Question

Hello everyone!! Quick dimensional analysis question.

I'm stuck at the multiplying step of this problem, because although I see the work, I won't be able to understand it to do it on my own. I know that they divided 540 by 60 to get 9, but why did they divide? I thought they were supposed to multiply solely.

And how did 60 cross out to become 1?? Somebody help, thank you! (picture of the problem in the replies).

Answer 1

You need to cancel off hours, and there's 60 min in 1 hour...\[\large \frac{ 540 \text{mi} }{ 4.5 \text{hr} }*\frac{ 1 \text{hr} }{ 60 \min }\]

Answer 2

Converting units is basically just multiplying by 1. But 1 here is written in the form of a fraction where what's in the numerator is the same as what's in the denominator. In this case, it's 1 h / 60 min. Since 1 hour equals 60 minutes, this fraction is equal to one, but it's carefully chosen so units in the numerator and denominator cancel so as to leave the answer in the units you need.

Answer 3

This is what they did... they divided the top by 60 and the bottom by 60.

That's always valid as long as you divide the top and bottom of a fraction by the same number.

They did it to simplify.

\[\large \frac{ 540 \text{mi}\div 60 }{ 4.5 \text{hr} }*\frac{ 1 \text{hr} }{ 60 \min \div 60}\]

Answer 4

I wish I could pick two best responses. Thank you @twwc960 for a great explanation and you as well, @agent0smith !! Now it makes a bit more sense seeing the work explained.

Answer 5

You realize you don't have to do it the way they do it? Do it in a way that makes sense to you.

Answer 6

I actually didn't really remember how to figure out dimensional analysis, so I needed a way that makes sense to me first to actually complete something. But you're 100% correct, that's what my 8th grade math teacher has always said.

Answer 7

Another way to do it, is just work out how many minutes there are in 4.5 hours.

Then you divide the miles driven, by the total number of minutes, to get miles per minute.

Answer 8

I rarely do things in 'dimensional analysis' way. I do things in a way that makes sense.

Answer 9

I think I'll start getting the hang of that, then. Thank you so much for answering my question, and even giving me more advice. You made so much more sense to me.

Answer 10

Just like if you had speed in miles per hour, and you want miles per second... if you know how far it goes in one hour, divide that by how many seconds in one hour.

Answer 11

Can I ask of you one more question, @agent0smith ?

Answer 12

Sure

Answer 13

How would I know to divide 24 by 8? I don't know if with what you explained previously could help me, but if so, I'm still a bit stuck, then.

Answer 14

Like, I know the different ways to do it now, but I don't know when to do it.

Answer 15

Why wouldn't you divide by 8? 8 is in the denominator.

Answer 16

Again. Don't follow their working. Do it in a way that makes sense to you.

Answer 17

No yeah, but...they're both different. Thinking now I could have made the connection of dividing the biggest number with 8 (probably?) But my first thought wouldn't have been to divide 24 from 24hrs by 8 from 8oz. I don't know if I'm making sense.

Answer 18

It's much harder trying to explain my thoughts by typing them than speaking them.

Answer 19

Ignore all the 1s in the fractions, and this is what you have.\[\large \frac{ 2*24*7 }{ 8}\]What is the EASIEST way to simplify this? Dividing 24 by 8.

Answer 20

Or, by steps, 2 times 24 which is 48, times 7, which is 336, then divide THAT by 8 to get 42.

Answer 21

Sure... if you want to make things much harder for yourself. I'd recommend trying to simplify first. Don't complicate and then simplify.

Answer 22

It actually seemed easier...Is that bad? lol

Answer 23

But both ways make 100% sense, though.

Answer 24

Can you do it in your head easily? because this is not hard to do in your head:\[\large \frac{ 2*24*7 }{ 8} = \frac{ 2*3*7 }{ 1}=\]compared to this\[\large \frac{ 2*24*7 }{ 8} = \frac{ 48*7 }{ 8}=\frac{ 336 }{ 8} = \]
It's even harder to write this way...48*7 is not something you instantly know in your head. Neither is 336/8.

2*3*7 is easy if you know times tables.

I'd recommend learning how to do things in simpler ways, than more complex ways that are far harder to do in your head.

Answer 25

True, true. 48*7 is something easy for me, at least. But not long division. So your point is obviously correct. But I was doing it on paper with my pen and it seemed as fast as dividing 24 by 8 first.

Answer 26

What about when you get to larger numbers? \[\large \frac{ 34 * 20 * 16 }{ 17*5*8}\]Would you rather do this the long way, or by simplifying\[\large \frac{ 34 * 20 * 16 }{ 17*5*8} = 2 *4 * 2 \]which of those looks like an easier method? :P

Answer 27

Probably the second one...xD

Answer 28

But the first one looks easier because I know how to get the answer straight away, even though it would be considered a "harder" way.

Answer 29

What if you were doing it on a no calculator test? Sure, you could do the multiplication part on paper... have fun with that long division, though.

The first one looks easier because you haven't practiced the second way.

Answer 30

True!! But if I had to do the long division, I had to do it. I'll have fun with it, though. Math is cool.

Answer 31

I'd force yourself to start looking for common factors, simplify things in the numerator and denominator as much as possible, it'll make things far easier when you're used to it.

Answer 32

Plus... it feels much cooler to do it the short way, you feel like a wizard.

Answer 33

As opposed to (which i used a calculator for, obviously)\[\large \frac{ 34 * 20 * 16 }{ 17*5*8} =\frac{10880 }{680}=\]