathematical Factorization

Mathematical factorization is a crucial process in algebra that involves breaking down a composite number or expression into a product of smaller or simpler factors. It's much like finding out which ingredients went into making a cake—the cake being the composite number or expression.

When factorizing, the goal is typically to rewrite the expression as a product of its factors as succinctly as possible. For instance, the number 12 can be factorized into 2 * 2 * 3 or, more neatly, into 2^2 * 3.

In the realm of algebraic expressions, factorization could involve:

• Taking out a common factor, as we did in the given exercise.
• Using special products formulas, such as squaring a binomial.
• Factoring by grouping, which involves rearranging terms and factoring in pairs or groups.
• Applying advanced techniques like completing the square or using the quadratic formula if applicable.

In our initial exercise, factorizing a + ant into a(1 + nt) simplifies the expression and reveals the common factor—much like uncovering the basic building blocks of an algebraic 'cake'. This not only tidies up the equation, making it more manageable and easier to work with, but it also highlights the relationship between components in the formula for the accumulated sum.