5 ms·

Wrt the example of fraction division, isn't this the standard way? Was anyone here taught to cross-multiply for division?

lacksconfidence 11y ago
fwiw, I was taught in the mid 90's about how (a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc)
enjo 11y ago
I'm struggling to remember how exactly I was taught, but cross-multiplying was the method I've used for as long as I can remember. I really believe it was how I was taught when we first encountered fractional division.
sleepychu 11y ago
Cross multiplication is a trick to multiply by the 'reciprocal' of the fraction. The failing here is that you don't understand why you get the right answer here. Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) * (4/1) = 2 that's just a 'convenient trick' to get the answer quickly. This is a great idea if your goal is to pass some maths exams which have a fixed format in the near future. It's a terrible idea if you want to be able to apply mathematics to anything.
- aptwebapps 11y ago
  Oh, I see. I thought cross multiplying was referring to what you do in order to add or subtract two fractions with different denominators. Should have just looked it up. The failing here is that you don't understand why you get the right answer here. By 'you' you mean the putative student mislead by shortcuts, right? I understand perfectly well what's happening with reciprocal multiplication and why it works. ;)
- IanCal 11y ago
  > Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half That sounds like looking for a "trick" to me, rather than the much simpler generic rearrangement. Trying to cram that phrase into more complex ones I'd start thinking "Wait, how many five eighths are there in seventy sixteenths?" If the question is "what is (1/2) / (1/4)" and you can't see it, then the idea of rearranging the equation and why rearranging it is OK is pretty important and general. (5/8) / (70/16) = x 5/8 = 70x / 16 5 . 16/8 = 70x 5 . 2 = 70x 10 = 70x 1/7 = x That seems simpler to me than "There is one seventh of seventy sixteenths in five eighths" and it requires knowing how basic multiplication and division works, and that the equals sign means that the things on both sides are equal, so if we multiply both sides by the same number or divide by the same number then that's OK. That can then lead into why you need to be careful about values being 0 or why sometimes you have to add ± to an answer. You can also just generally explore rearrangements, try out different modifications and see if it makes things simpler, go back and try others.
- plonh 11y ago
  Whaaa? 1/(1/x))=x is absolutely true for all x!=0 in a field. How does your method explain how to solve "1/2 / 3/4" ? How does talking about how many 4/3 are in 1/2 help?
  - ashark 11y ago
    How does it not? There's 2/3 of a 3/4 in a 1/2 (I'm assuming that last sentence was typoed and "talking about how many 3/4 in 1/2" was intended.)
- anthony_d 11y ago
  I've never heard of most of the tricks mentioned and agree they seem like terrible things to teach, but calling cross multiplication a trick seems bizarre. Cross multiplication is multiplying by the reciprocal. In neither case do you have to understand what you're doing. You can make this complaint about every arithmetic operation.
- pflats 11y ago
  Just for clarity on all my responses in these threads, I'm a (minor) contributor to Nix the Tricks and a personal friend and research partner of the author. Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) x (4/1) = 2 that's just a 'convenient trick' to get the answer quickly. No, both of these are true. Your logical answer is correct: 1/4 goes into 1/2 twice because there are two quarters in every half. This is how our ancestors conceived division and fractions, and that's how physically dividing things works in the real world. But division is not an independent operation; it's the inverse of multiplication. As plonh mentions downthread, division is, quite literally, defined on a field (like Q) as multiplying by the reciprocal. (1/2) / (1/4) = 2 because (4/1) = (1/4)^-1 and (1/2) * (4/1) = 2. Any other mathematical explanation is simply justifying this fact in more palatable terms.
ygra 11y ago
Just quickly skimmed a bit of the book, but I have never seen or heard any of those tricks (German here) with the exception of the formula triangle (still have that in mind for Ohm's law). That being said, I mainly need it for remembering the relationship and have no trouble solving the formula for a different quantity.

lacksconfidence 11y ago

fwiw, I was taught in the mid 90's about how (a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc)

enjo 11y ago

I'm struggling to remember how exactly I was taught, but cross-multiplying was the method I've used for as long as I can remember. I really believe it was how I was taught when we first encountered fractional division.

sleepychu 11y ago

Cross multiplication is a trick to multiply by the 'reciprocal' of the fraction. The failing here is that you don't understand why you get the right answer here. Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) * (4/1) = 2 that's just a 'convenient trick' to get the answer quickly. This is a great idea if your goal is to pass some maths exams which have a fixed format in the near future. It's a terrible idea if you want to be able to apply mathematics to anything.

aptwebapps 11y ago
Oh, I see. I thought cross multiplying was referring to what you do in order to add or subtract two fractions with different denominators. Should have just looked it up. The failing here is that you don't understand why you get the right answer here. By 'you' you mean the putative student mislead by shortcuts, right? I understand perfectly well what's happening with reciprocal multiplication and why it works. ;)
IanCal 11y ago
> Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half That sounds like looking for a "trick" to me, rather than the much simpler generic rearrangement. Trying to cram that phrase into more complex ones I'd start thinking "Wait, how many five eighths are there in seventy sixteenths?" If the question is "what is (1/2) / (1/4)" and you can't see it, then the idea of rearranging the equation and why rearranging it is OK is pretty important and general. (5/8) / (70/16) = x 5/8 = 70x / 16 5 . 16/8 = 70x 5 . 2 = 70x 10 = 70x 1/7 = x That seems simpler to me than "There is one seventh of seventy sixteenths in five eighths" and it requires knowing how basic multiplication and division works, and that the equals sign means that the things on both sides are equal, so if we multiply both sides by the same number or divide by the same number then that's OK. That can then lead into why you need to be careful about values being 0 or why sometimes you have to add ± to an answer. You can also just generally explore rearrangements, try out different modifications and see if it makes things simpler, go back and try others.
plonh 11y ago
Whaaa? 1/(1/x))=x is absolutely true for all x!=0 in a field. How does your method explain how to solve "1/2 / 3/4" ? How does talking about how many 4/3 are in 1/2 help?
- ashark 11y ago
  How does it not? There's 2/3 of a 3/4 in a 1/2 (I'm assuming that last sentence was typoed and "talking about how many 3/4 in 1/2" was intended.)
anthony_d 11y ago
I've never heard of most of the tricks mentioned and agree they seem like terrible things to teach, but calling cross multiplication a trick seems bizarre. Cross multiplication is multiplying by the reciprocal. In neither case do you have to understand what you're doing. You can make this complaint about every arithmetic operation.
pflats 11y ago
Just for clarity on all my responses in these threads, I'm a (minor) contributor to Nix the Tricks and a personal friend and research partner of the author. Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) x (4/1) = 2 that's just a 'convenient trick' to get the answer quickly. No, both of these are true. Your logical answer is correct: 1/4 goes into 1/2 twice because there are two quarters in every half. This is how our ancestors conceived division and fractions, and that's how physically dividing things works in the real world. But division is not an independent operation; it's the inverse of multiplication. As plonh mentions downthread, division is, quite literally, defined on a field (like Q) as multiplying by the reciprocal. (1/2) / (1/4) = 2 because (4/1) = (1/4)^-1 and (1/2) * (4/1) = 2. Any other mathematical explanation is simply justifying this fact in more palatable terms.

ygra 11y ago

Just quickly skimmed a bit of the book, but I have never seen or heard any of those tricks (German here) with the exception of the formula triangle (still have that in mind for Ohm's law). That being said, I mainly need it for remembering the relationship and have no trouble solving the formula for a different quantity.