Problem 60 - Prealgebra

Question

Perform the indicated operations. \(\frac{5(105-42)}{9}\)

Step-by-Step Solution

Answer

The result of the operation is 35.

Step 1: Simplifying Inside the Parentheses

Start by performing the subtraction inside the parentheses. Calculate \(105 - 42\). This simplifies to \(63\). Thus, the expression becomes \(\frac{5 \times 63}{9}\).

Step 2: Multiplying the Values

Next, multiply the remaining values in the numerator. Calculate \(5 \times 63\). This gives you \(315\). Now the expression looks like \(\frac{315}{9}\).

Step 3: Dividing the Values

Finally, perform the division. Calculate \(\frac{315}{9}\) to get \(35\).

Key Concepts

Simplifying Expressions

The first crucial step in solving math problems often involves simplifying expressions, which helps make complex problems more approachable. In the given problem, we start by simplifying inside the parentheses. Parentheses indicate that you need to perform the operations within them first. This is part of the order of operations, widely known by the acronym PEMDAS:

• P for Parentheses
• E for Exponents
• MD for Multiplication and Division (from left to right)
• AS for Addition and Subtraction (from left to right)

In our expression, we have \[105 - 42\] inside the parentheses. Simplifying this involves performing the subtraction: \[105 - 42 = 63\]Once simplified, the expression becomes \[\frac{5 \times 63}{9}\]. This simplification makes it easier to handle the expression in subsequent steps.

Multiplication

After simplifying the expression within the parentheses, the next operation is multiplication. Multiplication is a mathematical process of combining quantities. Here, the simplified expression \[5 \times 63\] means we need to multiply 5 by 63.Let's break down the multiplication:

• First, consider the number 63 as the sum of simpler products, such as \[(60 + 3)\]
• Multiply 5 by each of these components: \[5 \times 60 = 300\] and \[5 \times 3 = 15\]
• Add these results together: \[300 + 15 = 315\]

This step transforms our expression into \[\frac{315}{9}\]. Multiplying accurately ensures that the division in the following step is straightforward.

Division

The final operation in this problem is division, which is the process of determining how many times one number is contained within another. In our expression \[\frac{315}{9}\], division means finding how many groups of 9 are in 315.Here’s how to approach it:

• Think of division as reverse multiplication: we ask, "What number times 9 equals 315?"
• Based on multiplication tables, or by performing the division, we find that \[315 \div 9 = 35\]
• Thus, the final result of dividing 315 by 9 is 35.

The division step yields our solution, and helps in verifying our calculations from the multiplication step as well.