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Chapter 1: Problem 18

Simplify the given expression. $$ 27 \div 3 \cdot 3 $$

Short Answer

Expert verified
The simplified expression is 27.

Step by step solution

01

Division First

According to the order of operations (PEMDAS/BODMAS), we perform division and multiplication from left to right. Start by dividing 27 by 3: \[ 27 \div 3 = 9 \]
02

Multiplication Next

Now, multiply the result from the first step by 3: \[ 9 \cdot 3 = 27 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Division
Division is the process of splitting a number into equal parts. In arithmetic, it's one of the four basic operations, represented by the division symbol (梅) or a slash (/). To divide is to find how many times one number is contained within another, or to distribute evenly into groups. For example, in the expression 27 梅 3, we are determining how many groups of 3 fit into 27. This operation gives us an answer of 9.

When dealing with division, it's important to note:
  • You can鈥檛 divide by zero; division by zero is undefined.
  • Each division has three main components: the dividend, divisor, and quotient. In 27 梅 3 = 9, 27 is the dividend, 3 is the divisor, and 9 is the quotient.
  • Division can be seen as the inverse of multiplication.
Understanding division is crucial for working through problems involving multiple steps and operations, especially when order matters.
Multiplication
Multiplication is one of the fundamental operations in mathematics, often seen as repeated addition. It's expressed by the multiplication symbol (脳) or sometimes by a dot (路). When you multiply numbers, you're essentially adding a number to itself a specified number of times. For example, in the expression 9 脳 3, you interpret this as adding 9 three times: 9 + 9 + 9 = 27.

Key points to remember about multiplication include:
  • Any number multiplied by one remains unchanged (known as the identity property).
  • Any number multiplied by zero is zero.
  • It is commutative, meaning the order of the numbers does not affect the result (e.g., 3 脳 9 is the same as 9 脳 3).
In the provided exercise, after performing the division, multiplying 9 by 3 gives the final simplified result of 27.
PEMDAS/BODMAS
PEMDAS/BODMAS is an acronym to help remember the order of operations in mathematics. Each letter represents a specific operation that needs to be performed in a particular sequence to get the correct result.

The letters stand for:
  • P (or B) - Parentheses or Brackets: Solve anything in parentheses or brackets first.
  • E (or O) - Exponents or Orders: Calculate exponents or orders (such as powers and roots) next.
  • MD (or DM) - Multiplication and Division: Perform these operations from left to right.
  • AS (or MA) - Addition and Subtraction: Lastly, carry out addition and subtraction from left to right.
This rule ensures that everyone solves equations consistently and correctly. In the exercise example, PEMDAS/BODMAS guides us to handle division (27 梅 3) before multiplication (result 脳 3), adhering to the left-to-right rule within the same precedence level.

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