Problem 99 - Prealgebra

Question

Find the product of 827 and 25

Step-by-Step Solution

Answer

The product of 827 and 25 is 20675.

Step 1: Understand the Problem

We need to calculate the product of two numbers: 827 and 25. This involves multiplying these two numbers together.

Step 2: Break Down the Multiplication

Let's use the distributive property to handle the multiplication: We can express 25 as the sum of 20 and 5. So the multiplication becomes:\( 827 \times 25 = 827 \times (20 + 5) \)This expression can be split into two simpler multiplications: \( 827 \times 20 \) and \( 827 \times 5 \).

Step 3: Multiply by 20

Calculate \( 827 \times 20 \):First multiply 827 by 2:\( 827 \times 2 = 1654 \)Then multiply the result by 10:\( 1654 \times 10 = 16540 \)Thus, \( 827 \times 20 = 16540 \).

Step 4: Multiply by 5

Now calculate \( 827 \times 5 \):\( 827 \times 5 = 4135 \).You can verify this calculation by multiplication: 800 times 5 is 4000, 20 times 5 is 100, and 7 times 5 is 35. Their sum is 4135.

Step 5: Add Both Products

Now, add the results from Step 3 and Step 4:\( 16540 + 4135 = 20675 \).So, the product of 827 and 25 is 20675.

Key Concepts

Understanding the Distributive Property

The distributive property is a handy tool in mathematics that allows us to break complex calculations into simpler steps. Imagine you're unpacking groceries; separating items makes the task more manageable. Similarly, the distributive property helps handle multiplication by distributing one number across the sum of other numbers.
In our example, multiplying 827 by 25 seems daunting at first glance. By using the distributive property, we can break the number 25 into two parts—20 and 5. This gives us the equation:

• \( 827 \times (20 + 5) \)

The distributive property lets us expand this into:

• \( 827 \times 20 + 827 \times 5 \)

This way, rather than doing one large multiplication, we perform two simpler calculations. This not only makes the process easier but also reduces the chance of errors.

Implementing Step-by-Step Calculation

Breaking down a problem into step-by-step parts makes solving it clearer and more methodical. Let's walk through each piece of the calculation process for our problem. Start by focusing on the number 20 in the expression \( 827 \times (20 + 5) \).
First, multiply 827 by 2:

• \( 827 \times 2 = 1654 \)

By then multiplying 1654 by 10, we scale it back up to what happens when multiplying by 20:

• \( 1654 \times 10 = 16540 \)

Next, move on to the 5 in the expression. Perform a direct multiplication:

• \( 827 \times 5 = 4135 \)

These calculations show how breaking the numbers allows for simpler arithmetic, reducing complexity one step at a time.

Finding the Product of Numbers

When we talk about the product of numbers, we're discussing the result that comes from multiplying two numbers together. In this exercise, after breaking down the problem into manageable parts using the distributive property and detailed steps, we look at adding the products.
First, add the product from multiplying by 20:

• \( 16540 \)

And the product from multiplying by 5:

• \( 4135 \)

Thus, the final product of 827 and 25 is the sum of these two results:

• \( 16540 + 4135 = 20675 \)

Understanding the method behind finding the product of numbers helps ensure accurate calculations while deepening comprehension of the multiplication process itself.