Problem 259 - Prealgebra

Question

In the following exercises, multiply. $$275\cdot 6$$

Step-by-Step Solution

Answer

1650

Step 1: Understand the Problem

We need to multiply 275 by 6.

Step 2: Set Up the Multiplication

Write the numbers one below the other, aligning the digits by place value. \[ 275 \timesewline 6 \]

Step 3: Multiply Each Digit

Start from the rightmost digit of the lower number and multiply by each digit of the upper number:- Multiply 6 by 5 (units place): \[6 \times 5 = 30 \]Write down 0 in the units place and carry over 3.- Multiply 6 by 7 (tens place) and add the carry over: \[6 \times 7 = 42 \]Then add the carried 3: 42 + 3 = 45. Write down 5 in the tens place and carry over 4.- Multiply 6 by 2 (hundreds place) and add the carry over: \[6 \times 2 = 12 \] Then add the carried 4: 12 + 4 = 16. Write down 6 in the hundreds place and 1 in the thousands place.

Step 4: Combine the Results

After multiplying and carrying over, we have:\[1650 \]

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations fundamental to understanding math. It's about finding the total of one number added to itself a certain number of times. For example, multiplying 275 by 6 is the same as adding 275 six times. When you perform multiplication, it's crucial to follow step-by-step procedures to avoid errors. Remember that multiplication is commutative, meaning the order of numbers does not matter, but it’s important to follow rules when writing them for place value alignment.

Place Value

Place value is the value of a digit based on its position in a number. Understanding place value helps in organizing numbers correctly for multiplication. In our example, 275 represents:

• 2 in the hundreds place (200)
• 7 in the tens place (70)
• 5 in the units place (5)

When multiplying, each digit of the bottom number (6) interacts with the corresponding place value of the top number (275). Always align numbers by their place values to ensure accurate calculation. Begin multiplication from the rightmost digit and move to the left. This systematic approach simplifies handling larger numbers and ensures that carryovers are managed correctly.

Carrying in Multiplication

Carrying in multiplication is adding digit overflows to the next column. It's like borrowing in subtraction but in reverse. For instance, in our multiplication problem:

• Multiply 6 by 5 to get 30. Write down 0 and carry over the 3.
• Multiply 6 by 7 to get 42. Add the carried over 3 to get 45. Write down 5 and carry over the 4.
• Multiply 6 by 2 to get 12. Add the carried over 4 to get 16.

Each time you carry over, you ensure that the extra value is not lost and is appropriately added to the next higher place value. This practice helps in keeping calculations accurate and organized as you work through the problem.