How smart are you????

[Destroyer]

Ok, this one was bugging the crap outta me.. so as I usually do I did research until I understood the problem. Math needs to make sense, and it seems like this wasn't doing that. The question really is why does the division of fractions produce a larger number?

What we normally think about is that when you divide, your answer should be smaller. Twenty-four divided by six is four. Four is lower than twenty-four.

But when you divide fractions, like 1/4 divided by 1/4, the answer is a
larger number than 1/4. Why is this? I try telling myself that fractions are less than 1 and therefore the math may be different. But in math the answer always must make sense. This was simply not making sense

Then I found an answer that makes sense. It's important for our math to make sense. If our number sense disagrees with the answer, either the answer or our sense has to be fixed. If we leave them both as they are, we'll be in trouble. In this case, what seems natural and obvious when we're used to working with whole numbers has to be modified when we introduce fractions.

Let's think about why dividing by a whole number produces a smaller
result. Dividing 24 by 4 means I want to find out how many 4's it
takes to make 24. If I have 24 sticks

||||||||||||||||||||||||(24)

And I divide 24 by 4 by counting out 6 groups of 4 sticks:

||||(1) ||||(2) ||||(3) ||||(4) ||||(5) ||||(6)

Since each group contains more than one stick, there are fewer groups
than sticks.

Now I divide 6 by 1/4. That means I want to find out how many 1/4's it takes to make 6. If I have 6 sticks

||||||(6)

and I break each of them into 4 pieces:

||||(1) ||||(2) ||||(3) ||||(4) ||||(5) ||||(6)

I find that there are 24 quarter-sticks. Since each stick has been
turned into 4 pieces, there are more pieces than sticks - the result
of the division is greater than the original number of sticks.

(By the way, this also helps to explain why dividing by a fraction is the same as multiplying by the reciprocal.)

You're exactly right... when you divide by a number less than one,
things turn upside down, and the quotient is now larger than the
dividend.

In fact, you're multiplying by the reciprocal, which is a number greater
than one, and you therefore increase the number.

Similarly, multiplying by a number greater than one increases a number, and multiplying by a number less than one decreases it:

24 / 4 = 6 smaller.......6 x 4 = 24 bigger

6 / 1/4 = 24 bigger...... 24 x 1/4 = 6 smaller

So there you have it.. It's actually very simple once you understand the concept behind it... now it makes sense. You see, you can teach an old dog new tricks.

Yes RKC, I was wrong. It happens from time to time. But sometimes it's only by being wrong that you learn why something is right.

[RWilson2526]

Originally Posted by Destroyer

Ok, this one was bugging the crap outta me.. so as I usually do I did research until I understood the problem. Math needs to make sense, and it seems like this wasn't doing that. The question really is why does the division of fractions produce a larger number?

What we normally think about is that when you divide, your answer should be smaller. Twenty-four divided by six is four. Four is lower than twenty-four.

But when you divide fractions, like 1/4 divided by 1/4, the answer is a
larger number than 1/4. Why is this? I try telling myself that fractions are less than 1 and therefore the math may be different. But in math the answer always must make sense. This was simply not making sense

Then I found an answer that makes sense. It's important for our math to make sense. If our number sense disagrees with the answer, either the answer or our sense has to be fixed. If we leave them both as they are, we'll be in trouble. In this case, what seems natural and obvious when we're used to working with whole numbers has to be modified when we introduce fractions.

Let's think about why dividing by a whole number produces a smaller
result. Dividing 24 by 4 means I want to find out how many 4's it
takes to make 24. If I have 24 sticks

||||||||||||||||||||||||(24)

And I divide 24 by 4 by counting out 6 groups of 4 sticks:

||||(1) ||||(2) ||||(3) ||||(4) ||||(5) ||||(6)

Since each group contains more than one stick, there are fewer groups
than sticks.

Now I divide 6 by 1/4. That means I want to find out how many 1/4's it takes to make 6. If I have 6 sticks

||||||(6)

and I break each of them into 4 pieces:

||||(1) ||||(2) ||||(3) ||||(4) ||||(5) ||||(6)

I find that there are 24 quarter-sticks. Since each stick has been
turned into 4 pieces, there are more pieces than sticks - the result
of the division is greater than the original number of sticks.

(By the way, this also helps to explain why dividing by a fraction is the same as multiplying by the reciprocal.)

You're exactly right... when you divide by a number less than one,
things turn upside down, and the quotient is now larger than the
dividend.

In fact, you're multiplying by the reciprocal, which is a number greater
than one, and you therefore increase the number.

Similarly, multiplying by a number greater than one increases a number, and multiplying by a number less than one decreases it:

24 / 4 = 6 smaller.......6 x 4 = 24 bigger

6 / 1/4 = 24 bigger...... 24 x 1/4 = 6 smaller

So there you have it.. It's actually very simple once you understand the concept behind it... now it makes sense. You see, you can teach an old dog new tricks.

Yes RKC, I was wrong. It happens from time to time. But sometimes it's only by being wrong that you learn why something is right.

Destroyer, I think you have completely lost your mind....up at 1:15 AM counting sticks.

[Blue_Runner]

It helps if you turn 1/2 into a decimal. Or at least that is what I did.

1/2 = .5

30 / .5 = 60
60 + 10 = 70

[Destroyer]

Originally Posted by Blue_Runner

It helps if you turn 1/2 into a decimal. Or at least that is what I did.

1/2 = .5

30 / .5 = 60
60 + 10 = 70

Blue, you're exactly correct. What threw me though was something so simple as to be easily overlooked. Ferm actually hit the nail on the head. It was the / symbol. (Divide Divide = Multiply) The / symbol on a computer is or can be used as a divide symbol, since the divide sign that is on a calculator isn't on a computer keyboard. Since I rarely, if ever, use a computer for doing math, I simply forgot that / is divide and * is multiply on a computer keyboard. I completely missed the inversion.

(Truthfully, I think the question could have been written a little better, but the whole idea of this test is to trick us up, so I guess how they wrote it is fair)

And yes RW, sometimes I lose my mind. I usually find it again in a day or so.

(When I don't understand something is just bugs the hell outta me until I find the answer. I'm a night owl by habit, my normal bedtime is around 2am.. so at 1:15am I was just starting to get sleepy).

[Blue_Runner]

10-4 good buddy!

[nipper]

I got a 9 out of 11. Like many, I missed the California widow question and the Moses/Ark question. Ok, if the Ark question really was how many animals of each sex did NOAH bring on the ark, the answer would be "I don't know" wouldn't it? If he brought on a male and a female of each animal then existing in the world, we would not know that answer, would we? At least that was the way I was thinking, but of course ignoring the obvious that Moses did not do the ark thing.

[RWilson2526]

Originally Posted by nipper

I got a 9 out of 11. Like many, I missed the California widow question and the Moses/Ark question. Ok, if the Ark question really was how many animals of each sex did NOAH bring on the ark, the answer would be "I don't know" wouldn't it? If he brought on a male and a female of each animal then existing in the world, we would not know that answer, would we? At least that was the way I was thinking, but of course ignoring the obvious that Moses did not do the ark thing.

Hey Nipper where you been?