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Because multiplication and division are inverse operations. And the reciprocal of a number is its multiplicative inverse.
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Why does the method of dividing fractions work?
This is related to the fact that dividing by a number is the same as multiplying with the number's reciprocal.
Why does cross multiplying with fractions work?
When cross multiplying, finding the product of the means and extremes, you are technically getting a common denominator that reduces out.
Why don't you have to find a common denominator when multiplying fractions?
Multiplying fractions is quite different from adding them. You just multiply the numberators and the denominators separately. You can find the common denominator if you like, but in the end (after simplifying), you'll get the same result, and the additional work of finding the common denominator and converting the fractions turns out to be unnecessary. Try it out for some fractions!
Why does the way you divide fractions work?
When one number is divided by another it is the same as putting the first number over the second. If both the numbers are whole numbers, this creates a fraction. For example 4 ÷ 5 = 4/5 (four divided by five is the same as four fifths). But there is nothing from a mathematical point of view to prevent writing one fraction over another. Thus dividing fractions is the same as writing one fraction over another, which looks like a fraction. For example ½ ÷ ⅝ = (½)/(⅝) To create an equivalent fraction the top and bottom of a fraction are multiplied (or divided) by the same value. So having written a division of two fractions as the first over the second, by multiplying the top and bottom by the reciprocal of the bottom fraction results in a fraction with the same value. Any number multiplied by its reciprocal results in 1; thus the bottom number is equivalent to 1 and any value divided by 1 is that value. So the original division is the same as the first fraction multiplied by the reciprocal of the second. The reciprocal of a number is the value that when multiplied by the number results in 1. The reciprocal of a fraction is obtained by inverting it, ie swapping over the top and bottom numbers; for example the reciprocal of 4/5 is 5/4 since: 4/5 × 5/4 = (4×5)/(5×4) = 20/20 = 1. Example 2/3 ÷ 4/5 2/3 ÷ 4/5 = (2/3)/(4/5) ← writing the divide as the dividend over the divisor = (2/3 × 5/4)/(4/5 × 5/4) ← creating an equivalent fraction by multiplying top and bottom by 5/4 = (2/3 × 5/4)/(20/20) ← multiplying out the denominator = (2/3 × 5/4)/1 ← simplifying the new denominator gives the new denominator the value 1 = 2/3 × 5/4 ← fraction has the value of the numerator. If you have mixed numbers, convert them to improper fractions and then the method (above) works as you now have one number over another.
Does commutative property work for fractions?
Yes. Both the commutative property of addition, and the commutative property of multiplication, works:* For integers * For rational numbers (i.e., fractions) * For any real numbers * For complex numbers
Why does the method of dividing fractions work?
This is related to the fact that dividing by a number is the same as multiplying with the number's reciprocal.
Why does the butterfly method work when comparing fractions?
it works when comparing fractions by multiplying the fractions to see whitch one is greater not greater and equal
Why does cross multiplying with fractions work?
When cross multiplying, finding the product of the means and extremes, you are technically getting a common denominator that reduces out.
Why don't you have to find a common denominator when multiplying fractions?
Multiplying fractions is quite different from adding them. You just multiply the numberators and the denominators separately. You can find the common denominator if you like, but in the end (after simplifying), you'll get the same result, and the additional work of finding the common denominator and converting the fractions turns out to be unnecessary. Try it out for some fractions!
Why do you think it is helpful to convert mixed numbers into fractions before multiplying or dividing?
So that you can get your answer more faster and you don't need to do a lot of work
Adding and subtracting unlike denominators?
Fractions can only be added or subtracted if the denominators are the same. If the denominators are different, then the fractions need to be made into equivalent fractions with the same denominator. The new denominator can be found simply by multiplying the denominators together, but this can lead to some large fractions with which to work. A better new denominator is the lowest common multiple of (all the) denominators. (Once the new denominator is found, the fractions' new numerators are found by multiplying their current numerator by the new denominator divided by their current denominator to make their equivalent fractions with the new denominator.) Once all the fractions are converted into equivalent fractions with the new denominator then the fractions can be added or subtracted, with the result being simplified (if possible).
Why does the way you divide fractions work?
When one number is divided by another it is the same as putting the first number over the second. If both the numbers are whole numbers, this creates a fraction. For example 4 ÷ 5 = 4/5 (four divided by five is the same as four fifths). But there is nothing from a mathematical point of view to prevent writing one fraction over another. Thus dividing fractions is the same as writing one fraction over another, which looks like a fraction. For example ½ ÷ ⅝ = (½)/(⅝) To create an equivalent fraction the top and bottom of a fraction are multiplied (or divided) by the same value. So having written a division of two fractions as the first over the second, by multiplying the top and bottom by the reciprocal of the bottom fraction results in a fraction with the same value. Any number multiplied by its reciprocal results in 1; thus the bottom number is equivalent to 1 and any value divided by 1 is that value. So the original division is the same as the first fraction multiplied by the reciprocal of the second. The reciprocal of a number is the value that when multiplied by the number results in 1. The reciprocal of a fraction is obtained by inverting it, ie swapping over the top and bottom numbers; for example the reciprocal of 4/5 is 5/4 since: 4/5 × 5/4 = (4×5)/(5×4) = 20/20 = 1. Example 2/3 ÷ 4/5 2/3 ÷ 4/5 = (2/3)/(4/5) ← writing the divide as the dividend over the divisor = (2/3 × 5/4)/(4/5 × 5/4) ← creating an equivalent fraction by multiplying top and bottom by 5/4 = (2/3 × 5/4)/(20/20) ← multiplying out the denominator = (2/3 × 5/4)/1 ← simplifying the new denominator gives the new denominator the value 1 = 2/3 × 5/4 ← fraction has the value of the numerator. If you have mixed numbers, convert them to improper fractions and then the method (above) works as you now have one number over another.
Does the Distributive Property work with division and addition too not just multiplication and addition?
Technically it only works with multiplication. You can also use it with division, if the division appears on the RIGHT side, for example: (3 + 2) / 5 = 3 / 5 + 2 / 5. This can be justified by the fact that dividing by 5 is the same as multiplying by 1/5. To check other cases, convert the division to a multiplication, and see whether the distributive property applies to the corresponding multiplication.
How do you work out the reciprocal on a calculator?
To find the reciprocal of a number, calculate [1] [divided by] [the number].
Are the set of positive fractions closed under division?
Yes, because for any x and y that are positive fractions (y not equal to zero), x/y is also a positive fraction. Note that whole numbers are considered fractions with denominators of 1 -- otherwise it doesn't work.
Does commutative property work for fractions?
Yes. Both the commutative property of addition, and the commutative property of multiplication, works:* For integers * For rational numbers (i.e., fractions) * For any real numbers * For complex numbers
When you need to find the percentage of work in to work out what is that called?
It is the reciprocal of the efficiency.
