Distributive Property And Decimal Multiplication Explained

In this comprehensive article, we will delve into two fundamental concepts in mathematics: the distributive property and decimal multiplication. We will explore these concepts through detailed examples and step-by-step solutions, ensuring a thorough understanding for readers of all backgrounds. Our focus will be on verifying the distributive property with given values and performing decimal multiplications efficiently. This exploration will not only enhance your mathematical skills but also provide a solid foundation for more advanced topics. Whether you are a student looking to improve your grades or simply someone interested in mathematics, this article will offer valuable insights and practical knowledge.

The distributive property is a cornerstone of algebra, allowing us to simplify expressions and solve equations more effectively. It states that for any numbers a, b, and c, the following equation holds true:

a ( b + c ) = a × b + a × c

This property essentially means that multiplying a number a by the sum of two numbers (b + c) is the same as multiplying a by each of the numbers individually and then adding the products. To verify this property, we will substitute the given values and perform the calculations on both sides of the equation. This process will not only confirm the property but also provide a clear understanding of how it works in practice.

i) a = 5, b = -7, c = 3

Let's verify the distributive property for the first set of values: a = 5, b = -7, and c = 3. We will substitute these values into the equation and calculate both the left-hand side (LHS) and the right-hand side (RHS) separately.

Left-Hand Side (LHS)

The LHS of the equation is a ( b + c ). Substituting the given values, we get:

5 ( -7 + 3 )

First, we perform the addition inside the parentheses:

-7 + 3 = -4

Now, we multiply 5 by -4:

5 * × *(-4) = -20

So, the LHS is -20.

Right-Hand Side (RHS)

The RHS of the equation is a × b + a × c. Substituting the given values, we get:

5 * × *(-7) + 5 * × *3

First, we perform the multiplications:

5 * × *(-7) = -35 5 * × *3 = 15

Now, we add the products:

-35 + 15 = -20

So, the RHS is -20.

Verification

Since both the LHS and RHS are equal to -20, the distributive property is verified for a = 5, b = -7, and c = 3.

ii) a = -4, b = 11, c = -2

Now, let's verify the distributive property for the second set of values: a = -4, b = 11, and c = -2. We will follow the same process as before, calculating the LHS and RHS separately.

Left-Hand Side (LHS)

The LHS of the equation is a ( b + c ). Substituting the given values, we get:

-4 ( 11 + (-2) )

First, we perform the addition inside the parentheses:

11 + (-2) = 9

Now, we multiply -4 by 9:

-4 * × *9 = -36

So, the LHS is -36.

Right-Hand Side (RHS)

The RHS of the equation is a × b + a × c. Substituting the given values, we get:

-4 * × *11 + (-4) * × *(-2)

First, we perform the multiplications:

-4 * × *11 = -44 -4 * × *(-2) = 8

Now, we add the products:

-44 + 8 = -36

So, the RHS is -36.

Verification

Since both the LHS and RHS are equal to -36, the distributive property is verified for a = -4, b = 11, and c = -2. This confirms that the distributive property holds true for both sets of values, showcasing its universal application in algebraic manipulations. Understanding and applying the distributive property is crucial for simplifying complex expressions and solving equations effectively. By mastering this fundamental concept, students can build a strong foundation in algebra and excel in their mathematical studies.

Decimal multiplication is an essential arithmetic skill that extends the concept of multiplication to numbers with decimal points. It is widely used in various real-world applications, from calculating prices and measurements to scientific and engineering calculations. The key to performing decimal multiplication accurately is to understand the placement of the decimal point in the final product. In this section, we will explore several examples of decimal multiplication, providing a step-by-step guide to mastering this skill. Each example will highlight the importance of counting decimal places and correctly positioning the decimal point in the result, ensuring a clear and thorough understanding of the process.

a) 0.01 × 10

To multiply 0.01 by 10, we can think of 0.01 as one-hundredth. Multiplying by 10 shifts the decimal point one place to the right. Alternatively, we can perform the multiplication directly:

