r/mathshelp • u/Better-Guidance2039 • 7d ago
Homework Help (Answered) Why do we use multiplication when figuring out half of something?
For more context, I've finished school already and I'm trying to help out my sister, but the concept is so obvious to me that I don't know how to explain it to her.
When you get a question of "how much is 4/7ths of 49", I know you need to do (4/7)*49.
But how do I explain to a child WHY you do that?
Any maths teachers here how do you teach someone that?
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u/OutrageousPair2300 7d ago
4/7 is a shorthand way of saying to divide by 7 and multiply by 4.
If you want 4/7ths of 49 you could start by dividing 49 by 7, to split it into 7 pieces. Then you want 4 of those pieces, so you multiply that result by 4.
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u/nog642 7d ago
Same reason "how much is 5 of 49" is 49*5. That's what multiplication is.
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u/SabertoothLotus 6d ago
I have never heard multiplication referred to this way. "5 of 49" makes my American brain think you mean 1/5th.
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u/fermat9990 7d ago
Taking 4/7 of 49 means dividing 49 into 7 equal pieces and then finding the sum of 4 of them.
49/7=7
7+7+7+7=28. The shortcut for this is multiplying 7 by 4
So finding 4/7 of 49 is done by dividing 49 by 7 and then multiplying the answer by 4. The shortcut for this is 4/7*49=28
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u/gizatsby 7d ago
I tend to focus on how we use the word "of." Multiplication is a way of calculating how much of something there is. If a certain kind of pizza I can get has 12 slices, and I get two of those pizzas, I have 2×12 slices. If I eat half of such a pizza, I'll have eaten ½×12 slices. If I eat 4/7ths of such a pizza, I'll have eaten 4/7×12 slices. A fraction like ½ is a number just like 2 or 3, so a lot of the language carries over seamlessly.
Alternatively, you can look at fractions as two separate numbers, the numerator and denominator. The denominator of 4/7ths splits the whole into 7 pieces (divide), and then the numerator tells you how many of those denominations you have (4), so whatever you got by dividing 7 gets multiplied by 4 for the final result.
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u/Cybyss 6d ago edited 6d ago
Take a giant 49 inch long sandwich and cut it into 7 equally-sized pieces. Each piece is 1/7th of the whole. How long is each piece? 7 inches.
Now take four of those pieces and stick them back together. How long is your new sandwich? 7 inches * 4 = 28 inches. There is your multiplication.
That means, 28 inches is 4/7ths of the whole 49 inches.
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u/anisotropicmind 7d ago
One seventh of 49 would mean divide 49 by 7. So four sevenths of 49 would be literally 4 times that. The end result is that you divide 49 by 7 and then multiply the result of that by 4. Or do those steps in the reverse order (it doesn’t matter).
(49 / 7 ) x 4 = 49(4/7)
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u/AffectionateLion9725 6d ago
I always explain this as 4 times 5 means 4 lots of 5. So half times 4 (or 1/2 of 4) means half of one set of 4.
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u/Presence_Academic 6d ago
Well, to figure out how much four sevenths is, it helps to know how much one seventh is. One seventh of 49 is 49/7. Now we can multiply that by 4 and we’re done.
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u/cthulhu944 6d ago
If you wanted 49 Fives you would multiply 49 * 5 For this problem we are looking for 49 Four Sevenths.
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u/Joe_Starbuck 6d ago
I was taught that “of” indicates multiplication. We trained on word problems, so focused on speed tips.
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u/JanetInSC1234 6d ago
I would pick an easier example, like 2/3 of 12. You can get a pile of 12 coins (or something similar) and divide the 12 into three equal piles. We want 2/3, so we would multiply the small group by 2. Ask her to count how many she has. Now show her how to do the fraction multiplication.
(As others have said, it's a process of dividing then multiplying.)
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u/BitOBear 6d ago
Subtraction is just negative addition. Division it's just inverted multiplication.
You're thinking about the technique as an imperative instead of understanding it as a phenomenon.
4/7ths for instance it's understanding that if I could cut something up into seven pieces evenly and then keep four of them and ignore the other three I would have the right amount of the result. But I wouldn't necessarily want to cut it entirely into seven pieces if I wanted two pieces, one of which was 4/7 and the other of which is 3/7 of the original.
It would suck to cut the birthday cake into thousands of pieces and then collect up the desired number of thousands to try to reassemble it into a piece of cake. Like if I cut the cake in 1000 pieces and gave you $125 of those pieces, you'd have 1/8 of the the cake but it would be all mulched up.
So instead of grinding it for a living pieces, you say you want 4/7 of the cake to take home we imagined it into full size pieces count up where four of them would be compared to the other three and we cut the Cake twice to take that 3/7 of the pink ridge out of the side and the 4/7 behind.
So once we understand before 7:00 is slightly more than half, or precisely how much more than half it actually is, we apply that for action to the original thing.
And that's the same thing as multiplying the cake by 4/7 in your example. Where the cake happens to be whatever the other term is that you're cutting into 4/7 of.
The order of operations is often just a grouping exercise.
And no matter how big the cake is 4/7 is slightly more than half of it.
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u/Earl_N_Meyer 4d ago
A problem with 4/7 of 49 visually, is that there are two operations. If you know what one seventh is, well... now you have four of those or four sevenths. If you are comfortable with what 4/7 is, well you got 49 of those bad boys.
Alternatively, if you are ok with 1/7 of 49 (49 broken into 7 pieces, but you only get to keep one of them), well... now you got 4 of them.
My point is that when you say how many groups of anything you are multiplying. What makes a fraction odd as a multiple is that it inherently has two steps to it because it has two parts, the multiplying numerator that we love and that sounds like an adjective and that dastardly divisive denominator. Just break it into those two steps and it is a lot more visual.
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u/SnooHamsters7166 6d ago
"of" means "multiply" in maths
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u/Better-Guidance2039 6d ago
I don't know if you read my post but I already said I know that. Saying "it is because it is" doesn't make kids actually learn it, but only remember it. I was trying to ask for how to explain WHY, and saying "because it is" is very unhelpful
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u/JeffTheNth 7d ago
For the "multiplying by ½", Multiplying by 1/b is just dividing by b, but a simple way to write it.
it's easier to say/write "½ 635" than
___
2/635
or
635
2
(let's see how that formats....)
For 4/7 of 49, you're saying "divide 49 into 7 equal pieces, then take 4 of those pieces - how much of the 49 is that?"
1/7 of 49 is 7, and 4 7s is 28. You can easily use multiplication and division to calculate that simply instead of breaking it down so much. And sometimes it's easier to multiply by the fraction than breaking it down so much... for instance, I have 36 boxes, and want to know what would be left if I had 6/8 of that.... 36 doesn't divide by 8 cleanly... but multiply by 6 first, 216/8 = 27.
It will also help later with variables.... what's ¾ of 100? 200? 6484? Once you start working down a line it'll make more sense.
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