multiplication and division problem solving

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Multiplication Word Problems for Grade 3

Simple multiplication.

These worksheets contain simple multiplication word problems. Students should derive a multiplication equation from the word problem, solve the equation by mental multiplication and express the answer in appropriate units. Students should understand the meaning of multiplication before attempting these worksheets.

multiplication and division problem solving

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Math Word Problems

Welcome to the math word problems worksheets page at Math-Drills.com! On this page, you will find Math word and story problems worksheets with single- and multi-step solutions on a variety of math topics including addition, multiplication, subtraction, division and other math topics. It is usually a good idea to ensure students already have a strategy or two in place to complete the math operations involved in a particular question. For example, students may need a way to figure out what 7 × 8 is or have previously memorized the answer before you give them a word problem that involves finding the answer to 7 × 8.

There are a number of strategies used in solving math word problems; if you don't have a favorite, try the Math-Drills.com problem-solving strategy:

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Solving problems by multiplying and dividing fractions and mixed numbers, fraction word problems with interactive exercises.

Example 1: If it takes 5/6 yards of fabric to make a dress, then how many yards will it take to make 8 dresses?

Analysis: To solve this problem, we will convert the whole number to an improper fraction. Then we will multiply the two fractions.

Answer: It will take 6 and 2/3 yards of fabric to make 8 dresses.

Analysis: To solve this problem, we will multiply these two fractions.

Answer: Elena got 3/8 of the original box of cupcakes.

Analysis: To solve this problem, we will multiply these mixed numbers. But first we must convert each mixed number to an improper fraction.

Answer: The area of the classroom is 9 and 7/20 square meters.

Analysis: To solve this problem, we will divide the first fraction by the second.

Answer: 2 pieces

Analysis: To solve this problem, we will divide the first mixed number by the second.

Answer: The electrician has 2 and 5/8 pieces of wire.

Analysis: To solve this problem, we will divide the first mixed number by the second. First, we will convert each mixed number into an improper fraction.

Answer: The warehouse will have 2 and 2/25 pieces of tape.

Summary: In this lesson we learned how to solve word problems involving multiplication and division of fractions and mixed numbers.

Exercises

Directions: Subtract the mixed numbers in each exercise below. Be sure to simplify your result, if necessary. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

Note: To write the mixed number four and two-thirds, enter 4, a space, and then 2/3 into the form.

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Year 5 grade 4 unit 7 Multiplication and Division

Lesson 10: Problems Solving Involving Multiplication And Division

Objectives:

At the end of this lesson, students should be able to:

Word problems are general mathematical questions that have been translated into the form of a story. They are basically used to test one’s analytical skills as well as the understanding of a particular topic. To solve word problems, one must follow a number of steps as follows:

Some keywords and catch phrases are shown below:

Sometimes the problem may not give any of these keywords but rather it may suggest what should be done in the way the question is asked.

NB: Always bear in mind that multiplication and division are inverse to each other. Therefore, continuous addition means multiplication whereas continuous subtraction means inverse.

A packet of toffees contains 24 toffees. How many toffees will be in 3 packets?

1 packet contains 24 toffees

3 packets mean 24 + 24 + 24 which is continuous addition multiplication.

Therefore, 3 packets would contain: 24 toffees x 3

Hence 3 packets will contain 72 toffees.

A cinema holds 358 audiences each day. If there is a show for different audiences in 12 days, how many people altogether can the cinema hold?

Number of people held in 1 day = 358

Number of people held in 12 days = 358 x 12

Hence, in 12 days the cinema can hold 4296 audiences.

There are 858 audiences at a show. They were shared evenly into 6 auditoriums. How many audiences were placed in each auditorium?

Total number of audience = 858

No of auditoriums = 6

“Shared evenly means division”

143 audiences were placed in each auditorium.

Gail earns £150 a week and her little brother gets 5 times less. How much does her brother get in a week?

