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Multiply/divide fraction/fraction
I have been in automotive repair for 30 years, have taught electronics and automotive for years on the college level and have degrees in Digital electronics and also Computer service technology. I have problems understanding fractional multiplication or division. In whole numbers to multiply is to add repeatedly, to divide is to subtract repeatedly. When dealing with fraction to fraction these methods do not work (also served well as a reverse check of your work). I know 1/4 x 1/2=1/8, can someone explain to me why this is so and how you can prove it? Thanks!
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What you guys are saying makes sense but doesn't really answer my question. I recogonize some names with good recommendation but these answers don't help me. If I have 1/2 x 1/2 (aka 0.5 x 0.5) how does this answer to 1/4? Please tell me in the usual common repeated addition....0.5 +0.5=1.0 this does not equal the answer of 1/4 (0.25).
Please continue to try to help me understand this fractional problem?
B
Please continue to try to help me understand this fractional problem?
B
Whole numbers can be expressed as fractions; for example 2 could be expressed as 2/1 (2 divided by 1 equals 2), 7 could be expressed as 7/1 (7 divided by 1 equals 7).
Multiplying 2 by 7 can be expressed as 2/1 multiplied by 7/1 which would equal 14/1 (multiplying the numerators (2x7) and dividing this by the denominators multiplied (1x1) which gives 14).
When multiplying fractions, the method is exactly the same; multiplying the numerators and dividing this by the denominators multiplied.
So, for example, 1/2 multiplied by 1/4 equals 1/8 ( (1x1)/(2x4) ).
Don�t know if that answers the question you�re asking.
Multiplying 2 by 7 can be expressed as 2/1 multiplied by 7/1 which would equal 14/1 (multiplying the numerators (2x7) and dividing this by the denominators multiplied (1x1) which gives 14).
When multiplying fractions, the method is exactly the same; multiplying the numerators and dividing this by the denominators multiplied.
So, for example, 1/2 multiplied by 1/4 equals 1/8 ( (1x1)/(2x4) ).
Don�t know if that answers the question you�re asking.
Bruce
If what you mean by repeated addition is for example this:
2 x 3=2+2+2=6 then you can't use this for fractions unless one number is an integer (whole number):
1/4 x 3= 1/4 + 1/4 + 1/4= 3/4, which doesn't really get you too far
.As somebody has already pointed out , if you say 1/2 x 1/2 out loud as a half of a half then it should be clear that this is 1/4.
For something more complicated like 1/16 x 1/3 you can say one third of one sixteenth, but 1/16th doesn't divide easily by 3 so we need to multiply top and bottom by 3 and get 3/48ths. Then we can say one third of 3/48ths which is 1/48th, which is what you get if you apply the normal rule of multiplying the top numbers together and multiplying the bottom numbers together
1/16 x 1/3 =1/48.
If what you mean by repeated addition is for example this:
2 x 3=2+2+2=6 then you can't use this for fractions unless one number is an integer (whole number):
1/4 x 3= 1/4 + 1/4 + 1/4= 3/4, which doesn't really get you too far
.As somebody has already pointed out , if you say 1/2 x 1/2 out loud as a half of a half then it should be clear that this is 1/4.
For something more complicated like 1/16 x 1/3 you can say one third of one sixteenth, but 1/16th doesn't divide easily by 3 so we need to multiply top and bottom by 3 and get 3/48ths. Then we can say one third of 3/48ths which is 1/48th, which is what you get if you apply the normal rule of multiplying the top numbers together and multiplying the bottom numbers together
1/16 x 1/3 =1/48.
Go back to basics and think of it like slices of pie.
If you have some number y, then 2 y will be twice the amount of y. This much is obvious, hopefully.
But say you have the number (1/2) (half). If you then ask what (1/2) of y is, or (1/2) y, you know that it is half of number y.
If y is 10, then (1/2) y = 5.
Now, back to pies.
Suppose you have a whole pie, and take half of it away. You now have (1/2) of a pie.
But what is half of half a pie? This is, written as maths, (1/2) x (1/2). And if you think about the pie, half of the half you have is a quarter. Same with the fraction.
If you have some number y, then 2 y will be twice the amount of y. This much is obvious, hopefully.
