Word problems are fun and challenging to solve because they represent actual situations that happen in our world. As students, we are always wondering why we should learn one skill or another, and word problems help us see the practical value of what we are learning.
Read the tips and guidance and then work through the multiplication and division word problems in this lesson with your children. Try the three worksheets that are listed within the lesson (you will also find them at the bottom of the page.)
What is multiplication?
The process of finding out the product between two or more numbers is called multiplication. The result thus obtained is called the product. Suppose, you bought 6 pens on one day and 6 pens on the next day. Total pens you bought are now 2 times 6 or 6 + 6 = 12.
This can also be written as 2 x 6 = 12
Not the symbol used for multiplication. The symbol (x) is generally used to represent multiplication. Other common symbols that are used for multiplication are the asterisk (*) and dot (.)
Important terms in the multiplication
Some important terms used in multiplication are –
Multiplicand – The number to be multiplied is called the multiplicand.
Multiplier – The number with which we multiply is called the multiplier.
Product – The result obtained after multiplying the multiplier and the multiplicand is called the product.
The relation between the multiplier, multiplicand and the product can be expressed as –
Multiplier × Multiplicand = Product
Let us understand this using an example.
Suppose we have two numbers 9 and 5. We wish to multiply 9 by 5.
So, we express it as 9 x 5 which gives us 45.
Therefore, 9 x 5 = 45
Here, 9 is the multiplicand, 5 is the multiplier and 45 is the product.
What is division?
Division is the equal sharing of a given quantity.
For example, Alice wants to share 6 bananas equally with her friend Rose. So, she gives 3 of her bananas to Rose and she is also left with 3 bananas. This means that when we divide 6 by 2 we get 3.
Mathematically, we can write this as
6 ÷ 2 = 3
Symbol for Division
In mathematics, there is a symbol for every operation. The symbol for division is (÷). Other than the forward-slash (/) is also used to denote the division of two numbers, where, the dividend comes before the slash and the divisor after it. For instance, if we wish to write 15 is being divided by 3, we can write it as 15 ÷ 3 or 15 / 3. Both mean the same.
Important terms in Division
The number that is to be divided is called the Dividend. Here, 6 is the dividend.
The number by which the dividend is being divided is called the Divisor.
The result obtained by the process of division is called the Quotient.
The number that is left over after finding the quotient is called the Remainder.
Let us understand these by an example.
Suppose, we have a pack of 65 chocolates and we want to divide them equally among 7 children while keeping the remaining chocolates with us. How many chocolates does each child get and how many chocolates are we left with after dividing these chocolates?
Using multiplication tables, we have 7 x 9 = 63
Therefore, 7 x 9 + 2 = 65
This means that the quotient when 65 is divided by 7 will be 9 and the remainder will be 2.
As per the definition of the four terms of division, we have
Divisor = 7
Dividend = 65
Quotient = 9
Remainder = 2
Remember: The remainder is always smaller than the divisor.
Formula for Division
There are four important terms in the division, namely, divisor, dividend, quotient, and the remainder. The formula for divisor constitutes all of these four terms. In fact, it is the relationship of these four terms among each that defines the formula for division. If we multiply the divisor with the quotient and add the result to the remainder, the result that we get is the dividend. This means,
Dividend = Divisor x Quotient + Remainder
What are word problems?
A word problem is a few sentences describing a ‘real-life’ scenario where a problem needs to be solved by way of a mathematical calculation. In other words, word problems describe a realistic problem and ask you to imagine how you would solve it using math. Word problems are fun and challenging to solve because they represent actual situations that happen in our world.
How to Solve Word Problems involving multiplication and division?
The following steps are involved in the process of solving word problems involving multiplication and division of numbers –
- Read through the problem carefully, and figure out what it is about. This is the most important step as it helps to understand two things – what is given in the question and what is required to be found out.
- The next step is to represent unknown numbers using variables. Usually, these unknown numbers are the values that are required to be solved for.