Amount Gail earns in a week = £150

Amount Gail’s brother earns = 5 times less

If “5 times more” means multiplication then “5 times less” would imply division.

Gail’s brother gets £30 in a week.

Ian went shopping at a supermarket. He bought 6 cartons of wine at £13.98 each. How much did he pay at the counter?

Cost of each carton of wine = £13.98 = 1398p

No of cartons bought = 6

Continuous addition means multiplication.

Hence, Ian paid £83.88 at the counter.

Unit 7 Lesson 10: Exercise 1

Unit 7 Lesson 10: Exercise 2

Unit 7 Lesson 10: Exercise 3

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Math Article

Multiplication And Division

In Mathematics, multiplication and division are the two important arithmetic operations. Both the multiplication and division operations are closely related to each other just like addition and subtraction . All these operations are performed on all real numbers. The rules for multiplication and division of integers differ from rules for fractions and decimals.

Let us learn more about multiplication and division along with rules and examples. Also, learn how to multiply and divide integers, fractions and decimals.

What is Multiplication and Division in Maths?

In mathematics, there are four basic operations.

What is Multiplication?

Multiplication is the repeated addition of a number. If we multiply m by n, that means m is repeatedly added to itself for n times. The symbol for multiplication is ‘×’.

For instance, 8 multiplied by 4 is equal to 32. How? Adding 8, 4 times to itself, we get;

8 + 8 + 8 + 8 = 32

Therefore, we can write,

What is Division?

The division is a method of dividing or distributing a number into equal parts, For example, if 16 is divided by 4, then 16 is divided into 4 equal parts. Therefore, the resultant value is 4.

Parts of division

Dividend ÷ Divisor = Quotient

In the above example, there are three parts for division.

Multiplication and Division Relationship

Multiplication and division, are inverse operations of each other. If we say, a multiplied by b is equal to c, then c divided by b results in a. Mathematically, it can be represented as:

For example,

Multiplication and Division Rules

For every mathematical computation, we need to follow the rules. Thus even to multiply and divide the numbers, there are some rules which we need to follow.

Rule 1: Order of operations

The order of operations for multiplication does not matter. It means if we arrange the number in a different order while multiplying them, then the result will be the same.

Examples are:

In the above example, we can see, even if we have swapped the position of 3 and 4, the product of the two integers is equal to 12.

But this rule is not applicable for division. Let us take another example.

3 ÷ 12 ≠ 4 (it is equal to 0.25)

Thus, we cannot change the order of numbers in division method.

Rule 2: Multiplying and Dividing by Positive Numbers

If any real number is multiplied or divided by the positive real number, then the sign of the resulting number does not change.

-2 x 3 = -6

Since, 2 and 3 both are positive integers, therefore the product of 2 and 3 is also positive. But the product of -2 and 3 is a negative number.

-4 ÷ 2 = -2

Since, 4 and 2 both are positive, therefore, 4 divided by 2 is also a positive number. But -4 divided by 2 is a negative number.

Thus, we can conclude that:

Rule 3: Multiplying and Dividing by Negative Numbers

Multiplication and division of any real number by a negative number will change the sign of the resulting number. Examples are given below.

Multiply 5 by -2.

5 x -2 = -10

Multiply -5 by -2.

-5 x -2 = 10

Divide 10 by 2.

10 ÷ -2 = -5

-10 ÷ -2 = 5

Thus we can conclude that:

Summary of Multiplication and Division Rules

Multiplication and division of integers.

Integers are those values that are not fractions and can be negative, positive or zero. The integers can be easily represented on a number line. Thus, the arithmetic calculations on integers can be done in a simple manner.

Multiplying and dividing any integer with a whole number, or a fraction or an integer itself is given below with examples.

Multiplication and Division of Fractions

Here, we will learn how to multiply and divide fractions with examples.

