What is the Expanded Form of a Number?
The expanded form of a number is splitting up a number into tens, hundreds, thousands, ten thousands, and so on, based on the place value of its digits.
A number can be represented in an expanded form by expressing it as the sum of the products of its digits with their place value.
For example, the expanded form of 7453 can be written as 7000 + 400 + 50 + 3.
- Digit 7, is at the thousands place, hence 7000.
- Digit 4, is at hundreds place, hence 400.
- Digit 5, is at tens place, hence 50.
- Digit 3 is at ones place, hence 3.
How to Multiply using the Area Model and the Expanded Form?
We know that in an area model, a rectangular box is used for multiplication. The factors in this case, represent the length and width of the rectangle used in the model.
The area model and the expanded form can also be used together to multiply two numbers.
Let’s take an example. What is 359 x 4?
Here, 342 can be written in the expanded form as, 300 + 50 + 9.
So, there will be 3 area models, or rectangular boxes, each having a width of 4 units and a length of 300 units, 50 units, and 9 units, respectively.
The area of each rectangle is the product of its length and width. Hence, the calculation is written as:
300 units x 4 units = 1200 square units
50 units x 4 units = 200 square units
9 units x 4 units = 36 square units
The product of 359 x 4 will be the sum of the area of each rectangle.
359 x 4 = 1200 + 200 + 36 [Add]
So the product of 359 and 4 is 1436.
How do we Multiply Numbers using the Distributive Property and the Expanded Form
We know that the distributive property states that multiplying the sum of two or more addends by a number is the same as multiplying each addend separately by the number, and then adding the products.
This property can be represented by a generic equation as:
a × (b + c) = a × b + a × c
Let’s consider an example.
10 × (2 + 6) [distribute the 10 to the 2 and the 6]
= 20 + 60 [Simplify]
= 80 [Add]
The distributive property and the expanded form can be used together to multiply two numbers.
Let’s take an example: 452 × 8 = ?
Here, 452 can be written in the expanded form as: 400 + 50 + 2
Hence, 452 x 8 = (400 + 50 + 2) × 8
= (400 × 8) + (50 × 8) + (2 × 8) (distribute the 8 to the 400 ,50 and the 2)
= 3200 + 400 + 16 [Simplify]
= 3616 [Add]
Solved Multiplication of Single Digit Number Examples
Example 1: In a school, the students of 4 different grades go on a camping trip. There are 25 students in each grade. A single bus accommodates 50 students. Will 3 buses be able to accommodate all of the students?
Solution:
Number of students in each grade = 25
Total number of grades that will go on the trip = 4
Total number of students who will go on the trip = total number of grades × number of students in each grade
= 4 × 25
= 4 × (20 + 5) [expand 25]
= 4 × 20 + 4 × 5 (distribute the 4 to the 20 and 5)
= 80 + 20 [Simplify]
= 100 [add]
Hence, the total number of students who will go on the trip is 100.
Number of students that one bus can accommodate = 50
Number of students 3 buses can hold = 3 × the number of students one bus can hold
= 3 × 50
= 150 [Multiply]
Hence, the total number of students is 100, and as we know, three buses can accommodate 150 students, the three buses will be able to accommodate all the students for the camping trip.
Example 2: Use the expanded form and the distributive property to multiply, 115 and 6.
Solution:
115 × 6 = 6 × (100 + 10 + 5) (write 115 in expanded form)
= (6 × 100) + (6 × 10) + (6 × 5) (Distributive property)
= 600 + 60 + 30 [Simplify]
= 690 [add]
So, the product of 115 and 6 is 690.
Example 3: Find the product of 4552 and 4.
Solution:
4,552 × 4 = 4 × (4000 + 500 + 50 + 2) (write 4,552 in expanded form)
= (4 × 4000) + (4 × 500) + (4 × 50) + (4 × 2) (Distributive property)
= 16000 + 2000 + 200 + 8 [Simplify]
= 18,208 [add]
Hence we observe that the product of 4552 and 4 is 18,208.
Example 4: Rewrite the following equation as a product of two factors.
(7 × 80000) + (7 × 3000) + (7 × 500) + (7 × 1)
Solution:
(7 × 80000) + (7 × 3000) + (7 × 500) + (7 × 1)
= 7 × (80000 + 3000 + 500 + 1) [Simplify]
= 7 × 83,501 [add]
So, (7 × 80000) + (7 × 3000) + (7 × 500) + (7 × 1) can also be written as 83501 times 7.
The expanded form of a number is a way of representing it by splitting it into ones, tens, hundreds, and so on, based on the place value of the digits in the number.
For example, the expanded form of 435, is 400 + 30 + 5, where the digit 4 is in the hundreds place, the digit 3 is in the tens place, and the digit 5 is in the ones place.
The distributive property of multiplication states that multiplying the sum of two or more addends by a number is the same as multiplying each addend separately by the number and then adding the products together.
For example, the distributive property can be used to solve this equation:
2 (5 + 6) = 2 × 5 + 2 × 6 = 10 + 12 = 22.
The greatest single digit is 9.
So, the required product will be the product of the greatest single digit with itself. It is given as, 9 × 9 = 81.
Hence, the greatest possible product of two single digits is 81.