Problem 55 - Algebra 1: Concepts and Skills

Question

SIMPLIFYING EXPRESSIONS Simplify the expression without using a calculator. $$ \left(\frac{30}{10}\right)(5) $$

Step-by-Step Solution

Answer

The simplified expression is \(15\)

Step 1: Understand the basic arithmetic operation

The expression provided consists of the multiplication of a fraction \(\left(\frac{30}{10}\right)\) and an integer \(5\). It's important to remember that multiplication can be performed in any order.

Step 2: Simplify the fraction

Before multiplying, simplify the fraction \(\frac{30}{10}\) to reduce it to its lowest form. 30 divided by 10 equals 3.

Step 3: Multiply the simplified fraction with the integer

Now that the fraction is simplified down to 3, multiply it with the integer 5. The arithmetic rule for multiplication states that we simply multiply the two numbers together to get the result.

Step 4: Result

Multiplying 3 (the simplified fraction) by 5 (the given integer) equals 15

Key Concepts

Arithmetic Operations

When approaching any math problem, it's fundamental to understand the arithmetic operations involved. In mathematics, arithmetic operations typically include addition, subtraction, multiplication, and division.
These operations form the basis of mathematical calculations and are essential for simplifying expressions.

• **Addition** combines numbers to find a sum, while **subtraction** finds the difference between numbers.
• **Multiplication** involves finding the product by repeating addition and is used for scaling numbers.
• **Division** determines how many times one number is contained within another.

In our exercise, the expression involves multiplying a fraction and an integer. Recognizing this multiplication operation helps us know how to proceed with simplifying the expression. Understanding how different operations interact allows you to simplify expressions accurately and efficiently.

Fraction Simplification

Fractions consist of a numerator (top number) and a denominator (bottom number), and simplifying them helps to make calculations easier. Simplifying involves reducing the fraction to its smallest possible equal value by dividing both the numerator and the denominator by their greatest common factor (GCF).
In this exercise, the fraction \(\frac{30}{10}\) is simplified by dividing both 30 and 10 by their GCF, which is 10. Thus, \(\frac{30}{10}\) simplifies to 3.

• Identify the GCF of the numerator and denominator.
• Divide both numbers by this GCF.
• The result is a simplified fraction that retains the same value.

It's important to simplify fractions in calculations like these because it makes the arithmetic much simpler and helps reduce errors in further steps.

Multiplication Rules

Multiplication is one of the basic arithmetic operations and involves combining equal groups. When multiplying any two numbers, the order doesn't matter due to the commutative property of multiplication. That is, \(a \times b = b \times a\) for any numbers \(a\) and \(b\).
For multiplication involving fractions, the rule is straightforward. Once a fraction is simplified, multiplying it by an integer involves multiplying the simplified value with the integer.
In the exercise, after simplifying \(\frac{30}{10}\) to 3, we multiply this by 5, resulting in 15. Just follow these simple steps:

• Ensure the fraction is simplified.
• Multiply the simplified fraction by the integer by straightforward multiplication.
• Don't forget to double-check calculations for accuracy.

Being aware of these multiplication rules is crucial, as they help you easily and correctly solve problems involving both numbers and expressions.