Problem 70 - Prealgebra

Question

Perform the indicated operations. $$36 \times \frac{9}{1}$$

Step-by-Step Solution

Answer

The result is 324.

Step 1: Identify the Numbers and Operation

The problem involves multiplying two numbers: 36 and the fraction \( \frac{9}{1} \). This means we are asked to perform multiplication.

Step 2: Simplify the Fraction

The fraction \( \frac{9}{1} \) simplifies to the whole number 9. Mathematically it means the same, but it's easier to work with the whole number for multiplication.

Step 3: Perform the Multiplication

Now, multiply the whole numbers: 36 multiplied by 9. Calculate \( 36 \times 9 \).

Step 4: Calculate the Product

Perform the calculation \( 36 \times 9 = 324 \). By multiplying the numbers, we find that the product is 324.

Key Concepts

Whole Numbers

Whole numbers are the basic building blocks in arithmetic. They include all the non-negative numbers without fractions or decimals, such as 0, 1, 2, 3, and so on. In mathematics, whole numbers start from zero and go on infinitely. They are straightforward to work with when it comes to calculations like addition, subtraction, multiplication, and division.
When dealing with whole numbers, it’s vital to understand their properties:

• Whole numbers are infinitely countable.
• They don't include negative numbers or fractions.
• Every whole number can be part of operations like multiplication and division.

Understanding whole numbers is crucial when solving problems that involve them, like the exercise you encountered. For instance, 36 is a whole number that you multiplied with another whole number, 9, derived from the fraction \( \frac{9}{1} \). Knowing this can make operations more intuitive.

Simplifying Fractions

Simplifying fractions is a vital skill in mathematics, especially when you want to ease your calculations. Fractions consist of a numerator and a denominator, like \( \frac{9}{1} \), where 9 is the numerator and 1 is the denominator. Simplifying fractions means reducing them to their simplest form so they are easier to work with, or recognizing when they already present as a whole number.
Here's how you simplify fractions or recognize any whole numbers within them:

• If the numerator is divisible by the denominator, the fraction simplifies to a whole number.
• In the fraction \( \frac{9}{1} \), since any number divided by 1 is itself, this fraction simplifies to 9.
• Simplification helps in making multiplication straightforward, just as you saw in the solution where \( \frac{9}{1} \) was simplified to 9.

By simplifying fractions, you eliminate extra steps and make your calculations swifter and less error-prone.

Basic Arithmetic Operations

Basic arithmetic operations are the foundation of many math problems you'll encounter. These operations include addition, subtraction, multiplication, and division. In this exercise, you are dealing with multiplication, one of these basic operations.
Understanding basic operations is fundamental:

• Addition: Combining two or more numbers to get a sum.
• Subtraction: Finding the difference between two numbers.
• Multiplication: Repeated addition of a number, such as \( 36 \times 9 \), which means adding 36 nine times.
• Division: Splitting a number into equal parts.

For multiplication particularly, knowing your times tables can simplify tasks. For instance, multiplying 36 by 9 involves finding the repeated sum of 36. Following through with these fundamental skills allows you to perform operations accurately and efficiently. This exercise's computation, \( 36 \times 9 = 324 \), demonstrates the practicality of understanding multiplication within the realm of basic arithmetic operations.