Place Value
The place value of a digit in a number is the value it has when it is at a particular position in a number. The place value of the number 871429 is as shown below.
Associative Property
The associative property is a number property which indicates that the product of three or more numbers is the same regardless of how the multiplicands are grouped. In other words, if a, b and c are three numbers that need to be multiplied, then \(a\times (b\times c)=(a\times b)\times c\)
For instance ,
\(11\times (10\times 2)=(11\times 10)\times 2\)
= 220
Distributive Property
The distributive property can be understood as follows:
If we multiply the sum of two or more addends by any number, we will get the same result as multiplying each of the addends individually by the number and then adding the products together.
If a, b and c are three numbers, then
\(a(b+c)=a\times b+a\times c\)
Let’s examine the following example to understand this.
\(2\times (4+6)\)
\(=2\times 10\)
= 20
Or,
\(2\times 4+2\times 6\)
= 8 + 12
= 20
Area Model
We divide the components into ‘tens’ and ‘ones’, utilize them to represent the side lengths of our rectangle, and then multiply the parts when using the area model. The area, or product of the equation, is equal to the sum of the parts. In this case, we multiplied 10 and 10 to get 100, 10 and 3 to get 30, 2 and 10 to get 20, and 2 and 3 to get 6. The entire area of the rectangle, or the product, is then calculated by adding all four of these sections together. 100 + 20 + 40 + 8 = 168.
Partial Product
The partial product approach is multiplying one digit of a number with each digit of the other number and adding the products to get the answer. This method is based on relying on the expanded form of a number, which is an application of the distributive property.
For example, observe how the partial products of 24 and 32 can be obtained.
First the number is split into its place values and we get the following result.
\((24\times 32)=(20+4)(30+2)\)
\((20\times 30) + (20\times 2) + (4\times 30) + (4\times 2)\)
To get a final answer add the partial products together.
600 + 40 + 120 + 8
640 + 128 = 768
This is how the product can be obtained using partial products.
Regrouping
Regrouping used in multiplication can be derived from the distributive property learnt above.In this method, each of the digits is multiplied by the digits of the other number. The results that have two digits are carried over to the next place. Let’s understand how to find the product of 123 and 59.
Now, let us understand this in detail:
The first step is to multiply the digit in the ‘ones’ place of the first number with the digit in the ‘ones’ place of the second number. Seven is written below the ‘ones’ place of the multiplicands. The ‘2’ is carried over to the ‘tens’ place.
Next, nine is multiplied by the number in the ‘tens’ place of the other number. The result is added to the 2 that was previously carried over.
Finally, ‘9’ is multiplied to the ‘hundreds’ place of the other number.
The resulting number is added with ‘2’ as it was carried over from the previous place.
The same steps are repeated with 5 except that the numbers are placed below the ‘tens’ place and are progressively placed after the same. Finally, the products are added as shown below and we get the answer.
The result is 7257.
Rounding or Estimation
To calculate the product, round off the multiplier and multiplicand to the nearest ‘tens’, ‘hundreds’, or ‘thousands’ and then multiply the rounded figures. Multiplying rounded figures is much easier.
I. Estimate the products of 43 and 87.
43 ⟶ 40 43 is rounded down to 40
87 ⟶ 90 87 rounded up to 90
Calculate mentally; 40 × 90 = 3600
The estimated product is 3600.
In order to estimate products, we round the given components to the desired place value. Estimating products can help us figure out whether or not a result is close to the requirement in a circumstance.
Solved Examples
Example 1:
Using the area model and partial product, find the product of 14 and 22.
Solution:
First, we split 14 into 10 and 4.
22 is split into 10, 10 and 2.
\(14\times 22=(10+4)(20+2)\)
Splitting the number according to its place values, we get the following.
= (10 + 4) (20 + 2)
Using the distributive property, we get the following.
\(=(10\times 20)+(10\times 2)+(4\times 20)+(4\times 2)\)
Simplifying it further we get,
= 200 + 20 + 80 + 8
= 220 + 88
= 308
Example 2:
Using regrouping, find the product of 25 and 32. Compare the result by using estimation or rounding.
Solution:
Using regrouping:
First 2 is multiplied with 5, the result gives us 10 of which will be carried over. 2 is multiplied by 2, giving us 4. One is added to 4 which gives us 50. 3 is multiplied by 5, giving us 15. 1 is carried over to the next place. Note that five is placed in the ‘tens’ place column. 3 is later multiplied by 2 which gives us 6. The number that is carried over is added to 6 giving us 7. Finally the results are added and we get the final product as 800.
Using rounding or estimation:
First, numbers should be rounded to their nearest multiples of ten.
25 -> 30
32 -> 30
\(30\times 30=900\)
The result we get by multiplying 25 and 32 using partial products is 800 and the result obtained by multiplying using rounding is 900.
Example 3:
A video game has fifteen quests in each level. There are a total of 27 levels in the game. Ray has finished 23 levels in the game. How many quests did Ray complete? Also find the approximate number of quests the game has.
Solution:
To find the number of quests Ray completed, multiply 15 with 27.
The answer we get is 405.
Let’s find the approximate number of quests the game has by rounding the numbers to their nearest ‘tens’.
27-> 30
15-> 20
\(30\times 20=600\)
So, there are approximately 600 quests in the game.
Example 4:
Mandy brings thirty-two pouches of chocolates to school and she wants to distribute it among her classmates. Unfortunately, the number of pouches was not enough so she decided to open the pouch and distribute the chocolates individually. Each pouch has 12 chocolates. Determine the total number of chocolates Mandy has.
Solution:
1 pouch has 12 chocolates.
Number of chocolates in 32 pouches \(=12\times 32\)
Using partial products, the product of the two numbers are as follows
\(12\times 32=(10+2)(30+2)\)
\(=(10\times 30)+(10\times 2)+(2\times 30)+(2\times 2)\)
= (300) + (20) + (60) + (4)
= (320) + (64)
= 384
Example 5:
Find the areas of the spaces allocated by Mr Stevens for his new office. Also calculate the total area as well.
Solution:
The given figure is an area model, multiplying the length with the breadth for each of these smaller sections will give the answer. Adding the values of these smaller areas will give the total answer as well.
We will use the formula of Area of a rectangle to find all the required areas:
Area of a rectangle = length × width
Area covered by the working area = 10 × 10
= 100 squared feet
Area taken up by the washroom = 10 × 9
= 90 squared feet
Area taken up by the pantry = 8 × 10
= 80 squared feet
Area taken up by the lift and stairs = 9 × 8
= 72 squared feet
Therefore the total area = 100 + 90 + 80 + 72
= 342 squared feet.
Area models, partial products, regrouping and rounding or estimation are a few of the strategies that can be used to multiply two-digit numbers.
The multiplication properties such as Associative property, distributive property and so on, can break down large numbers into smaller ‘ones’. This will make problems easier to solve as it breaks down a large calculation into smaller ‘ones’.