Problem 65 - Prealgebra

Question

Perform the indicated operations. $$864 \div 144$$

Step-by-Step Solution

Answer

The result of dividing 864 by 144 is 6.

Step 1: Analyze the Division Problem

We need to divide 864 by 144. This can be expressed as \( \frac{864}{144} \). We aim to find how many times 144 fits completely into 864.

Step 2: Perform the Division

Divide 864 by 144 by calculating \( 864 \div 144 \). 144 goes into 864 six times completely. Therefore, the division yields: \( 6 \).

Step 3: Verify the Result by Multiplication

We verify the result by multiplying the quotient by the divisor: \( 6 \times 144 = 864 \). Since the product equals the original dividend, the solution is verified.

Key Concepts

Understanding the Quotient

When you perform a division operation like \( 864 \div 144 \), the answer you get is known as the quotient. Here, the quotient is \( 6 \). But what does this really mean? In simple terms, the quotient tells us how many times the divisor, \( 144 \), can fit completely into the dividend, \( 864 \).

Think of division as a way of splitting up a number into equal parts. The quotient represents how many of these parts we can create. For example, if you have 864 candies and want to distribute them into groups of 144, you can form 6 full groups. That's how the quotient helps in practical scenarios.

Also, keep in mind that a quotient is often a whole number. However, if there's a remainder, the quotient will include fractions or decimal values as well.

The Role of a Divisor

In the division process, the divisor is the number by which we divide the dividend. For the division \( 864 \div 144 \), 144 is the divisor. This tells us we are trying to see how many times 144 can "fit" into 864.

Divisors play a critical role in division. They essentially determine the size of each part or group formed when dividing the dividend. When the divisor is large, there are usually fewer parts, and when it's smaller, there are more. For instance, dividing 864 by a larger number would result in fewer groups compared to dividing it by 144.

Understanding the role of a divisor can help you with mental math and improve your intuitive grasp on dividing numbers, especially when dealing with large numbers.

Multiplication Verification

After obtaining a quotient, it's a good practice to verify it using multiplication. This is an important step as it confirms the accuracy of your division.

In our example, we found that the quotient of \( 864 \div 144 \) is \( 6 \). To check this, we multiply the quotient \( 6 \) by the divisor \( 144 \). This operation should give us back the original dividend: \( 6 \times 144 = 864 \). Since our product matches the original dividend, it verifies that the division was done correctly.

This process of multiplication verification is helpful not only for confirming results but also for strengthening your understanding of the relationship between multiplication and division. It also provides a great opportunity to practice your multiplication skills.