Sunday, July 10, 2016

Chapter 5.12 - Properties of Equivalent fractions

In the previous section we learned about equivalent fractions and cross multiplication. In this section we will see some advanced forms of fractions.

Consider the following patterns:

 In the first pattern, we have two equivalent fractions: 1/2 and 2/4.
• Based on them, a new fraction is formed.
    ♦ The numerator of the new fraction is the sum of the numerators of the equivalent fractions: 1 + 2
    ♦ The denominator of the new fraction is the sum of the denominators of the equivalent fractions: 2 + 4
• What is the result? The new fraction is 3/6, which is another equivalent fraction of the original fractions 1/2 and 2/4
 In the second pattern, we have two equivalent fractions: 2/4 and 3/6.
• Based on them, a new fraction is formed.
    ♦ The numerator of the new fraction is the sum of the numerators of the equivalent fractions: 2 + 3
    ♦ The denominator of the new fraction is the sum of the denominators of the equivalent fractions: 4 + 6
• What is the result? The new fraction is 5/10, which is another equivalent fraction of the original fractions 2/4 and 3/6
 In the third pattern, we have two equivalent fractions: 2/7 and 4/14.
• Based on them, a new fraction is formed.
    ♦ The numerator of the new fraction is the sum of the numerators of the equivalent fractions: 2 + 4
    ♦ The denominator of the new fraction is the sum of the denominators of the equivalent fractions: 7 + 14
• What is the result? The new fraction is 6/21, which is another equivalent fraction of the original fractions 2/7 and 4/14

What we saw above is a very useful property of equivalent fractions, which finds application in many areas of science and engineering. We need to prove the property algebraically:




Consider another pattern:


 In the first pattern, we have two equivalent fractions: 5/2 and 10/4.
• Based on them, two new fractions are formed.
• The first new fraction is formed from the first equivalent fraction 5/2:
    ♦ The numerator is (numerator + denominator): 5 + 2
    ♦ The denominator is (numerator - denominator): 5 - 2
• The second new fraction is formed from the second equivalent fraction 10/5:
    ♦ The numerator is (numerator + denominator): 10 + 4
    ♦ The denominator is (numerator - denominator): 10 - 4
• Result: First new fraction = 7/3 and second new fraction = 7/3. 
• That is., First new fraction = Second new fraction
 In the second pattern, we have two equivalent fractions: 4/2 and 16/8
• Based on them, two new fractions are formed.
• The first new fraction is formed from the first equivalent fraction 4/2:
    ♦ The numerator is (numerator + denominator): 4 + 2
    ♦ The denominator is (numerator - denominator): 4 - 2
• The second new fraction is formed from the second equivalent fraction 16/8:
    ♦ The numerator is (numerator + denominator): 16 + 8
    ♦ The denominator is (numerator - denominator): 16 - 8
• Result: First new fraction = 6/2 and second new fraction = 6/2. 
• That is., First new fraction = Second new fraction
 In the third pattern, we have two equivalent fractions: 9/4 and 18/8
• Based on them, two new fractions are formed.
• The first new fraction is formed from the first equivalent fraction 9/4:
    ♦ The numerator is (numerator + denominator): 9 + 4
    ♦ The denominator is (numerator - denominator): 9 - 4
• The second new fraction is formed from the second equivalent fraction 18/8:
    ♦ The numerator is (numerator + denominator): 18 + 8
    ♦ The denominator is (numerator - denominator): 18 - 8
• Result: First new fraction = 13/5 and second new fraction = 13/5 
• That is., First new fraction = Second new fraction


What we saw above is another useful property of equivalent fractions, which finds application in many areas of science and engineering. We need to prove the property algebraically:




In the next section we will see yet another property.

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