Lesson 27 Section 2
Parts of Fractions
Example 1 . A recipe calls for 6 cups of flour, and you are going to make a third of the recipe. How much flour will you use?
Answer. A third of 6 cups is 2 cups. But how much is a third of ?
A third of is . Simply take a third of the numerator.
A third of 6 cups is 2 cups.
But say that the recipe calls for 6 cups of flour. Since the numerator 1 does not have a third part, how will you take a third of ?
If we divide each into thirds, then the whole 1 will be in three times as many parts. A third of is .
To take a third of a fraction, simply multiply the denominator by 3.
A third of 6cups is 2 cups.
There are two ways, then, to take a part of a fraction.
Answer. Half of 25%, which is 12½%. (Lesson 16, Example 4.)
Now, each eighth will be another 12½%.
See Problem 15.
Answer. Five times larger. Because 25 is 5 × 5.
Example 9. The following problem appeared in a recent textbook:
The writer no doubt intend it to be translated as
-- thus making it a division problem:
However, if we make verbal sense of the problem --
1-fifth is half of what number?
-- then the answer is obvious. Just as 1 apple is half of 2 apples, so
1-fifth is half of 2-fifths.
Example 12. To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of orange and green are mixed, what fraction of the new mixture is yellow dye?
Solution. For convenience, let
O = Orange dye, R = Red dye, Y = Yellow dye,
G = Green dye, B = Blue dye.
Now, the orange dye consists of a total of five parts: 3 parts red and 2 parts yellow. That is,
The green dye consists of three parts: 2 parts blue and 1 part yellow.
To add equal amounts of O and G, we may simply add half of each. Therefore, half of O is
And half of G is
Upon adding those equal parts of O and G, the new mixture will consist of
The LCM of those fractions is 30. (Lesson 23.) The fractions of Y --
We now see why we multiply the numerators and multiply the denominators:
It follows, as it must, from what the symbols mean.
"2 goes into 6 three (3) times."
And so we have arrived where we began (Lesson 26, Question 3), at the formal rule for multiplying fractions:
Multiply the numerators and multiply the denominators.
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Copyright © 2016 Lawrence Spector
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