Introduction
From addition and subtraction, we introduce another basic arithmetic operation called multiplication. But what is multiplication? How do we use it?
Through this article, we will learn about the concept of multiplying numbers, some properties related to this operation, and then we proceed with working with numbers through multiplication.
What is Multiplication?
When we refer to multiplication, we mean to add numbers in groups. Intuitively, we can think of multiplication as putting together equal groups of things:
If we have a group of four balls and combine it with two other groups of four balls, we are multiplying the number of balls by the number of groups we put together:
What are the Operands in Multiplication?
In the previous example, we have the following operation involving multiplication:
4×3=12
Here, the first number being multiplied, 4, is called the multiplicand.
The second number, 3, is called the multiplier, this is the number of times the multiplicand will be added to itself.
We generally refer to both multiplicand and multiplier as factors in multiplication.
Lastly, the result of the multiplication is called the product. In this example, 12 is the product obtained by multiplying 4 by 3.
What are the Symbols Used for Multiplication?
When we multiply numbers, we commonly use the cross symbol “” for denoting multiplication. As we have seen in the previous example, the cross symbol was used in multiplying 4 and 3.
In some cases, we might encounter the dot symbol “ ” while performing arithmetic. For numbers, this also means to multiply them:
4⋅3=12
Lastly, we can also use a pair of parentheses “( )” to denote multiplication between numbers:
(4)(3)=12
Properties of Multiplication
Aside from its definition, we also have some properties that are useful when performing multiplication. We list below the following properties of multiplication:
Commutative Property of Multiplication
The Commutative Property of Multiplication allows us to multiply numbers regardless of the order they are evaluated. It states that either of the factors can be designated as the multiplicand or multiplier, but the same product would still be obtained:
4×3=3×4
We observe that reversing the order of operators has the same result, since putting together three groups of four balls is the same as putting together four groups of three balls to have a total of twelve balls.
Associative Property of Multiplication
The Associative Property of Multiplication allows us to multiply numbers regardless of the grouping they are evaluated. It states that for multiplying three factors, the order in which they are multiplied does not matter, and the same product will be obtained:
4×(3×2)=(4×3)×2
We can show that this equality holds by evaluating the two different orders of multiplication performed:
4×(3×2)=4×6=24
(4×3)×2=12×2=24
∴4×(3×2)=(4×3)×2
Distributive Property of Multiplication
The Distributive Property of Multiplication allows us to carry over multiplication over addition. It states that multiplying one number to a sum of two numbers is also equal to the sum of the product of one number and each addend:
4×(3+2)=(4×3) + (4×2)
We can show that this equality holds by evaluating both sides of the equation individually:
4×(3+2) = 4×5 = 20
(4×3) + (4×2) = 12 + 8 = 20
∴4×(3+2) = (4×3) + (4×2)
Identity Property of Multiplication
The Identity Property of Multiplication gives us a special case of multiplication by the number 1. It states that any number multiplied by 1 results to the same number:
4×1=4
3×1=3
Intuitively, we can think of it as having one group of objects and we are simply counting that group without adding anything.
Zero Property of Multiplication
The Zero Property of Multiplication is another special case of multiplication by the number 0. It states that any number multiplied by 0 results to 0:
4×0=0
3×0=0
Multiplication Tables
Before we proceed, we can familiarize ourselves first with some prepared illustrations. A Multiplication Table is a tabular list of factors put together with their products. This is a helpful tool when multiplying small numbers, as we can look back to this section to save time while solving.
Below is a multiplication table showing products between 0 to 10:
Multiplying Numbers
Now that we have learned all about multiplication and its properties, we can explore the different ways we can multiply numbers. We will discuss three different methods for multiplication: Repeated Addition, Long Multiplication, and Divide and Conquer. Each method has advantages over the other, and we will see which scenarios are these best used.
Multiplication as Repeated Addition
Multiplication, in principle, can be expanded as a series of repetitive additions. This simplifies the multiplication operation into an addition operation. We note that this comes handy for smaller numbers, however it becomes tedious adding up larger numbers.
Suppose we have the following expression where we wish to multiply 5 by 3:
5×3= ?
We can think of the multiplier 3 as the number of times the multiplicand will be added by itself. In this example, we will add the multiplicand 5 three times:
5×3=5+5+5
By addition, we know that the sum of adding the multiplicand three times is given by:
5+5=10
10+5=15
∴5+5+5=15
Finally, from the sum obtained by repeated addition we know that the product of the expression will be:
5×3=15
Generally, for any two numbers a and b, the product a×b can be evaluated as a series of repeated additions where a is added by itself b times:
a×b=a+a+a+⋯ (b times)
Long Multiplication
For larger numbers, we can multiply them using Long Multiplication. To guide us, let us work on an example while explaining the steps to performing long multiplication.
