Problem 17 - Fundamentals of Mathematics

Question

Use the distributive property to compute each product. \(35 \cdot 12\)

Step-by-Step Solution

Answer

420

Step 1: Break Down the Numbers

To use the distributive property, we first break down one of the numbers into a sum. Here, we can write 12 as \(10 + 2\). Therefore, the expression becomes \(35 \cdot (10 + 2)\).

Step 2: Apply the Distributive Property

According to the distributive property, \(a(b+c) = ab + ac\). We use this property to distribute 35 over the sum \(10 + 2\). Thus, \(35 \cdot (10 + 2) = 35 \cdot 10 + 35 \cdot 2\).

Step 3: Calculate Each Product

Now, compute each product separately. First, \(35 \cdot 10 = 350\). Then, \(35 \cdot 2 = 70\).

Step 4: Add the Results

Finally, add the results of the two products to find the final answer. So, \(350 + 70 = 420\).

Key Concepts

Arithmetic Operations

Arithmetic operations are fundamental skills in mathematics that involve basic numerical computations. They include operations such as addition, subtraction, multiplication, and division. These operations form the building blocks for more complex mathematical concepts.
In the given exercise, we focus on multiplication. Multiplication is essentially repeated addition. For example, multiplying 35 by 12 means adding 35 to itself 12 times. However, performing such a calculation directly can be cumbersome for large numbers, which is why we use certain strategies to simplify it.
By understanding and mastering arithmetic operations, students can carry out these calculations with ease, setting a strong foundation for advanced math topics.

Multiplication Strategies

Multiplication strategies are techniques used to simplify the process of multiplying numbers, especially those that are larger or more complex. These strategies are crucial in mathematical problem-solving.
One of the most effective strategies is using the distributive property. It allows us to break down a difficult multiplication task into simpler ones. In our example problem, we break down the number 12 into 10 and 2, making it easier to multiply by 35.

• The distributive property states that \( a(b+c) = ab + ac \).
• In application, this means we can distribute 35 over the sum \( 10 + 2 \).

Thus, the original expression \( 35 \times 12 \) turns into smaller problems: \( 35 \times 10 \) and \( 35 \times 2 \). This strategy makes it much easier to handle large numbers mentally or on paper.

Mathematics Problem Solving

Mathematics problem-solving involves using a variety of strategies and logical reasoning to find solutions to mathematical problems. The distributive property, used in our exercise example, plays a key role in efficient problem-solving.
Steps in problem-solving often include breaking a complex problem into manageable parts, applying mathematical properties, and verifying results. In this exercise, each step is designed to methodically reduce complexity: breaking down numbers, applying properties, and calculating results.

• Break down numbers to simplify calculations.
• Apply known mathematical properties, like the distributive property.
• Calculate and verify the results step-by-step.

Such structured approaches not only provide the correct answer but also enhance one's understanding of mathematical concepts and their interconnectedness.