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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.
When you need to find the percentage of work in to work out what is that called?
It is the reciprocal of the efficiency.
