Problem 4 - Prealgebra and Introductory Algebra

Question

Fill in the blank to make a true statement. Two people completed a job. If one person completed \(\frac{t}{30}\) of the job and the other person completed \(\frac{t}{20}\) of the job, then \(\frac{t}{30}+\frac{t}{20}=\) _________.

Step-by-Step Solution

Answer

\(\frac{t}{12}\)

Step 1: Determine the Least Common Multiple (LCM)

The first step is to work out the least common multiple of the denominators 30 and 20. The least common multiple of 20 and 30 is 60. This is because 60 is the smallest number that both 20 and 30 can divide into.

Step 2: Rewrite fractions with common denominator

Now, rewrite the fractions using the common denominator. The fraction \(\frac{t}{30}\) becomes \(\frac{2t}{60}\) and the fraction \(\frac{t}{20}\) becomes \(\frac{3t}{60}\)

Step 3: Add fractions with common denominator

Now simply add these two fractions together: \(\frac{2t}{60}+\frac{3t}{60}=\frac{5t}{60}\)

Step 4: Simplify the added fraction

Finally, simplify the result. Here, the numerator and the denominator can both be divided by 5. So, \(\frac{5t}{60}\) simplifies to \(\frac{t}{12}\)

Key Concepts

Least Common Multiple

When adding fractions, the Least Common Multiple (LCM) is essential because it helps find the smallest shared multiple between denominators. It's crucial for combining fractions that have different denominators into a single fraction. Finding the LCM is the first step in many algebra problems involving addition or subtraction of fractions. Let’s say you have two numbers, 'a' and 'b'. The LCM of 'a' and 'b' is the smallest number that is a multiple of both. For instance, for the numbers 20 and 30, the multiples of 20 are 20, 40, 60, 80, and so on, while the multiples of 30 are 30, 60, 90, etc. The smallest common multiple is 60. Identifying the LCM ensures that you can rewrite fractions to have the same denominator, thus allowing them to be added together correctly.

Common Denominator

The common denominator is a shared denominator between two or more fractions, allowing them to be compared or combined. It's equivalent to the LCM of the original denominators. In algebra, when adding or subtracting fractions, each fraction must be expressed with this common denominator. For instance, if you're working with the fractions \(\frac{t}{30}\) and \(\frac{t}{20}\), the common denominator would be 60, based on the LCM of 30 and 20. Once you rewrite the fractions as \(\frac{2t}{60}\) and \(\frac{3t}{60}\), they can easily be combined because they're like fractions – fractions with the same denominator.

Simplify Fractions

To simplify fractions means to reduce them to their simplest form where the numerator and the denominator are as small as possible while still having the same value. This is achieved by finding a common factor of the numerator and the denominator and dividing both by that number. For example, in the fraction \(\frac{5t}{60}\), the number 5 is a common factor of both the numerator and denominator. By dividing the top and bottom by 5, you simplify the fraction to \(\frac{t}{12}\). Simplifying makes it easier to understand the value of the fraction and at the same time, keeps the expression neat and more manageable. It's like cleaning up your work to make it clear and precise!

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like add, subtract, multiply, and divide) to represent values. When working with algebraic expressions, it's important to follow the rules of mathematical operations and properly manage terms and factors. In the context of adding fractions, algebraic expressions require careful treatment. For instance, in our initial problem, we have the variable 't' in both fractions. Even though the denominators are different, the algebraic expression allows us to find a common denominator and combine the fractions. When \(t\) is divided by the common denominator, we obtain an algebraic expression in its simplest form: \(\frac{t}{12}\). Handling algebraic expressions adeptly is a fundamental skill in algebra.