Suppose we want to multiply the numbers 123×45. We begin by writing the multiplicand on top of the multiplier, while aligning them according to their decimal places:
We then work on getting the partial product 123×5 by taking two numbers at a time, starting from the ones digit of the multiplicand and the ones digit of the multiplier:
Next, we move to the tens digit of the multiplicand and the ones digit of the multiplier:
Then, we move to the hundreds digit of the multiplicand and the ones digit of the multiplier:
Afterwards, we repeat the same procedure solving for the partial product 123×40 by working on the tens digit of the multiplier. We note that we start writing the result on a new column, with the ones digit being 0:
Similarly, we move to the tens digit of the multiplicand:
Then, we move to the hundreds digit of the multiplicand:
Now, we add up the partial products taking two numbers at a time, starting from the ones digit:
Finally, we determine the product to be:
123×45=5535
Splitting Multipliers
We also have a more flexible method of multiplying numbers that makes use of the Distributive Property of Multiplication. For this approach, we break down the multiplier into a sum of its digits to simplify a complicated multiplication problem:
587×33 = 587×(30+3)
By the Distributive Property of Multiplication, we can further expand the expression as a sum of products:
587×(30+3) = (587×30) + (587×3)
Hence, we can solve for the simpler products. For this, we can use Long Multiplication to get their products:
(587×30) = 17610
(587×3) = 1761
We can then substitute these computed values into the original equation:
587×(30+3)=17610 + 1761
As such, we can solve for the product by taking the sum of the simpler products obtained:
587×33=19371
Problem-Solving Examples
We can now proceed to solve sample problems to apply what we have learned so far. Each problem tackles different formulas discussed and gives us a challenge on how to solve through the information given to us.
Using Repeated Addition
Sample Problem 1:
Annie went to a candy shop and bought four bags of jellybeans. If each bag contains exactly 10 jellybeans, how many jellybeans did she buy in total?
Solution:
We know that each bag contains 10 jellybeans, and that Annie bought four bags of it. We can express this mathematically by multiplying the number of jellybeans per bag and the number of bags she bought:
10 jellybeans/bag×4 bags
We can expand this expression into a series of repetitive additions so that we get:
10 jellybeans/bag×4 bags=(10+10+10+10) jellybeans
Adding together the number of jelly beans, we get:
10+10=20 jellybeans
20+10=30 jellybeans
30+10=40 jellybeans
Hence, we can solve for the product 10×4:
10 jellybeans/bag×4 bags=40 jellybeans
Therefore, we conclude that Annie bought a total of 40 jellybeans.
Sample Problem 2:
A taxi driver needs to load up three liters of gasoline for his trip home. When he went to the gas station, he was charged two dollars per liter of gasoline. How much does he need to pay for his refill?
Solution:
We know that the taxi driver needs to fill three liters of gasoline, and that it costs two dollars per liter of gasoline. Hence, we can compute for the total cost for the refill by taking the product of the amount of gasoline and the cost per liter:
3 liters×2 dollars/liter
We can apply the Commutative Property of Multiplication to rewrite this into:
3 liters×2 dollars/liter = 2 dollars/liter×3 liters
Then, we can expand this multiplication by adding two dollars by the number of liters the taxi driver needs:
2 dollars/liter×3 liters = 2 dollars + 2 dollars + 2 dollars
Adding the costs per liter, we then obtain:
2 dollars + 2 dollars = 4 dollars
4 dollars + 2 dollars = 6 dollars
Hence, we can solve for the product 2×3:
2 dollars/liter×3 liters=6 dollars
Therefore, we conclude that the taxi driver needs to pay 6 dollars for the refill.
Sample Problem 3:
A cook went shopping for ingredients at a supermarket. In his cart, he put three bags of potatoes and two dozen eggs. If each bag contains 8 potatoes, how many items in total is in the cook’s cart?
Solution:
We are given three bags of potatoes, each containing eight potatoes, and two dozen eggs. Mathematically, the total number of items, denoted by T, inside the cart can be expressed using:
T=(8 potatoes/bag×3 bags)+(12 eggs/dozen×2 dozen)
We first solve for the number of potatoes by taking the product 8×3 as a series of repetitive additions:
(8 potatoes/bag×3 bags) = 8 potatoes + 8 potatoes + 8 potatoes
Adding the numbers together, we get:
8 potatoes + 8 potatoes = 16 potatoes
16 potatoes + 8 potatoes = 24 potatoes
As such, we know that the total number of potatoes is given by:
(8 potatoes/bag×3 bags) = 24 potatoes
Next, we multiply the number of eggs by taking the product 12×2 as follows:
(12 eggs/dozen×2 dozen) = 12 eggs+12 eggs
Adding the numbers together, we obtain:
12 eggs + 12 eggs = 24 eggs
Hence, we know the total number of eggs to be equal to:
(12 eggs/dozen × 2 dozen) = 24 eggs
Finally, we can solve for the total number of items by combining the computed number of potatoes and eggs:
T = 24 potatoes + 24 eggs
∴T = 48 items
Therefore, we conclude that there are 48 items inside the cook’s cart.
