Problem 28 - Prealgebra

Question

Find each sum or product. Explain your reasoning. $$11 \cdot 9 \cdot 10$$

Step-by-Step Solution

Answer

The product is 990.

Step 1: Multiply First Pair of Numbers

Start by multiplying the first two numbers in the sequence: 11 and 9. Calculate \(11 \times 9\). To do this, you can use the distributive property: \(11 \times 9 = 11 \times (10 - 1) = 11 \times 10 - 11 \times 1 = 110 - 11 = 99\).

Step 2: Multiply Result with Last Number

Take the result from Step 1 and multiply it by the remaining number, 10. So, calculate \(99 \times 10\). Multiplying by 10 is simple: just add a zero to the end of the original number, \(99 \times 10 = 990\).

Key Concepts

Understanding the Distributive Property

The distributive property is a useful math tool, especially when working with multiplication. It allows us to break down complex problems into simpler parts. This property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. In mathematical terms, for any numbers \(a\), \(b\), and \(c\), the formula is \(a \times (b + c) = a \times b + a \times c\). When subtracting, it works similarly: \(a \times (b - c) = a \times b - a \times c\).
In the given exercise, we use the distributive property to simplify the multiplication \(11 \times 9\). By rewriting 9 as \(10 - 1\), we can express this multiplication as \(11 \times (10 - 1)\). Further calculating gives \(11 \times 10 = 110\) and \(11 \times 1 = 11\). Finally, by subtracting, you find \(110 - 11 = 99\). This use of the distributive property makes the multiplication more manageable.

Solving with a Step-by-Step Solution

A step-by-step solution can help clarify the process of solving a problem, making it easier to understand. First, identify each component as outlined in the exercise.

Here's the breakdown of the problem \(11 \times 9 \times 10\):

• Step 1: Begin with the first multiplication, which is \(11 \times 9\). Use the distributive property to handle this calculation with ease: \(11 \times 9 = 11 \times (10 - 1) = 110 - 11 = 99\).
• Step 2: Multiply the result from step 1 with the last number, 10. Since multiplying by 10 is straightforward, you can easily add a zero to the end of 99, giving \(99 \times 10 = 990\).

Following these steps ensures clarity and accuracy in reaching the final answer.

Mastering Arithmetic Operations

Arithmetic operations like addition, subtraction, multiplication, and division are fundamental to mathematics. Understanding how to perform these operations is crucial for solving problems efficiently and effectively.

Here, multiplication plays a central role. It involves combining equal groups into a single total. The exercise \(11 \times 9 \times 10\) uses multiplication to find the product of these numbers.

• Multiply the numbers in pairs to simplify the calculation.
• Start with smaller, more manageable numbers before tackling larger results.
• Rewriting complex problems using properties like the distributive property aids in simplification.

Once familiar with these operations, students can tackle more complex math problems with confidence.