- Once the numbers have been represented as variables, the next step is to translate the rest of the problem in the form of a mathematical expression.
- Once this expression has been formed, the last step is to solve this expression for the variable and obtain the desired result.
Let us understand it through an example.
Example
A hawker delivers 148 newspapers every day. How many newspapers will he deliver in a non-leap year?
Solution
We have been given that a hawker delivers 148 newspapers every day. We need to find out the total number of newspapers that he will deliver in a non-leap year. Let us summarise the given information as
Number of newspapers delivered by the hawker in a day = 148
Number of newspapers that he will deliver in a non-leap year = ?
Now, we know that a non-leap year consists of 365 days. This means that we need to find out the total number of newspapers that the hawker will deliver in 365 days. Therefore,
Total number of days on which hawker delivers the newspapers = 365
Now, to find the total number of newspapers delivered by the hawker in 365 days we will have to multiply the Number of newspapers delivered by the hawker in a day by the total number of days in a year. So, we have,
Number of newspapers that he will deliver in a non-leap year = (Number of newspapers delivered by the hawker in a day ) x (total number of days in a year ) ……….. ( 1 )
Substituting the given values in the above equation, we have
Number of newspapers that he will deliver in a non-leap year = 148 x 365
Now, 148 x 365 = 54020
Hence, the number of newspapers that he will deliver in a non-leap year = 54020
Let us consider another example.
Example
In a school, a fee of £ 345 is collected per student. If there are 240 students in the school, how much fee is collected by the school?
Solution
We have been given that in a school a fee of £ 345 is collected per student. Also, there are 240 students in the school. We need to find out the total fee collected by the school from all students. Let us first summarise this information
Amount of fee collected by the school from each student = £ 345
Number of students in the school = 240
Total amount of fee collected by the school = ?
This can be calculated by multiplying the fee collected for each student by the number of students in the school. Therefore we have,
Total amount of fee collected by the school = (Amount of fee collected by the school from each student ) x (Number of students in the school ) …….. ( 1 )
Substituting the given information in the above equation, we get
Total amount of fee collected by the school = £ ( 345 x 240 )
Now, 345 x 240 = 82800
Hence, Total amount of fee collected by the school = £ 82800
Solving Multiplicative Comparison Word Problems
Multiplication as Comparing
In multiplicative comparison problems, there are two different sets being compared. The first set contains a certain number of items. The second set contains multiple copies of the first set.
Any two factors and their product can be read as a comparison. Let’s look at a basic multiplication equation: 4 x 2 = 8.
| 8 is the same as 4 sets of 2 or 2 sets of 4. 8 is 4 times as many as 2, and 2 times as many as 4. |
What Operation to Use: Multiply? Divide? Add? Subtract?
The hardest part of any word problem is deciding which operation to use. There can be so many details included in a word problem that the question being asked gets lost in the whole situation. Taking time to identify what is important, and what is not, is essential.
Use a highlighter on written problems to identify words that tell you what you are solving, and give you clues about which operations to choose. Make notes in the margins by these words to help you clarify your understanding of the problem.
Remember: If you don’t know what’s being asked, it will be very difficult to know if you have a reasonable answer.
Different Types of Problem
There are three kinds of multiplicative comparison word problems (see list below). Knowing which kind of problem you have in front of you will help you know how to solve it.
- Product Unknown Comparisons
- Set Size Unknown Comparisons
- Multiplier Unknown Comparisons
The rest of this lesson will show how these three types of math problems can be solved.
Multiplication Problems: Product Unknown
In some multiplicative comparison word problems, you are given the number of items in one set, and you are given the “multiplier” amount. The multiplier amount tells you how many times bigger (or more) the second set is than the first. “Bigger” can also mean “longer,” or “wider,” or “taller” in problems involving measurement, or “faster” in problems involving a rate of speed.
These problems in which you know both the number in one set, and the multiplier are called “Product Unknown” comparisons, because the total is the part that is unknown.