0. 01 * × *10 = 0.1

In this case, 0.01 has two decimal places, and 10 has zero decimal places. The product should have 2 + 0 = 2 decimal places. However, since multiplying by 10 shifts the decimal one place to the right, the result is 0.1, which has one decimal place. This is a simple yet important example that demonstrates the effect of multiplying decimals by powers of 10. Such multiplications are common in everyday calculations, such as converting units or scaling quantities. Understanding this concept helps in performing quick mental calculations and estimating results, which are valuable skills in various practical scenarios.

b) 2.13 × 20

To multiply 2.13 by 20, we can first multiply 2.13 by 2 and then multiply the result by 10. Multiplying by 2:

2. 13 * × *2 = 4.26

Now, multiply 4.26 by 10, which shifts the decimal point one place to the right:

4. 26 * × *10 = 42.6

Alternatively, we can perform the multiplication directly:

2. 13 * × *20 = 42.60

Here, 2.13 has two decimal places, and 20 has zero decimal places. The product should have 2 + 0 = 2 decimal places. However, since the trailing zero doesn't affect the value, we can write the result as 42.6. This example demonstrates a slightly more complex multiplication involving a decimal and a whole number. Breaking down the multiplication into smaller steps, such as multiplying by 2 and then by 10, can make the process more manageable. This approach is particularly useful when dealing with larger numbers or more complex decimals, as it simplifies the calculation and reduces the likelihood of errors.

c) 0.06 × 100

To multiply 0.06 by 100, we can think of 0.06 as six-hundredths. Multiplying by 100 shifts the decimal point two places to the right. Alternatively, we can perform the multiplication directly:

0. 06 * × *100 = 6

In this case, 0.06 has two decimal places, and 100 has zero decimal places. The product should have 2 + 0 = 2 decimal places. However, multiplying by 100 shifts the decimal two places to the right, resulting in 6, which is a whole number. This multiplication highlights the efficiency of multiplying decimals by powers of 10. When multiplying by 100, the decimal point moves two places to the right, effectively scaling the number up by a factor of 100. This concept is fundamental in various applications, such as converting units of measurement (e.g., centimeters to meters) or calculating percentages. Understanding the relationship between decimal places and powers of 10 simplifies many mathematical operations and enhances problem-solving skills.

d) 0.234 × 100

To multiply 0.234 by 100, we multiply the number by 100 which shifts the decimal point two places to the right. Let's perform the multiplication:

0. 234 * × *100 = 23.4

In this example, 0.234 has three decimal places, and 100 has zero decimal places. The product should have 3 + 0 = 3 decimal places. However, multiplying by 100 shifts the decimal two places to the right, resulting in 23.4. This demonstrates how multiplying decimals by powers of 10 affects the position of the decimal point. Understanding this principle is crucial for accurate decimal multiplication and is applicable in various contexts, such as calculating areas, volumes, or financial transactions. The ability to quickly and accurately multiply decimals by powers of 10 is a valuable skill that enhances mathematical proficiency and problem-solving capabilities.

This problem involves understanding the relationship between speed, time, and distance. The fundamental formula that connects these three quantities is:

Distance = Speed × Time

In this scenario, James is riding a bike at a constant speed, and we need to determine the distance he covered in a given time. The problem provides the speed in miles per hour, and we need to calculate the distance in miles. This requires a straightforward application of the formula, emphasizing the importance of using consistent units. Rate problems like this are common in real-life situations, such as planning travel routes, calculating travel times, or determining the efficiency of transportation methods. By solving this problem, we reinforce the practical application of mathematical concepts and develop skills in problem-solving and critical thinking.

Throughout this article, we have explored the distributive property and decimal multiplication through detailed examples and step-by-step solutions. We have verified the distributive property with given values, demonstrating its fundamental role in algebraic manipulations. We have also performed decimal multiplications, highlighting the importance of accurate decimal placement and the effect of multiplying by powers of 10. Furthermore, we have touched upon a rate problem, illustrating the practical application of mathematical concepts in real-world scenarios. By mastering these concepts and techniques, readers can enhance their mathematical proficiency and build a strong foundation for future studies. The examples and explanations provided in this article aim to foster a deeper understanding of mathematical principles and their relevance in everyday life, encouraging a more confident and capable approach to problem-solving.