A fraction is a part of a whole. For example, ½ is a fraction that represents half of a whole number or any value. Here, the upper part is called the numerator and the lower part is called the denominator. Let us multiply and divide fractions with examples.

¼ x ½ = (1 x 1)/(4 x 2) = ⅛

¼ ÷ ½ = (1 x 2)/(1 x 4) = 2/4 = ½

Multiplication and Division of Decimals

Decimals are numbers with a decimal point, (Eg: 2.35). They represent the fraction of something or some value, such as ½ = 0.5. A decimal notation or point differentiates an integer part from fractional part (e.g. 2.35 = 2 + 7/20).

Multiplication of Decimals

When a decimal is multiplied by any real number, then the position of the decimal point (.) changes.

For example: 0.33 x 2= 0.33 + 0.33 + 0.33. Multiplication of decimal numbers is similar to the multiplication of whole numbers. Steps for multiplication of decimals are given below with an example.

Consider multiplication of two numbers, 2.32 and 3 for example.

Step 1: Take the count of the total number of places (digits) to the right of the decimal point in both numbers.

Here, in 2.32 there are two digits to the right of the decimal point and 3 is a whole number with no decimal point. Therefore, the total number of digits to the right of the decimal is 2.

Step 2: Now forget about the decimal point and just multiply the numbers without the decimal point.

Step 3: After multiplication, put the decimal point in answer 2 places (step 1) from the right i.e. answer (2.32 x 3) will be 6.96.

Simply, just multiply the decimal numbers without decimal points and then give decimal point in the answer as many places same as the total number of places right to the decimal points in both numbers.

Division of Decimals

We can use the same trick we used in the multiplication of decimals i.e. remove the decimal points and divide the numbers like whole numbers.

Let’s divide 40.5 by .20. Methods to divide these decimal numbers are as follows:

Method 1: Convert the decimal numbers into whole numbers by multiplying both numerator and denominator by the same number. The denominator must be always a whole number.

(Multiply both the numerator and denominator by 5)

Method 2: Alternatively, one can convert decimal numbers into whole numbers by multiplying with numbers having powers of 10 (10, 100, 1000, etc.).

Consider the denominator, count the number of places (digits) right to the decimal point.

Here, the denominator is 0.20 and the number of digits right to the decimal point is 2.

40.5 ÷ 0.20 = 40.5 / 0.20

Multiplication and Division of Decimal Numbers by 10, 100 and 1000

Multiplication and division of decimals by numbers that have powers of 10 is easier than that by a whole number. Rules for multiplication and division of decimal numbers by 10, 100 and 1000:

Multiplication and Division Equations

The equations are the expression that includes integers, variables, equality signs and arithmetic operations. For example,

If we solve the above equation for a, then,

2a = 7 – 9

a = -2÷2 = -1

Thus, we can see, the above solution involved both multiplication and division methods.

Solved Examples – Multiplication and Division

Q.1: Find the product of:

Q.2: Find the division of:

Solution: The divisions are:

Word Problems on Multiplication and Division

Q.1: There are 90 pencils in 1 box. How many pencils are there in 3 boxes?

Solution: Given, 1 box has 90 pencils.

So, in 3 boxes, number of pencils = 3 x 90 = 270

Therefore, there are total of 270 pencils in 3 boxes.

Q.2: Raju has 1615 candies stored in a box. If there are 85 such boxes, then how many candies are there in each box?

Solution: Total number of candies = 1615

Number of boxes = 85

Therefore, each box contains = 1615 ÷ 85 candies

= 19 candies.

Practice Questions – Multiplication and Division

1. Fill in the blanks:

2. Find the value of:

3. Each toffee cost Rs. 2. If there are 120 toffees, then what is the total cost?

4. A school is planning a trip. There are 1729 students and each bus has 19 seats. How many buses are required to go for the trip?