In order to answer the question you are being asked, you need to multiply the number in the set by the multiplier to find the product.
Product Unknown Comparisons
In some multiplicative comparison word problems, you are given the number of items in one set, and you are given the “multiplier” amount. The multiplier amount tells you how many times bigger (or more) the second set is than the first. “Bigger” can also mean “longer,” or “wider,” or “taller” in problems involving measurement, or “faster” in problems involving a rate of speed. These problems in which you know both the number in one set, and the multiplier are called “Product Unknown” comparisons, because the total is the part that is unknown.
Let us understand it through an example.
Example
Mary is saving up money to go on a trip. This month, she saved three times as much as money as she saved last month. Last month, she saved £ 24.00. How much money did Mary save this month?
Solution
We have been given that Mary is saving up money to go on a trip. This month, she saved three times as much as money as she saved last month. Last month, she saved £24.00. we need to find out how much money did Mary save this month?
Now, as much as tells you that you have a comparison. Three times is the multiplier. 24 is the amount in the first set. The question being asked is how much money did Mary save this month? To find the answer, we multiply 24 by 3. Therefore, we have 24.00 x 3 = 72.
It is important to clearly show that you understand what your answer means. Instead of writing just 72, we will write it as Mary saved £ 72 this month.
Note that whenever we finish a math problem of any kind, we always go back to the original problem. Think: “What is the question we are being asked?”
Make sure that our final answer is a reasonable answer for the question we are being asked.
We were asked, “How much money did Mary save this month?”
Our answer is: Mary saved $72.00 this month. Our answer is reasonable because it tells how much money Mary saved this month. We multiplied a whole number by a whole number, so the amount of money Mary saved this month should be more than she saved last month. Seventy-Two is more than 24 . Our answer makes sense.
Multiplication Problems: Set Size Unknown
In some multiplicative comparison word problems, the part that is unknown is the number of items in one set. You are given the amount of the second set, which is a multiple of the unknown first set, and the “multiplier” amount, which tells you how many times bigger (or more) the second set is than the first. Remember, “bigger” can also mean “longer,” or “wider,” or “taller” in problems involving measurement, or “faster” in problems involving a rate of speed.
These problems in which you know both the number in the second set, and the multiplier are called “Set Size Unknown” comparisons, because the number in one set is the part that is unknown.
In order to answer the question you are being asked, you need to use the inverse operation of multiplication: division. This kind of division is “partition” or “sharing” division. Dividing the number in the second set by the multiplier will tell you the number in one set, which is the question you are being asked in this kind of problem.
Set Size Unknown Comparisons
In some multiplicative comparison word problems, the part that is unknown is the number of items in one set. You are given the amount of the second set, which is a multiple of the unknown first set, and the “multiplier” amount, which tells you how many times bigger (or more) the second set is than the first. Remember, “bigger” can also mean “longer,” or “wider,” or “taller” in problems involving measurement, or “faster” in problems involving a rate of speed. These problems in which you know both the number in the second set and the multiplier are called “Set Size Unknown” comparisons because the number in one set is the part that is unknown.
Let us understand it through an example.
Example
Jeff read 12 books during the month of August. He read four times as many books as Paul. How many books did Paul read?
Solution
We have been given that Jeff read 12 books during the month of August. He read four times as many books as Paul. We need to find out how many books did Paul read?
“As many as “ tells you that we have a comparison. Four times is the multiplier. 12 books is the amount in the second set. How many books did Paul read? This is the question we are being asked. To solve, divide 12 by 4. Now 12 ÷ 4 = 3. It is important to clearly show that we understand what our answer means. Instead of just writing 3, we write complete sentence that Paul read three books.
Note that whenever we finish a math word problem, always go back to the original problem. Think: “What is the question we are being asked?” Make sure that our final answer is a reasonable answer for the question you are being asked. We were asked, “How many books did Paul read?” Our answer is: Paul read three books. Our answer is reasonable because it tells how many books Paul read. We divided a whole number by a whole number, so the number of Paul’s books should be less than the number of Jeff’s books. Three is smaller than 12. My answer makes sense.
