Problem 28 - Algebra 1

Question

\(\frac{18}{t}\) when \(t=3\)

Step-by-Step Solution

Answer

The value of the fraction \(\frac{18}{t}\) when \(t=3\) is 6.

Step 1: Understand the Given Problem

We are given a fraction \(\frac{18}{t}\), where \(t\) is a variable. We are asked to find the value of this fraction when \(t=3\). Do not forget, division by zero is undefined in mathematics, hence, \(t\) should never equal zero.

Step 2: Substitute the Given Value of \(t\)

Since we are told that \(t=3\), we will substitute 3 for \(t\) in the fraction \(\frac{18}{t}\). After substitution, our fraction becomes \(\frac{18}{3}\). Now, we only need to do the division.

Step 3: Perform the Division

Now, simply divide the numerator (18) by the denominator (3). When you divide 18 by 3, you get 6.

Key Concepts

Variable Substitution

When solving algebraic expressions that include variables, a common technique used is variable substitution. This method involves replacing the variable with a specific value that is given or determined by the context of the problem. It's like stepping into the shoes of the variable and seeing the world from its perspective!

For example, when you're given an equation like \(\frac{18}{t}\) and tasked to evaluate it at \(t=3\), you're essentially being asked to 'swap out' the \(t\) with 3. This is a straightforward process but it's important to follow the rules of algebra meticulously to avoid errors. Remember, before you substitute, always ensure that you're not dividing by zero, as this is undefined and can lead to mathematical contradictions. Once the substitution is done correctly, the equation can be simplified further.

Simplifying Fractions

A fraction represents a part of a whole, and simplifying fractions is a fundamental skill in algebra. This process involves reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than one, making the expression cleaner and easier to understand.

In simplifying the fraction \(\frac{18}{3}\) after variable substitution, you look for a common factor that divides both the numerator (18) and the denominator (3). Here, the number 3 is a common factor. Dividing both the top and bottom by 3, the fraction simplifies to 6, which is the fraction in its simplest form. It's essential to fully simplify fractions to achieve the most precise and straightforward form of the answer.

Performing Division in Algebra

When you work with algebraic equations, you often need to perform division among numbers and variables. Division in algebra follows the same rules as arithmetic division: you simply divide the numerator by the denominator. It's like sharing a pie equally among a number of friends!

In the case of our example, \(\frac{18}{3}\), we're sharing 18 units equally among 3 parts. Doing the division gives us 6, meaning each part receives 6 units. It's crucial to understand division well because it lays the foundation for more advanced concepts in algebra, such as factorizing expressions and solving equations. Whenever performing division, ensure that the denominator is never zero, as this would make the division undefined and thus, not solvable within the realm of real numbers.