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.

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. |

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.

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.

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.

The problem below includes color-coding to help analyze the Product Unknown Comparison. Notice also how the importance of fully stating the answer and also of checking if the answer makes sense.

Mary is saving up money to go on a trip. This month, she saved three times as much money as she saved last month. Last month, she saved $24.00. How much money did Mary save this month? |

Mary is saving up money to go on a trip. This month, she saved three times as much money as she saved last month. Last month, she saved $24.00. How much money did Mary save this month? |

As much as tells you that you have a comparison. Three times is the multiplier. $24.00 is the amount in the first set. How much money did Mary save this month? is the question you are being asked. To solve, multiply $24.00 x 3. |

$24.00 x 3 = $72.00. It’s important to clearly show that you understand what your answer means. Instead of just writing $72.00, write: Mary saved $72.00 this month. |

Whenever you finish a math problem of any kind, always go back to the original problem. Think: “What is the question I am being asked?” Make sure that your final answer is a reasonable answer for the question you are being asked. I was asked, “How much money did Mary save this month?” My answer is: Mary saved $72.00 this month. My answer is reasonable because it tells how much money Mary saved this month. I 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 . My answer makes sense. |

Try the word problems on the worksheet below (the worksheet is also listed towards the bottom of this page)

- Multiplicative Comparison - Product 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? |

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

As many as tells you that you have a comparison. Four times is the multiplier. 12 books is the amount in the second set. How many books did Paul read? is the question you are being asked. To solve, divide 12 by 4. |

12 ÷ 4 = 3 It’s important to clearly show that you understand what your answer means. Instead of just writing 3, write: Paul read three books. |

Remember, whenever you finish a math word problem, always go back to the original problem. Think: “What is the question I am being asked?” Make sure that your final answer is a reasonable answer for the question you are being asked. I was asked, “How many books did Paul read?” My answer is: Paul read three books. My answer is reasonable because it tells how many books Paul read. I 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. |

Try the word problems on the worksheet below (the worksheet is also listed towards the bottom of this page)

- Multiplicative Comparison - Set Size 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.

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? |

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? |

Taller than tells you that you 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? is the question you are being asked. To solve, divide 18 feet by six feet. |

18 ÷ 6 = 3 It’s important to clearly show that you understand what your answer means. Instead of just writing 3, write: The giraffe is three times taller than the gorilla. |

Remember, whenever you finish a math word problem, always go back to the original problem. Think: “What is the question I am being asked?” Make sure that your final answer is a reasonable answer for the question you are being asked. I was asked, “How much taller than the gorilla is the giraffe?” My answer is: The giraffe is three times taller than the gorilla. My answer is reasonable because it tells how much taller the giraffe is, compared to the gorilla. I divided a whole number by a whole number, so my quotient should be less than my dividend. Three is less than 18, so my answer makes sense. |

Try the word problems on the worksheet below (the worksheet is also listed towards the bottom of this page)

- Multiplicative Comparison - Multiplier Unknown

- Multiplicative Comparison - Product Unknown
- Multiplicative Comparison - Set Size Unknown
- Multiplicative Comparison - Multiplier Unknown
- Multiplicative Comparison - Product, Set Size, Multiplier Unknown (mixed)