Problem 34 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(30 t\left(\frac{t-7}{10 t}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(3t - 21\).

Step 1: Distribute the Multiplier

Start by distributing the multiplier, which in this case is 30t, across each term inside the parentheses. The expression is given as:\[ 30t \times \left( \frac{t-7}{10t} \right) \]Now distribute the 30t to the fraction:\[ \frac{30t \times (t-7)}{10t} \] This will allow us to simplify later.

Step 2: Simplify the Fraction

Now, simplify the expression from Step 1 by focusing on common terms. In the fraction \( \frac{30t \times (t-7)}{10t} \), the \( t \) in the numerator and the denominator cancel out. This results in:\[ \frac{30 \times (t-7)}{10} \]

Step 3: Simplify the Remaining Terms

With \( \frac{30(t-7)}{10} \), continue simplifying by canceling common factors between the numerator and the denominator. The 30 and 10 have a greatest common divisor of 10:\[ \frac{30}{10} = 3 \]Thus, the expression simplifies to:\[ 3(t-7) \]

Step 4: Expand and Simplify

Expand the expression \( 3(t-7) \) to remove the parentheses:\[ 3 \times t - 3 \times 7 \]This gives:\[ 3t - 21 \]Therefore, the final simplified expression is \( 3t - 21 \).

Key Concepts

Distributive Property in Polynomial Multiplication

The distributive property is a fundamental concept you will often use in algebra. It allows you to multiply terms outside of parentheses by each term within. Think of it like sharing a task across all participants; each term gets a piece of the action.

In this exercise, we had an expression: \( 30t \left( \frac{t-7}{10t} \right) \). Here, "30t" is distributed across the terms inside the parentheses. This means you will multiply "30t" by everything inside: both "t" and "-7". Doing so brings us to: \[ \frac{30t \times (t-7)}{10t} \]

This step sets the foundation for simplifying later. Remember, the beauty of the distributive property is how it breaks down complex expressions into manageable pieces.

Simplifying Fractions in Algebra

Simplifying fractions in algebra can often seem tricky, but it follows basic principles. You are looking for common factors in the numerator and the denominator that can be canceled out. This reduces the expression to its most understandable form.

In the original step, after using the distributive property, we arrived at: \( \frac{30t \times (t-7)}{10t} \). Notice both the numerator and the denominator have a "t". This means you can cancel out those 't' terms, leaving you with a simpler expression: \[ \frac{30 \times (t-7)}{10} \]

Think of fraction simplification as tidying up your algebraic work. You're ensuring unnecessary parts are removed. This makes it straightforward and easy to interpret.

Understanding the Greatest Common Divisor

The greatest common divisor (GCD) is a powerful tool when simplifying expressions. It represents the largest number that divides two or more numbers without a remainder.

In our problem, once we simplified to \( \frac{30(t-7)}{10} \), the focus shifted to the numbers "30" and "10" outside of any variables. By identifying that the GCD of 30 and 10 is 10, you can reduce the fraction. Dividing both the numerator and the denominator by their GCD, 10, results in the expression: \[ 3(t-7) \]

Harnessing the GCD effectively reduces expressions to their simplest form, making it easier to work with and understand algebraic equations.