Frequently Asked Questions on Multiplication and Division

What is the order of operations of multiplication and division, what do we get when we multiply an integer with an integer, how do you divide a number by a fraction, how to multiply decimal by a fraction, is multiplication and division applicable to all the real numbers.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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Multiplication and Division KS2

This collection is one of our Primary Curriculum collections - tasks that are grouped by topic.

One Wasn't Square

Age 7 to 11 challenge level.

Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.

All the Digits

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

Cycling Squares

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

Can you replace the letters with numbers? Is there only one solution in each case?

Multiplication Square Jigsaw

Can you complete this jigsaw of the multiplication square?

Shape Times Shape

These eleven shapes each stand for a different number. Can you use the number sentences to work out what they are?

What Do You Need?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

How Do You Do It?

This group activity will encourage you to share calculation strategies and to think about which strategy might be the most efficient.

Table Patterns Go Wild!

Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

Journeys in Numberland

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Ordering Cards

Age 5 to 11 challenge level.

This problem is designed to help children to learn, and to use, the two and three times tables.

Let Us Divide!

Look at different ways of dividing things. What do they mean? How might you show them in a picture, with things, with numbers and symbols?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Sweets in a Box

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Round and Round the Circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Highest and Lowest

Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number.

Zios and Zepts

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

Abundant Numbers

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

Find at least one way to put in some operation signs (+ - x Ã·) to make these digits come to 100.

Flashing Lights

Norrie sees two lights flash at the same time, then one of them flashes every 4th second, and the other flashes every 5th second. How many times do they flash together during a whole minute?

The Moons of Vuvv

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

Mystery Matrix

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

Four Goodness Sake

Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?

Multiplication Squares

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

Factor Lines

Age 7 to 14 challenge level.

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Two Primes Make One Square

Can you make square numbers by adding two prime numbers together?

Cubes Within Cubes

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Which Is Quicker?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

A Square of Numbers

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Odd Squares

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?

Up and Down Staircases

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Carrying Cards

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

An Easy Way to Multiply by 10?

Do you agree with Badger's statements? Is Badger's reasoning 'watertight'? Why or why not?

Multiples Grid

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Factors and Multiples Game

Age 7 to 16 challenge level.

This game can replace standard practice exercises on finding factors and multiples.

Music to My Ears

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

What's in the Box?

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

Factor-multiple Chains

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

This challenge is a game for two players. Choose two of the numbers to multiply or divide, then mark your answer on the number line. Can you get four in a row?

Picture a Pyramid ...

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

The Remainders Game

Play this game and see if you can figure out the computer's chosen number.

Which Symbol?

Choose a symbol to put into the number sentence.

Times Tables Shifts

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Counting Cogs

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

Light the Lights Again

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

Follow the Numbers

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

Curious Number

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

Factor Track

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

So It's Times!

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

Square Subtraction

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

This Pied Piper of Hamelin

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Multiply Multiples 1

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

Multiply Multiples 2

Can you work out some different ways to balance this equation?

Multiply Multiples 3

Have a go at balancing this equation. Can you find different ways of doing it?

Division Rules

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Always, Sometimes or Never? Number

Are these statements always true, sometimes true or never true?

Satisfying Four Statements

Can you find any two-digit numbers that satisfy all of these statements?

Picture Your Method

Can you match these calculation methods to their visual representations?

Compare the Calculations

Can you put these four calculations into order of difficulty? How did you decide?

Dicey Array

Watch the video of this game being played. Can you work out the rules? Which dice totals are good to get, and why?

4 by 4 Mathdokus

Can you use the clues to complete these 4 by 4 Mathematical Sudokus?

Helping with Math

Multiplication and Division Word Problems

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

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

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 –

Let us understand it through an example.

A hawker delivers 148 newspapers every day. How many newspapers will he deliver in a non-leap year?

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?

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.

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

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.

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.

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?

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.

Jeff read 12 books during the month of August. He read four times as many books as Paul. How many books did Paul read?

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

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.

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.

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?

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

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