Using Long Multiplication
Sample Problem 4:
Solve for the product of two two-digit numbers 34 and 21:
34×21
Solution:
We begin by writing the multiplicand on top of the multiplier as shown below:
Then, we solve for the partial product 34×1. By the Identity Property of Multiplication, we know the answer to be 34:
Next, we solve for the partial product 34×20. We write the result on the next column, starting with a 0 on the ones digit:
Afterwards, we add the partial products to get the result of the multiplication:
Therefore, we conclude that the product of 34 and 21 is 714.
Sample Problem 5:
Solve for the product of two three-digit numbers 379 and 107:
379×107
Solution:
We begin by writing the multiplicand on top of the multiplier as shown below:
Then, we solve for the partial product 379×7. We note that we carry over the results to the next digit as shown:
Next, we solve for the partial product 379×0. By the Zero Identity of Multiplication, we know the answer to be 0:
Lastly, we solve for the partial product 379×100:
After getting the partial products, we can then add them to get the product of the given numbers. Again, we note that we carry over to the next digit when adding:
Therefore, we conclude that the product of 379 and 107 is 40553.
Using Split Multipliers
Sample Problem 6:
We consider the previous example where we want to multiply 123×45. Solve for their product using Split Multipliers.
Solution:
We begin by expressing the multiplier 45 as a sum of its tens and ones digit:
123×45=123×(40+5)
Then, using the Distributive Property we can rewrite the expression as a sum of products:
123 × (40 + 5) = (123×40) + (123×5)
We then proceed to solve for the value of the first expression (123×40). By Long Multiplication, we have:
∴123×40=4920
Similarly, we solve for the second expression (123×5) by Long Multiplication:
∴123×5=615
We can then substitute the results into the equation:
123 × (40+5) = 4920 + 615
Finally, by adding the terms together we can solve for the product of 123×45:
123×45 = 123 × (40+5) = 5535
Sample Problem 7:
Solve for the product between 363 and 87:
363×87
Solution:
We begin by expressing the multiplier as a sum of its digits:
363×87=363×(80+7)
A applying the Distributive Property, we can rewrite the expression as a sum of products:
363 x (80+7) = (363 x 80) + (363 x 7)
We then proceed to solve for the value of the first expression (363×80). By Long Multiplication, we have:
363×80 = 29040
Similarly, we solve for the second expression (363×7) by Long Multiplication:
∴363 x 7 = 2541
We can then substitute the results into the equation:
363 × (80+7) = 29040 + 2541
Finally, by adding the terms together we can solve for the product of 363×87:
363 × 87 = 363 × (80+7) = 31581
Summary
Multiplication is a basic arithmetic operation where we add numbers in groups.
For an expression involving multiplication between two numbers a and b, we have the following:
a×b=ab
Here, a is the multiplicand or the first number being multiplied, b is the multiplier or the number of times the multiplicand will be added by itself, and both can be generally referred to as factors. The result of the operation ab is called the product.
The symbols used for multiplication are the following: cross symbol “”, dot symbol “ ”, and parentheses “( )”
There are also useful properties when multiplying numbers a, b, and c:
The Commutative Property of Multiplication allows us to multiply numbers regardless of the order they are evaluated:
a×b=b×a
The Associative Property of Multiplication allows us to multiply numbers regardless of the grouping they are evaluated:
a×(b×c)=(a×b)×c
The Distributive Property of Multiplication allows us to carry over multiplication over addition:
a×(b+c) = (a×b) + (a×c)
The Identity Property of Multiplication states that any number multiplied by 1 results to the same number:
a×1=a
The Zero Property of Multiplication states that any number multiplied by 0 results to 0:
a×0=0
A Multiplication Table is a tabular list of factors put together with their products. This tool can be used when multiplying small numbers by looking at the values listed in this table.
When multiplying numbers, we can perform the following methods:
Repeated Addition treats multiplication between numbers as adding the multiplicand by itself by the multiplier:
a×b=a+a+a+⋯ (b times)
Long Multiplication can be used to multiply larger numbers by writing the factors in columns and then taking the sum of their partial products to obtain the result of the operation.
Splitting Multipliers breaks down the multiplier into a sum of its digits to simplify a complicated multiplication problem. This makes use of the Distributive Property of Multiplication.
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