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How is a plus b over a minus b equals c plus d over c minus d?
(a + b)/(a - b) = (c + d)/(c - d) cross multiply(a + b)(c - d) = (a - b)(c + d)ac - ad + bc - bd = ac + ad - bc - bd-ad + bc = -bc + ad-ad - ad = - bc - bc-2ad = -2bcad = bc that is the product of the means equals the product of the extremesa/b = b/c
What do you do after cross multiplying when comparing fractions with unlike denominators?
Suppose the two fractions are a/b and c/d ad that b, d > 0. Then cross multiplication gives ad and bc. If ad > bc then a/b > c/d, If ad = bc then a/b = c/d, and If ad < bc then a/b < c/d
Can you always find another fraction in between ant two fractions why?
Yes. The simple answer is that rational fractions are infinitely dense. A longer proof follows:Suppose you have two fractions a/b and c/d where a, b, c and d are integers and b, d are positive integers.Without loss of generality, assume a/b < c/d.The inequality implies that ad < bc so that bc-ad>0 . . . . . . . . . . . . . . . . . . . (I)Consider (ad + bc)/(2bd)Then (ad+bc)/2bd - a/b = (ad+bc)/2bd - 2ad/2bd = (bc-ad)/2bdBy definition, b and d are positive so bd is positive and by result (I), the numerator is positive.That is to say, (ad+bc)/2bd - a/b > 0 or (ad+bc)/2bd > a/b.Similarly, by considering c/d - (ad+bc)/2bd is can be shown that c/d > (ad+bc)/2bd.Combining these results,a/b < (ad+bc)/2bd < c/d.
How do you tell a proportion is true?
You appear to be referring to the fact that if a/b = c/d then ad = bc. You know this is true because of the axioms of equality (If you do the same thing to both sides of an equation, then equality is maintained). a/b = c/d multiply by d ad/b = c multiply by b ad = bc If you wish to prove a specific proportion is true, you can test it with the equation ad = bc
What are 2 ways you can tell that two ratios can form a proportion?
1. They are equivalent fractions.2. They cross-multiply to the same value (a/b = c/d if and only if ad = bc).1. They are equivalent fractions.2. They cross-multiply to the same value (a/b = c/d if and only if ad = bc).1. They are equivalent fractions.2. They cross-multiply to the same value (a/b = c/d if and only if ad = bc).1. They are equivalent fractions.2. They cross-multiply to the same value (a/b = c/d if and only if ad = bc).
