Problem 108 - Fundamentals of Mathematics

Question

Compute each product using the distributive property. \(35 \cdot 202\)

Step-by-Step Solution

Answer

7070

Step 1: Break Down 202 Into Accessible Parts

To simplify the multiplication, we use the distributive property. First, break down 202 into two parts that are easy to work with: 200 and 2. So, 202 can be written as \[202 = 200 + 2\]

Step 2: Use the Distributive Property

Now apply the distributive property, which states that \(a(b+c) = ab + ac\). In this case, our \(a\) is 35, \(b\) is 200, and \(c\) is 2. \[35 imes 202 = 35 imes (200 + 2)\]

Step 3: Multiply 35 by Each Component

Calculate each part separately:- First, multiply 35 by 200: \[35 imes 200 = (35 imes 2) imes 100 = 70 imes 100 = 7000\]- Next, multiply 35 by 2: \[35 imes 2 = 70\]

Step 4: Add the Two Products Together

Now, add the results from the two individual multiplications:\[7000 + 70 = 7070\]

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation that involves combining groups of equal sizes. In simple terms, when you multiply, you are adding a certain number multiple times. For instance, if you multiply 3 by 4, it means you add three groups of 4 together: 4 + 4 + 4 = 12. Multiplication is not only used in basic arithmetic but also extensively in algebra and higher mathematics to solve equations and problems.
Understanding multiplication's role is essential in learning more complex operations like the distributive property. In the context of this exercise, multiplying 35 by 202 directly can seem challenging. However, by employing strategies and properties like the distributive property, we can simplify such multiplication tasks effectively. This method helps us work with smaller, more manageable numbers.

Breaking Down Numbers

Breaking down numbers into parts is an essential strategy in mathematics, especially when dealing with large numbers. It involves decomposing a number into smaller, simpler components that are easier to handle individually.
This technique is crucial because performing calculations with smaller numbers can make the process quicker and less error-prone. During the exercise, the number 202 was broken down into two parts: 200 and 2. This decomposition takes advantage of the simplicity of multiplying by powers of 10, like 200.

• Breaking down involves finding components that, when added together, give back the original number.
• This technique is widely used in various mathematical concepts like factoring, division, and solving algebraic equations.

Using this approach prepares the ground for applying other mathematical properties efficiently, such as the distributive property, making computations much smoother.

Addition in Multiplication

Addition in multiplication is a technique that leverages the fundamental operations of addition and multiplication together. It is best highlighted by the distributive property, which states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.
This property can be written as: \[a(b + c) = ab + ac\]In our exercise, it ensures that multiplication can be broken into simpler steps:

• First, compute \(35 \times 200\), which simplifies using basic multiplication and understanding of powers of ten, resulting in 7000.
• Next, calculate \(35 \times 2\), getting 70.
• Finally, add the two products: \(7000 + 70 = 7070\).

This method shows how addition is inherently part of multiplication operations and can simplify complex problems. It's a powerful tool because it allows tackling difficult problems by transforming them into smaller, more straightforward operations.