multiplication and division of whole numbets

Basic Multiplication Steps
- Start by multiplying the last digit of the second number.
- Example:
  - 5 times 4 equals 20, with 2 remaining.
  - Next, multiply 5 by 2 equals 10, adding the 2 remaining gives 12 (1 is remaining).
  - Thus, compute for each digit and carry over as necessary.
Structured Multiplication Process
- When multiplying larger numbers, proceed digit by digit from right to left.
- For example, multiply by 7:
  - 7 times 9 = 63 (write down 3 and carry over 6).
  - 7 times 0 + 6 = 6.
  - 7 times 4 = 28.
  - 7 times 2 = 14 + 2 = 60.
  - Add these results together for the final answer.
Recording Results
- Write results under the corresponding place value.
- For instance, for tens, write under the tens place of the first number's result.

Example 1: 290 x 49
- Start with 9 times:
  - 9 x 6 = 54 (5 remaining).
  - 9 x 9 = 81 + 5 = 86 (8 remaining).
  - 9 x 2 = 18 + 8 = 26.
- Now multiply by 4:
  - 4 x 6 = 24 (2 remaining).
  - 4 x 9 = 36 + 2 = 38.
  - 4 x 2 = 8 + 3 = 11.
- Add results together across columns.

Method
- Multiply each digit of the four-digit number by each digit of the other number.
- Place results correctly according to place value.
- Example: 2105 x 154.
  - Start with least significant digit, moving towards most significant (right to left).
  - 4 times: 4 x 5 = 20 (0 placement).
  - 4 x 0 = 0 + 2 = 2.
  - 4 x 1 = 4.
  - 4 x 2 = 8.
- Continue with next digit (5) by adjusting placement under tens and hundreds.

Identifying Zeros
- Multiply the non-zero components while keeping track of trailing zeros.
- Example: 62 x 300 = 62 x 3; then add the two zeros back to the result: 18600.
Understanding Significance of Trailing Zeros
- Zeros signify multiplication by ten, impacting the resultant value's place.

Relation of Multiplication and Division
- Division is the inverse of multiplication.
- If you multiply to find a total, division helps to find how many of one quantity fit into that total.
Basic Division Steps
- Start from the first digit; if smaller than the divisor, incorporate the next digit.
- For example: 732 divided by 2.
  - 7 divided by 2 = 3 (3 x 2 = 6, remainder 1).
  - Bring down the next digit 3 to make 13.
  - 13 divided by 2 = 6 (6 x 2 = 12, remainder 1).
  - Bring down next digit 2 for final computation.
  - Result: 366 with no remainder.
- This is called the quotient.

Examples
- Example: 70872 divided by 6.
  - Start from leftmost digit, 8 is divided by 6 to gauge fit.
  - This isolates the portion of the number to apply division methodically per digit.
  - Example calculations yield a quotient and potential remainder.
Further Practice
- Additional divisions of numbers like 457 by 3 to practice the method fully.
- Division by two-digit numbers is generally simpler and avoids very large quotients.
- Steps include continuously breaking down by digits, assessing component fit for the divisor.

Multiplication and division are related operations; mastery of one assists in understanding the other.
Recognizing place values is essential for accurate arithmetic operations.
Practice is essential for confidence and proficiency in multiplication and division skills irrespective of number size.