Elsewhere us looked right into what could be dubbed an organic development to enhancement of fractions. Because that example, in stimulate to describe why 1/3 + 1/4 = 7/12, we considered the task of separating 7 apples in between 12 guys under the restriction that an apple could be at most cut into 5 same pieces. The systems was to divides 3 apples right into 12 quarters and also 4 apples into 12 thirds letting each boy have 1/3 and also 1/4 of one apple. Clearly, the procedure leads to all boys having equal amounts. Due to the fact that there were 7 apples and 12 boys, every of the boys obtained 7/12 of one apple. The conclusion the 1/3 + 1/4 = 1/12 follows. Curiously, the addition result so expressed has been obtained without any type of recourse come the meaning of the enhancement of fractions. A nearby idea have the right to serve to define the id of equivalence the fraction.
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How plenty of halves room in the whole? Two,One,Two,Three,Four,Many, the course. How numerous thirds room in the whole? Three,One,Two,Three,Four,Many, that course. What does this say? very simple: 1 = 2 × 1/2 and also 1 = 3 × 1/3.
Now let"s count. If, say, one bag contains three apples, in 2 bags there space 6 apples,3 apples,6 apples,9 apples,12 apples; in 3 bags there space 9 apples,3 apples,6 apples,9 apples,12 apples; in 4 bags there room 12 apples,3 apples,6 apples,9 apples,12 apples, and so on. Similarly, in 1 apple there are 3 , 3 , 6 , 9 , 12 thirds; in 2 apologize there are 6 , 3 , 6 , 9 , 12 thirds; in 3 apologize there are 9 , 3 , 6 , 9 , 12 thirds; in 4 apples there room 12 , 3 , 6 , 9 , 12 thirds. In a concise form, us have
1 = 3 × 1/3, 2 = 6 × 1/3, 3 = 9 × 1/3, 4 = 12 × 1/3.
The an initial of these means neither more nor much less than 1/3 = 1/3; the second tells united state that 2/6 = 1/3; next we have 3/9 = 1/3 and also 4/12 = 1/3. Let"s emphasize those:
1/3 = 1/3, 2/6 = 1/3, 3/9 = 1/3, 4/12 = 1/3.
So there appear to be at least two ways to introduce indistinguishable fractions.
The an ext common one is well portrayed with the following problem. A cake reduced into 6 piece is to it is in divided between three boys. Exactly how much will each of the boys receive? due to the fact that there are 6 pieces and also three boys, every of the boys will acquire 2 pieces, i.e. 2/6 that the cake. ~ above the other hand, in the way described, the three boys have divided evenly one whole cake between themselves, each gained 1/3 of the whole, implying that 2/6 = 1/3.
The second approach is to inquire exactly how to divide 2 cakes between 6 boys. A simple way of doing the is to division the cakes into thirds and give 1/3 of a cake to each boy. Because each of the six boy it s okay an equal part of the two cakes we naturally conclude that 1/3 = 2/6.
In both situations the dark component is a take-home part of a solitary boy.
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Algebraically, 1/3 is derived from 2/6 by dividing both the numerator (2) and also the denominator (6) the 2/6 through the very same number (2). Together an procedure is dubbed reduction.
DefinitionDividing the numerator and also the denominator of a portion by their typical factor is referred to as reduction.The procedure wherewith both the numerator and the denominator are multiplied by the very same number is called expansion.
Both reduction and also expansion command to tantamount fractions. An ext generally, two fractions are equivalent if both have the right to be diminished to the very same fraction: