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Why Do We Multiply Numerator & Denominator When Multiplying Fractions?

• Context: High School
• Thread starter Thread starter bballwaterboy
• Start date Start date Sep 2, 2014
• Tags Tags Fractions

Click For Summary

Discussion Overview

The discussion centers around the question of why we multiply the numerators and denominators when multiplying fractions, exploring the underlying reasoning and conceptual understanding of this mathematical operation. Participants share their thoughts on the differences between multiplication and addition of fractions, along with illustrative examples.

Discussion Character

• Exploratory
• Conceptual clarification
• Debate/contested
• Mathematical reasoning

Main Points Raised

• Some participants express confusion about why a common denominator is not needed when multiplying fractions, contrasting it with the necessity of a common denominator for addition.
• One participant suggests that multiplying fractions can be understood through the analogy of dividing a pie among people, illustrating the concept with a specific example.
• Another participant emphasizes the importance of getting the correct answer when multiplying fractions, suggesting that alternative methods would not align with physical reality.
• A different perspective is presented, where the denominator is described as a "unit," drawing parallels between the multiplication of fractions and calculating the area of a rectangle.
• One participant introduces a more formal mathematical interpretation of fractions, discussing the definitions and operations involved in multiplying them, and how this relates to the distributive law in addition.
• Participants correct each other on minor errors in examples, indicating a collaborative effort to refine the discussion.

Areas of Agreement / Disagreement

Participants generally agree on the confusion surrounding the multiplication of fractions compared to addition, but multiple competing views and explanations remain regarding the reasoning behind the operations. The discussion does not reach a consensus on a singular explanation.

Contextual Notes

Some explanations rely on specific interpretations of fractions and mathematical operations, which may depend on the definitions used. The discussion includes various illustrative examples that may not cover all potential scenarios or assumptions.

Who May Find This Useful

This discussion may be useful for students grappling with the concepts of fraction multiplication, educators seeking to understand common misconceptions, and anyone interested in the foundational reasoning behind basic arithmetic operations.

bballwaterboy Messages 85 Reaction score 3 This is kind of a "dumb" question, but why do we multiply the numerators and denominators when multiplying fractions? For example: 1/5 x 2/3 = 2/15 Intuitively, I know why we need a common denominator when adding and subtracting fractions. We need to add apples to apples and oranges to oranges for it to logically make sense. But why do we suddenly not need a common denominator when multiplying fractions? Wouldn't the same analogy apply here? Don't we need to do an apples to apples kind of operation? Mathematics news on Phys.org

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Mark44 Mentor Insights Author Messages 38,095 Reaction score 10,646

bballwaterboy said: This is kind of a "dumb" question, but why do we multiply the numerators and denominators when multiplying fractions? For example: 1/5 x 2/3 = 2/15 Intuitively, I know why we need a common denominator when adding and subtracting fractions. We need to add apples to apples and oranges to oranges for it to logically make sense. But why do we suddenly not need a common denominator when multiplying fractions? Wouldn't the same analogy apply here? Don't we need to do an apples to apples kind of operation?

Let's make a slightly easier problem, namely 1/5 X 1/3. Your problem is only a bit harder. Imagine a pie cut into three large pieces, two of which have already been eaten, so that we have 1/3 of the pie. If we want to divide this piece of pie among five people equally, what fraction of the pie will each get? I'm using the idea that dividing by 5 is the same as multiplying by 1/5. There are many different forms for the fraction 1/3, such as 2/6, 3/9, and so on. A form with a 5 in the numerator would be helpful - 5/15 would be a good choice. So one-fifth of 5/15 would be 1/15. After doing many such problems, you might get the idea that the answer could have been calculated more quickly simply by multiplying the numerators (getting 1) and the denominators (getting 15). Last edited: Sep 3, 2014 Matterwave Science Advisor Homework Helper Gold Member Messages 3,971 Reaction score 329

Mark44 said: There are many different forms for the fraction 1/3, such as 2/6, 3/92, and so on.

