Defining Multiplication as an Operation
Multiplication is another representation of repeated addition. Instead of long equations to add equal groups or numbers, we use multiplication to quicken the process. For example, 3 + 3 + 3 + 3, which results in 12 as the answer, can also be expressed as, 34, which also gives us 12 as the answer. Multiplication involves the factors or values being multiplied as well as the product. The product can be divided evenly by each of the factors without a remainder.
Defining Box Multiplication
The box method of multiplication is another form of long multiplication. In this method we split the factors and then continue with the multiplication.
For example, let us multiply 23 and 45.
Split 23 as 20 + 3 and 45 as 40 + 5.
Fill the cells in the first row with 20 and 3 and the cells in the first column with 40 and 5.
Find the product of numbers in each cell in the row and each cell in the column, that is,
\(40 \times 20\text{ }=\text{ }800\)
\(40 \times 3\text{ }=\text{ }120\)
\(5 \times 3\text{ }=\text{ }15\)
These products are called partial products.
20 | 3 | |
40 | 800 | 120 |
5 | 100 | 15 |
In the next step, we add all the partial products that are in the boxes.
800 + 120 + 100 + 15 = 1035
To double check this product, \(23 \times 45 =\text{ } 1035\)
In this article we will look at solved examples that involve multiplying two digit and three digit numbers using the box method.
Rapid Recall
Solved Examples
Example 1:
Find the product of the following numbers with the help of the box multiplication method:
- \(25 \times 56\)
- \(234\times 24\)
- \(98\times 567\)
- \(78\times 165\)
Solution
- \(25\times 56\)
50 | 6 | |
20 | 1000 | 120 |
5 | 250 | 30 |
Adding the partial products in the boxes:
\(1000 + 250 + 120 + 30 = 1,400\)
Hence, \(25 \times 56\text{ }is\text{ } 1,400\)
2. \(234 \times 24\)
Applying the box multiplication method:
200 | 30 | 4 | |
20 | 4000 | 600 | 80 |
4 | 800 | 120 | 16 |
Adding the partial products in the boxes:
4000 + 600 + 80 + 800 + 120 + 16 = 5,616
Hence, \(234 \times 24\text{ }is\text{ }5,616.\)
3. \(98 \times 567\)
Applying the box multiplication method:
500 | 60 | 7 | |
90 | 45000 | 5400 | 630 |
8 | 4000 | 480 | 56 |
Adding the partial products in the boxes:
45000 + 5400 + 630 + 4000 + 480 + 56 = 55,566
Hence, \(98 \times 567\text{ } is \text{ }55,566.\)
4. \(78 \times 165\)
Applying the box multiplication method:
100 | 60 | 5 | |
70 | 7000 | 4200 | 350 |
8 | 800 | 480 | 40 |
Adding the products in the boxes:
7000 + 4200 + 350 + 800 + 480 + 40 = 12,870
Hence, \(78 \times 165\text{ }is\text{ }12,870.\)
Example 2:
Jamie wants to calculate the total number of books he has to start a library. There are 56 boxes with 134 books in each box. The library has 8 rooms. Find the number of books that can be placed in each room.
Solution
Number of boxes = 56
Number of books in a box = 134
Applying the box method of multiplication to find the total number of books.
100 | 30 | 4 | |
50 | 5000 | 1500 | 200 |
6 | 600 | 180 | 24 |
Adding the partial products in the boxes:
5000 + 1500 + 200 + 600 + 180 + 24 = 7,504
So, Jamie has 7,504 books in total.
Number of rooms = 8
Total number of books = 7,504
Number of books in each room \(=\text{ } \frac{7504}{8} = 938 \)
Hence, 938 books can be placed in each room of the library.
There are four properties that are associated with multiplication.
- Commutative Property
- Associative Property
- Distributive Property
- Identity Property
The commutative property states that the product of the factors remains the same even if the order of the factors are altered. That is, ‘a ✕ b = b ✕ a’.
The associative property is similar to the commutative property and in this case the grouping of factors will not affect the product. That is, ‘a ✕ b ✕ c = (a ✕ b) ✕ c = a ✕ (b ✕ c)’.
The distributive property states that multiplying a number with the sum or difference of two or more numbers, can also be written as the number being multiplied to individual addends, minuends and subtrahends, and then calculating the sum or difference. That is, ‘a ✕ (b ± c ± d) = (a ✕ b) ± (a ✕ c) ± (a ✕ c)’