But say you have the number (1/2) (half). If you then ask what (1/2) of y is, or (1/2) y, you know that it is half of number y.
If y is 10, then (1/2) y = 5.
Now, back to pies.
Suppose you have a whole pie, and take half of it away. You now have (1/2) of a pie.
But what is half of half a pie? This is, written as maths, (1/2) x (1/2). And if you think about the pie, half of the half you have is a quarter. Same with the fraction.
As far as division by a fraction is concerned I would explain it as follows:
Just as 12 divided by 3 means "How many fours are there in 12?" or "How many fours make 12?" so
4 divided by 1/3 means "How many thirds are there in 3?"
Now since there are 3 thirds in 1 by definition, then there must be 3 x 4 = 12 thirds in 4.
That is why the rule is to turn the fraction you are dividing by upside down and multiply. So
4/(1/3)=4 x 3/1= 4 x 3=12
Just as 12 divided by 3 means "How many fours are there in 12?" or "How many fours make 12?" so
4 divided by 1/3 means "How many thirds are there in 3?"
Now since there are 3 thirds in 1 by definition, then there must be 3 x 4 = 12 thirds in 4.
That is why the rule is to turn the fraction you are dividing by upside down and multiply. So
4/(1/3)=4 x 3/1= 4 x 3=12
Your method of repeated addition does work but you have to keep the two sets of calculations, those above and those below the line, separate. So, for example the calculation 2/3 x 3/4 which gives the answer 1/2 can be calculated/ checked as 2x3 = 2+2+2 = 6 divided by 3x4 = 3+3+3+3 = 12 so 6/12 which simpifies to 1/2.
I'm not sure it's helpful to think of division as subtracting. Division is sharing. And whereas subtracting a positive number always makes a number smaller, division by a fraction between zero and one makes the answer bigger not smaller.
Think of division as the inverse of multiplication
4 x 2 = 8
8 � 2 = 4
Similarly
4 x � = 2
2 � � = 4
Think of division as the inverse of multiplication
4 x 2 = 8
8 � 2 = 4
Similarly
4 x � = 2
2 � � = 4
Suppose you put one litre of water into one empty bucket and it makes it one cm deep.
That�s 1 � 1 = 1 (One litre of water in one bucket is one cm deep)
Split the water equally between two buckets � it�ll be about half a cm deep in each.
1 � 2 = � (One litre of water in two buckets is half a cm deep)
If your bucket is like one of those recycling bins which have a central division, so that you can put cans one side and bottles the other, you can put the litre of water in just one half of the bucket � it�ll be about 2 cm deep.
1 � 0.5 = 2 (One litre of water in half of one bucket is two cm deep)
If you started with only half a litre of water the levels would only be half as high.
Writing � instead of 1 at the start of those equations, and halving the numbers after the = signs, we get:-
� � 1 = � (Half a litre of water in one bucket is half a cm deep)
� � 2 = � (Half a litre of water in two buckets is a quarter of a cm deep)
� � � = 1 (Half a litre of water in half of one bucket is one cm deep)
[Ignore the words �about�. They are just to stop purists from complaining that the buckets might taper, that deeper water might be fractionally more compressed, or several other trivialities.]
That�s 1 � 1 = 1 (One litre of water in one bucket is one cm deep)
Split the water equally between two buckets � it�ll be about half a cm deep in each.
1 � 2 = � (One litre of water in two buckets is half a cm deep)
If your bucket is like one of those recycling bins which have a central division, so that you can put cans one side and bottles the other, you can put the litre of water in just one half of the bucket � it�ll be about 2 cm deep.
1 � 0.5 = 2 (One litre of water in half of one bucket is two cm deep)
If you started with only half a litre of water the levels would only be half as high.
Writing � instead of 1 at the start of those equations, and halving the numbers after the = signs, we get:-
� � 1 = � (Half a litre of water in one bucket is half a cm deep)
� � 2 = � (Half a litre of water in two buckets is a quarter of a cm deep)
� � � = 1 (Half a litre of water in half of one bucket is one cm deep)
[Ignore the words �about�. They are just to stop purists from complaining that the buckets might taper, that deeper water might be fractionally more compressed, or several other trivialities.]
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