Multiplication
Properties of multiplication
1. Commutative Property
– Definition: You can multiply two numbers in any order, and the product will be the same.
– Example:
– 12,345 x 54321 = 54321 x 12,345
– Explanation: Even if the numbers are large, you can switch them around, and the answer remains the same.
2. Associative Property
– Definition: When multiplying three or more numbers, how you group them doesn’t affect the result.
– Example:
– 10,000 x 2 x 6 = 10,000 x 2 times 6 = 120,000
– Explanation: You can group the numbers in different ways, and the result will still be the same.
3. Identity Property (Multiplication by 1)
– Definition: When you multiply any number by 1, the result is the same number.
– Example**:
– 65,432 x 1 = 65,432
– Explanation: Even for large numbers like 5-digit and 6-digit numbers, multiplying by 1 doesn’t change the number.
4. Zero Property (Multiplication by 0)
– Definition: When you multiply any number by 0, the result is always 0.
– Example:
– 123,456 x 0 = 0
– **Explanation**: No matter how large the number is, multiplying by 0 will always give 0.
5. Distributive Property
– Definition: A number multiplied by a sum is the same as multiplying the number by each addend and then adding the results.
– Example:
– 25,000 x (3 + 4) = (25,000 x 3) + (25,000 x 4) = 75,000 + 100,000 = 175,000
– Explanation: You can break down the large numbers into smaller parts (like breaking up a sum inside the parentheses), multiply each part, and then add the results together.
Multiplication by expanding the multiplicand
In multiplication by expanding the multiplicand we will break the larger numbers into smaller, manageable parts to make multiplication simpler.
Example 1: Multiplying a 5-digit number
Let’s say we are multiplying 34,526 × 3.
Step 1: Expand the multiplicand (34,526)
We can break 34,526 into place values, like this:
34,526 = 30,000 + 4,000 + 500 + 20 + 6
Step 2: Multiply each part by 3
Now, multiply each part of the expanded number by 3 (the multiplier):
– 30,000 × 3 = 90,000
– 4,000 × 3 = 12,000
– 500 × 3 = 1,500
– 20 × 3 = 60
– 6 × 3 = 18
Step 3: Add all the products together
Now, we just add all the results:
90,000 + 12,000 + 1,500 + 60 + 18 = 103,578
So, 34,526 × 3 = 103,578.
Step 1: Expand the multiplicand (456,789)
We break 456,789 into place values, like this:
456,789 = 400,000 + 50,000 + 6,000 + 700 + 80 + 9
Step 2: Multiply each part by 4
Now, multiply each part of the expanded number by 4 (the multiplier):
– 400,000 × 4 = 1,600,000
– 50,000 × 4 = 200,000
– 6,000 × 4 = 24,000
– 700 × 4 = 2,800
– 80 × 4 = 320
– 9 × 4 = 36
Step 3: Add all the products together
Now, add all the results:
1,600,000 + 200,000 + 24,000 + 2,800 + 320 + 36 = 1,827,156
So, 456,789 × 4 = 1,827,156.
Multiplication by a 2-digit and a 3-digit multiplier
Example 1: Multiplying a 5-digit number by a 2-digit multiplier
Let’s multiply 23,456 × 12.
Step 1: Expand the 2-digit multiplier (12)
We can break the multiplier into 10 + 2.
So, now the multiplication looks like this:
23,456 × (10 + 2)
Step 2: Multiply the 5-digit number by each part of the multiplier
– First, multiply 23,456 × 10:
– This is simple! Just add a zero: 23,456 × 10 = 234,560
– Now, multiply 23,456 × 2:
– 23,456 × 2 = 46,912
Step 3: Add the two products
Now, we add the two results together:
234,560 + 46,912 = 281,472
So, 23,456 × 12 = 281,472.
Example 2: Multiplying a 6-digit number by a 3-digit multiplier
Let’s multiply 654,321 × 234.
Step 1: Expand the 3-digit multiplier (234)
We break it down as:
234 = 200 + 30 + 4
Step 2: Multiply the 6-digit number by each part of the multiplier
– First, multiply 654,321 × 200:
– To multiply by 200, multiply by 2 and then add two zeros:
– 654,321 × 2 = 1,308,642, and then 1,308,642 × 100 = 130,864,200
– Next, multiply 654,321 × 30:
– To multiply by 30, multiply by 3 and add one zero:
– 654,321 × 3 = 1,962,963, and then 1,962,963 × 10 = 19,629,630
– Finally, multiply 654,321 × 4:
– 654,321 × 4 = 2,617,284
Step 3: Add all the products together
Now, add all the results:
– 130,864,200 + 19,629,630 + 2,617,284 = 153,111,114
So, 654,321 × 234 = 153,111,114.
Summary of the Steps:
1. Break down the multiplier (whether it’s 2 digits or 3 digits) into its parts.
– For example, break 12 into 10 + 2, or break 234 into 200 + 30 + 4.
2. Multiply the large number (the multiplicand) by each part of the multiplier separately.
3. Add all the results to get the final answer.
This method makes large multiplications easier because you break them into smaller, more manageable steps.
Multiplication in real life: word problems
Word Problem 1: Multiplying a 5-digit number
Problem:
A toy factory makes 23,456 toys every month. How many toys does the factory produce in 12 months?
Solution:
We know the factory makes 23,456 toys in 1 month. To find out how many toys it makes in 12 months, we multiply:
23,456 × 12
Now, let’s solve it step by step.
– 23,456 × 12 can be broken down as 23,456 × (10 + 2).
– First, multiply 23,456 × 10 = 234,560.
– Then, multiply 23,456 × 2 = 46,912.
– Now, add the two results:
234,560 + 46,912 = 281,472
So, the factory makes 281,472 toys in 12 months.
Word Problem 2:A large stadium has 124,678 seats. If 235 concerts are held in the stadium in a year, and each concert sells all the seats, how many total tickets are sold?
Solution:
Each concert has 124,678 seats. To find out how many tickets are sold for 235 concerts, we multiply:
124,678 × 235
Let’s solve this step by step.
– Break 235 into 200 + 30 + 5.
– First, multiply 124,678 × 200. Multiply by 2 and add two zeros:
– 124,678 × 2 = 249,356
– 249,356 × 100 = 24,935,600
– Next, multiply 124,678 × 30. Multiply by 3 and add one zero:
– 124,678 × 3 = 374,034
– 374,034 × 10 = 3,740,340
– Finally, multiply 124,678 × 5 = 623,390.
– Now, add all the products together:
24,935,600 + 3,740,340 + 623,390 = 29,299,330
So, the total number of tickets sold in 235 concerts is 29,299,330.