Problem 55 - Prealgebra

Question

Find each product mentally. Example $$\begin{aligned}15 \cdot 12 &=15(10+2) \\\&=150+30 \text { or } 180\end{aligned}$$. $$8 \cdot 23$$

Step-by-Step Solution

Answer

8 times 23 equals 184.

Step 1: Break Down the Second Number

We need to find the product of 8 and 23. Start by breaking down 23 into a sum of two numbers that are easier to multiply. We can express 23 as 20 + 3.

Step 2: Distribute 8

Now distribute the 8 to both parts of the sum (20 and 3) separately. This means we calculate 8 times 20 and 8 times 3.

Step 3: Calculate Each Product

Calculate 8 times 20: \[8 \times 20 = 160\]Then, calculate 8 times 3: \[8 \times 3 = 24\]

Step 4: Add the Results

Now add the two partial products together. This means adding 160 and 24 together: \[160 + 24 = 184\]

Key Concepts

Understanding the Distributive Property

The distributive property is a cornerstone concept in arithmetic and algebra that helps make calculations easier. When you multiply a number by a sum, you can distribute the multiplication over each addend separately. In formula terms, this can be expressed as:

• For any numbers a, b, and c: \(a(b + c) = ab + ac\)

This principle means you can "break apart" complex multiplications into simpler ones. For the problem \(8 \times 23\), by expressing 23 as \(20 + 3\), you can multiply 8 by 20 and 8 by 3 separately. Breaking a problem down in this way allows you to work with smaller, more manageable numbers.
This method not only simplifies calculations but also ensures accuracy when working mentally with bigger numbers.

Breaking Down Numbers for Easier Multiplication

Breaking down numbers is a strategy that is particularly useful for mental math. It involves expressing a number as a simple sum of numbers that are easier to handle. For example, in our exercise, the number 23 is split into 20 and 3.

This method makes it possible to apply the distributive property effectively because each resulting number is simpler to multiply.

• By breaking down numbers, you can focus on straightforward multiplications.
• For example, multiplying 8 by 20 (a simple rounded number) and 8 by 3 (a small single-digit number) each becomes a quick, easy task.

It's important to recognize the types of numbers that simplify most conveniently; typically, this involves looking for numbers that round or neatly combine to tens or fives, aiding the mental math process.

Calculating Partial Products

Calculating partial products involves completing individual multiplications separately before adding them together to find the total product. This is a powerful tool in mental math, especially when combined with the distributive property.

• In our example with \(8 \times 23\), once 23 is broken into 20 and 3, you calculate:
  • \(8 \times 20 = 160\)
  • \(8 \times 3 = 24\)
• These separate results, 160 and 24, are the "partial products."
• The final step is to add these partial products together to get the complete answer: \(160 + 24 = 184\)

Using partial products, complex multiplication problems become a series of simpler steps. Each step requires only simple multiplication, making mental computation much more manageable. This technique also builds a stronger intuitive understanding of number operations, aiding future math skills.