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Lesson 21 EQUIVALENT FRACTIONSIn this Lesson, we will answer the following:




in the number line. 



The student who has not learned about ratios (Lesson 16) may nevertheless benefit from the summary below.
denominator, namely half. Any fraction where the numerator is half of the denominator will be
1 is half of 2. 2 is half of 4. 3 is half of 6. And so on. These are all at the same place on the number line. Each number is half of 1. In terms of equal ratios, if a pie is cut into 2 pieces and you eat 1 of them, you will get just as stuffed as when the pie is in 4 pieces and you eat 2; or if you eat 4 out of 8, or 5 out of 10, and so on. In each case, you have eaten an equivalent "fraction" of the pie, namely half. Example 3. Write the missing numerator:
Answer. 7 is a third of 21. A third of 12 is 4.
7 is to 21 as 4 is to 12.
Example 4. Write the missing numerator:
Answer. Here, we will look alternately (Lesson 17, Question 2). To make 35, 7 has been multiplied by 5. Therefore, 3 also must be multiplied by 5:
This is the theorem of the same multiple. Everything that we know about ratios carries over into fractions, because each numerator has a ratio to its denominator.
Answer. A decimal is a fraction whose denominator is understood to be a power of 10: 10, 100, 1000, etc. We can easily make the denominator 50 into 100 by multiplying it by 2. Therefore we must also multiply 9 by 2:
Hundredths is two decimal places. (Lesson 2, Question 5.) 



Answer. 20 and 24 have a common divisor, 4.
This is the Theorem of the Common Divisor. (Lesson 16, Question 7.)
Answer. When the terms have the same number of 0's, we may ignore them.
Effectively, we have divided 500 and 1500 by 100. (Lesson 1, Question 11.)
Solution. Divide 20 by 8. "8 goes into 20 two (2) times (16) with 4 left over."
Alternatively, we could reduce first. 20 and 8 have a common divisor 4:
Notice that we are free to interpret the same symbol
20 divided by 8.
Any fraction in which the numerator and denominator are equal, is equal to 1.
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