Problem 76 - Prealgebra

Question

Perform the indicated operations. $$15,000 \times \frac{1}{1,000}$$

Step-by-Step Solution

Answer

15

Step 1: Understanding the operation

The problem is asking us to multiply 15,000 by the fraction \( \frac{1}{1,000} \). When multiplying by a fraction, you essentially perform the multiplication for the numerator and then divide by the denominator.

Step 2: Multiplying the numbers

Multiply the whole number 15,000 by the numerator of the fraction, which is 1. This gives us:\[ 15,000 imes 1 = 15,000 \]

Step 3: Dividing the result

Now we divide the result from Step 2 by the denominator of the fraction, which is 1,000. So we have:\[ \frac{15,000}{1,000} = 15 \]

Key Concepts

Multiplication

Multiplication is one of the core operations in arithmetic, and it involves finding the total number of items when you have a certain number of groups, each containing the same number of items. In essence, it's repeated addition. For instance, if you have 3 groups containing 4 apples each, multiplying 3 by 4 gives you the total number of apples.

Here's the basic idea:

• The numbers you multiply are called "factors." In our example, 3 and 4 are factors.
• The result is called the "product." So, the product of 3 and 4 is 12.

In the exercise we started with, we multiplied 15,000 (a large number) by a fraction. When multiplying by a fraction, you typically multiply the whole number by the fraction's numerator first. This is straightforward as you treat the numerator like any other whole factor.

Whether you're multiplying whole numbers or fractions, the fundamentals of the process remain similar. Remembering the order of operations can greatly aid in tackling more complex problems.

Division

Division is essentially the process of finding out how many times a number (the divisor) is contained within another number (the dividend). It's the inverse operation of multiplication. If we're dividing 20 by 4, we're asking how many groups of 4 exist within 20, and the result, or "quotient," is 5.

In our original problem, after multiplying 15,000 by 1 (which didn't change the value), we moved on to division.

• This involves taking the result of the multiplication (15,000) and dividing it by 1,000.
• Here's a simple trick: Dividing by 1,000 is the same as moving the decimal point three places to the left. So, 15,000 becomes 15.

Division can deal with remainders as well (for cases where numbers don't divide evenly), but in this specific example, since 15,000 is a multiple of 1,000, it divided perfectly, leaving no remainder. Understanding division is critical in achieving numerical accuracy in arithmetic.

Fractions

Fractions are a way of representing parts of a whole. A fraction has two components:

• The "numerator," which is the top number, tells you how many parts are being considered.
• The "denominator," the bottom number, tells you the total number of equal parts the whole is divided into.

In our exercise, the fraction was \( \frac{1}{1,000} \), which means 1 part out of 1,000.

Fractions can represent division implicitly. For example, \( \frac{1}{1,000} \) signifies 1 divided by 1,000.

When performing operations with fractions, such as multiplication and division, it's useful to remember that multiplying by a fraction is like scaling down, and dividing by a fraction is the opposite. Working with fractions often requires simplifying or converting them into forms that are easy to work with, reinforcing their versatility and utility in many mathematical contexts.