Question

Find each product without using a calculator \(100,000 \times 754.01\)

Short answer

75,401,000

Step-by-step solution

Write the equation in standard form

Understand that the equation to solve is: hundred thousand times seven hundred fifty-four and one-hundredth.Write it as: \( 100,000 \times 754.01 \)

Separate the parts of the second number

Break down the second number into its whole number and the decimal part:\( 754.01 = 754 + 0.01 \)

Use distributive property

Apply the distributive property to multiply the broken parts separately:\( 100,000 \times (754 + 0.01) = (100,000 \times 754) + (100,000 \times 0.01) \)

Perform the multiplications separately

First, multiply the whole number part:\( 100,000 \times 754 = 75,400,000 \)Then, multiply the decimal part:\( 100,000 \times 0.01 = 1,000 \)

Sum the results from the previous step

Add the two results together:\( 75,400,000 + 1,000 = 75,401,000 \)

Key concepts

Distributive Property

The distributive property is a helpful strategy in multiplication. It allows us to break down a complex problem into simpler parts.
Here's how it works: If you have a multiplication equation like \( a \times (b + c) \), you can distribute the outer multiplication to both terms inside the parenthesis. This means you can write \( a \times \ b + a \times \ c\) . For example:
\[ 100,000 \times (754 + 0.01) = (100,000 \times 754) + (100,000 \times 0.01) \].
This makes it easier to handle each part separately.
By breaking down larger numbers or complex decimals into manageable chunks, the distributive property simplifies multiplication.

• Break down the larger number into smaller parts.
• Multiply each part separately.
• Add the results together.

This method helps avoid mistakes and develops a deeper understanding of multiplication.

Decimal Multiplication

Multiplying decimals may seem tricky, but with practice, it becomes straightforward. Let's use the example from the exercise:
First, consider multiplying two numbers where one or both are decimals. Separate the decimal number into its whole and fractional parts like this:
\[754.01 = 754 + 0.01 \]
Handle each multiplication separately before combining them.
For the example:

• Multiply the whole number part: \ 100,000 \times 754 = 75,400,000 \.
• Multiply the decimal part: \ 100,000 \times 0.01 = 1,000 \.

Decimals involve place value, and it's essential to count the decimal places accurately when multiplying. Finally, sum the partial products:
\[75,400,000 + 1,000 = 75,401,000 \].
This step-by-step approach ensures accuracy in decimal multiplication.
.

Place Value

Place value is crucial in understanding and performing multiplications, especially with larger numbers and decimals. Each digit in a number represents a different place value.
For instance, in the number 100,000:

• 1 is in the hundred thousand's place.
• Each 0 holds a specific place value as well.

When dealing with decimals, place value is just as important. The number 754.01 can be broken into:
\[754 = 7 \times 100 + 5 \times 10 + 4 \times 1 \]
\[0.01 = 1 \times 0.01 \]
Understanding place value helps simplify multiplication.
By keeping track of where each digit falls, you avoid errors and gain a better grasp of how numbers work.
This is especially highlighted when you apply the distributive property and handle each part of the number separately.