I think you typo'd an extra "2" there. Mark44 Mentor Insights Author Messages 38,095 Reaction score 10,646 It's fixed now. I originally wrote 3/12, and neglected to get rid of the 2 when I changed the fraction to 3/9. Thanks for the correction! NascentOxygen Staff Emeritus Science Advisor Homework Helper Messages 9,253 Reaction score 1,080

bballwaterboy said: This is kind of a "dumb" question, but why do we multiply the numerators and denominators when multiplying fractions?

The most persuasive reason for doing it that way: so that you get the right answer! Do it any other way, and the answer won't agree with physical reality. No, I'm not kidding! HallsofIvy Science Advisor Homework Helper Messages 42,895 Reaction score 983 One way to look at it is that the denominator of a fraction is a "unit". Just as saying the length of a line segment is "3 meters" means that I am using "meter" as my unit of length and the line segment is three of them, so the fraction "3/4" means that we are dealing with units of "one fourth" and we have three of them. So just as a rectangle with sides "3 meters" and "5 meters" has area "15 square meters" or "15 m^2" so the fraction "3/4= 3 fourths" multiplied by the fraction "3/5= 3 fifths" is 9 (fourths x fifths)= 9/20. Fredrik Staff Emeritus Science Advisor Homework Helper Insights Author Gold Member Messages 10,876 Reaction score 423

bballwaterboy said: This is kind of a "dumb" question, but why do we multiply the numerators and denominators when multiplying fractions? For example: 1/5 x 2/3 = 2/15 Intuitively, I know why we need a common denominator when adding and subtracting fractions. We need to add apples to apples and oranges to oranges for it to logically make sense. But why do we suddenly not need a common denominator when multiplying fractions? Wouldn't the same analogy apply here? Don't we need to do an apples to apples kind of operation?

It's not a dumb question at all. This is one way of looking at it: 1/5 is by definition the real number x such that 5x=1. 1/3 is defined similarly. 2/3 should be interpreted as ##2\cdot\frac{1}{3}##. To multiply the fractions 1/5 and 2/3 is to solve the equation ##\frac 1 5 \cdot \frac 2 3 =x##. If you multiply both sides by 5, you get ##\frac{2}{3}=5x##. If you multiply both sides of that by 3, you get 2=3·5·x. If you multiply both sides of that by ##\frac{1}{3\cdot 5}##, you get ##\frac{2\cdot 1}{3\cdot 5}=x##. You also asked about the common denominator when we're doing addition. I would say that the reason is that we would like to use the distributive law: a(b+c)=ab+ac. If we see a sum ab+ac with a common factor (in this case a) in both terms, the distributive law tells us that we can rewrite the sum as a(b+c). The point of rewriting ##\frac 2 3+\frac 4 5## with a common denominator is that it enables us to identify a common factor in each term: \begin{align} &\frac 2 3+\frac 4 5 =\frac 2 3\cdot 1+\frac 4 5\cdot 1 =\frac 2 3\cdot \frac 5 5 +\frac 4 5\cdot\frac 3 3 =\frac{2\cdot 5}{3\cdot 5}+\frac{4\cdot 3}{5\cdot 3}\\ & =\frac{1}{15}\cdot 10+\frac{1}{15}\cdot 12 =\frac{1}{15}(10+12)=\frac{1}{15}{24}=\frac{24}{15}. \end{align} The first calculation I did shows that we don't need a common denominator when we multiply fractions. The second calculation should explain why: The distributive law is used only when both addition and multiplication are involved. Last edited: Sep 3, 2014 bballwaterboy Messages 85 Reaction score 3 Thanks guys for the examples. I do have something to ask and follow-up on, but will write back this weekend. Need to cram for a quiz today (not in math though)! It's odd, though, how a simple math concept can be so confusing at times. I'll write back more once I get the chance. Thanks for now everyone! symbolipoint Homework Helper Education Advisor Gold Member Messages 7,663 Reaction score 2,090 Put my post back!

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