Multiplicative Comparison Problems: Multiplier Unknown
In some multiplicative comparison word problems, you are given the number of items in one set, and you are given the number of items in the second set, which is a multiple of the first set. The “multiplier” amount is the part that is unknown.
The multiplier amount tells you how many times bigger (or more) the second set is than the first. “Bigger” can also mean “longer,” or “wider,” or “taller” in problems involving measurement, or “faster” in problems involving a rate of speed.
These problems in which you know both the number in one set, and the number in the second set are called “Multiplier Unknown” comparisons, because the multiplier is the part that is unknown.
In order to answer the question you are being asked, you need to use the inverse operation of multiplication: division. This kind of division is called “measurement” division.
Multiplier Unknown Comparisons
In some multiplicative comparison word problems, you are given the number of items in one set, and you are given the number of items in the second set, which is a multiple of the first set. The “multiplier” amount is the part that is unknown. The multiplier amount tells you how many times bigger (or more) the second set is than the first. “Bigger” can also mean “longer,” or “wider,” or “taller” in problems involving measurement, or “faster” in problems involving a rate of speed. These problems in which you know both the number in one set and the number in the second set are called “Multiplier Unknown” comparisons because the multiplier is the part that is unknown. In order to answer the question you are being asked, you need to use the inverse operation of multiplication: division. This kind of division is called “measurement” division.
Let us understand it through an example.
Example
The gorilla in the Los Angeles Zoo is six feet tall. The giraffe is 18 feet tall. How many times taller than the gorilla is the giraffe?
Solution
We have been given that the gorilla in the Los Angeles Zoo is six feet tall. The giraffe is 18 feet tall. We need to find out how many times taller than the gorilla is the giraffe?
Taller than tells us that we have a comparison. Six feet is the amount in the first set. 18 feet is the amount in the second set. How many times taller than the gorilla is the giraffe? This is the question we are being asked. To solve this we divide18 feet by six feet. Now, 18 ÷ 6 = 3. It is important to clearly show that we understand what our answer means. Instead of just writing 3, we write the complete sentence that the giraffe is three times taller than the gorilla.
Note that whenever we finish a math word problem, always go back to the original problem. Think: “What is the question we are being asked?” Make sure that your final answer is a reasonable answer for the question you are being asked. We were asked, “How much taller than the gorilla is the giraffe?” Our answer is: The giraffe is three times taller than the gorilla. Our answer is reasonable because it tells how much taller the giraffe is, compared to the gorilla. We divided a whole number by a whole number, so our quotient should be less than my dividend. Three is less than 18, so our answer makes sense.
Solved Examples
Example 1 There are 287 rows in a stadium. How many students can be seated in this stadium if each row has 165 seats to be occupied?
Solution We are given that,
Number of rows in a stadium = 287
Number of seats in each row = 165
Total number of students that can be seated in the stadium = 287 x 165 = 47335.
Example 2 Henry bought 15 packets of cookies. Each packet contains 35 cookies. How many cookies in all does Henry have?
Solution We are given that
Number of packets of cookies bought by Henry = 15
Number of cookies in each packet = 35
Total number of cookies that Henry has = 15 x 35 = 525
Key Facts and Summary
- The process of finding out the product between two or more numbers is called multiplication. The result thus obtained is called the product.
- Division is the equal sharing of a given quantity.
- A word problem is a few sentences describing a ‘real-life’ scenario where a problem needs to be solved by way of a mathematical calculation.
Recommended Worksheets
Fact Families for Multiplication and Division (Summer Themed) Math Worksheets
Multiplication and Division of Fractions (Veterans’ Day Themed) Math Worksheets
Multiplication and Division Problem Solving (Halloween Themed) Math Worksheets
