I am reteaching myself math, but something is bugging me soooo bad and I can’t find the answer. What is a real life example of multiplying a fraction by a fraction? I was wondering why .05 to the 5th exponent would get smaller not bigger. This is driving me bonkers.
Sure 1/2 makes sense, but how about 1/2 times 3/5 in real life?!?
Edit: OMFG. Math is cool and makes sense. Finally, I’m 28. Thank you all!!!!
Edit: I was given an AP Scholar award, but it was not for math.
* * * The best explanation goes to the person who explained “times” and “of” were synonomous!!!! * * *
NOW EXPLAIN THIS: How am I in the 99.9th percentile for arithmetic, but suck at math?! Do I have potential? Am I still gifted in “math” or are math and arithmetic too separate things. A professor told me they are different parts of the brain.
In: Mathematics
A fraction is a way to say you want to multiply by one number *and* also divide by another number.
90 * (2/3) means start with 90, then multiply by 2, and then divide by 3.
Multiplying more than one fraction is to apply each fraction’s multiplication and division. (2/3) * (4/5) means 2 / 3 * 4 / 5. And because the order of multiplications and divisions is not significant, you can reorder this as 2 * 4 / 3 / 5.
Then that can be combined and simplified to (2 * 4) / (3 * 5), or 8 / 15.
A REAL life example:
You’ve been feeding your dog 2/3 cup of brand A kibbles each meal according to the instructions. Then here comes brand B kibbles, a higher quality food for your dog and you’re supposed to give 2/3 cup each meal as well.
Since you don’t want to shock your dog’s digestive system (and result in horrific diarrhoea), you ought to gradually decrease the proportion of brand A kibbles and increase the proportion of brand B kibbles over the course of 5 days or longer.
So here’s the meal plan gonna look–
Day 1: (2/3)X(4/5) cup of brand A + (2/3)X(1/5) cup of brand B
Day 2: (2/3)X(3/5) cup of brand A + (2/3)X(2/5) cup of brand B
Day 3: (2/3)X(2/5) cup of brand A + (2/3)X(3/5) cup of brand B
Day 4: (2/3)X(1/5) cup of brand A + (2/3)X(4/5) cup of brand B
Day 5: (2/3)X(0/5) cup of brand A + (2/3)X(5/5) cup of brand B
Yup, this is a real life example… None of those cutting up pizza/pie crap. Who doesn’t eat the entire thing in one sitting?
I have not seen this being addressed, but my comment here would be that you are trying to continuously anchor mathematics in intuition. That eventually only gets you so far. From a certain point on in mathematics you inevitably leave the grounds of the innately familiar, and move to using the power of applying rules that will give you the right answer, WITHOUT being intuitive. I would argue that separation from intuition is what makes mathematics so powerful.
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I’m surprised I didn’t see any visual representations of multiplication here. Check out this website to see what multiplication of fractions looks like.
https://www.origoeducation.com/blog/focus-on-fractions-a-visual-model-to-teach-multiplication-and-division-of-fractions/
It makes a little more sense if you think about what multiplying is doing in general: it’s sorting things into groups.
When you multiply 2 X 4 what you’re saying is “I want 2 groups of 4.”
So with your example of (1/2) X (3/5) you’re saying “I want half a group of 3/5.” And that’s why the answer (3/10) is smaller and than the original 3/5.
And bonus round: when you divide what you’re asking is “how many groups of one thing does it take to make another?”
So example, if you’re dividing 4 by 2, you’re asking how many groups of 2 are there in 4.
So if we’re dividing (1/2) by (1/4), we’re asking “how many groups of 1/4 does it take to make 1/2?” And this is why the answer is 2.
This is why fractions get smaller when you multiply and larger when you divide.
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The answer you’re looking for is something you haven’t really thought to ask: what does the operation “times” mean in everyday English?
Say “of” whenever you see the times symbol.
Hope that clears up your confusion 😉
Lets look at integers to begin with….x * 4 == x + x + x + x; “x multiplied by four is the same as four of itself added together”
Similarly if we turn that multiplier down to 1 we see a problem like this….x * 1 == x + 0; “x multiplied by one is itself with nothing else added to it.”
Regarding fractions-
If we turn that multiplier down even more it becomes a fraction. Consider how we would add less than nothing or start out with less than the original number.
x * (2/3) == ((x*2)/3) + 0 == ***x + (- (x/3))***; “x multiplied by half is the same as x ***plus a third of itself less than nothing added to it***.
I’ve written the section in bold because I assume the regular “just multiply it by the numerator” solution wasn’t doing it for you. Instead, we’re adding like in the first to examples I gave but we are multiplying less than 1, and we are adding less than zero than x, we are adding a negative number.
[edit: incorrectly said “half”, as the equation is written we should add a negative third (subtract one third)]
This used to bug me a lot too, it clicked when I was thinking about areas. So, let’s say you multiply 2 lengths.
2m x 2m you get 4m2 (2 meters multiplied by 2 meters is 4 meters squared).
So, here the unit is changing from meters to meters squared (so, you are going from length to area). So, you are essentially making a big square with a side of 2m which is the same as 4 1mx1m squares = 4m2
Now, let’s think about fractions.
0.5mx0.5m is 0.25m2. Here, you are making a square with a 0.5m side, and if you think about a square with a side equal to 0.5 m, it would just take a quarter of space as a square with a 1m side which lines up perfectly.
So, we understand that the magnitude is going down, but forget that the units are changing in ways which increases the difference between them dramatically.
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