{"id":646,"date":"2017-10-28T22:26:09","date_gmt":"2017-10-28T22:26:09","guid":{"rendered":"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/?post_type=chapter&p=646"},"modified":"2018-04-05T18:21:21","modified_gmt":"2018-04-05T18:21:21","slug":"the-key-fraction-rule","status":"publish","type":"chapter","link":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/chapter\/the-key-fraction-rule\/","title":{"raw":"The Key Fraction Rule","rendered":"The Key Fraction Rule"},"content":{"raw":"

[latexpage]<\/p>\r\n

We know that\u00a0$\\frac a b$\u00a0is the answer to a division problem: \\[ \\frac a b \\]\u00a0represents the amount of pie an individual child receives when\u00a0$a$\u00a0pies are shared equally by\u00a0$b$\u00a0children.<\/p>\r\n

What happens if we double the number of pie and double the number of kids? Nothing! The amount of pie per child is still the same: \\[ \\frac{2a}{2b} = \\frac a b.\\]<\/p>\r\n

For example, as the picture shows,\u00a0$\\frac 6 3$\u00a0and\u00a0$\\frac{12}6$\u00a0both give two pies for each child.<\/p>\r\n

\"\"<\/p>\r\n

\"\"<\/p>\r\n

And tripling the number of pies and the number of children also does not change the final amount of pies per child, nor does quadrupling each number, or one trillion-billion-tupling the numbers! \\[\u00a0\\frac{6}{3}\r\n\\quad\r\n=\r\n\\quad\r\n\\frac{12}{6}\r\n\\quad\r\n=\r\n\\quad\r\n\\frac{18}{9}\r\n\\ =\\ \\dots \\ =\\ \\text{ two pies per child.}\\]<\/p>\r\n

\"\"<\/p>\r\n

This leads us to want to believe:<\/p>\r\n\r\n

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<\/a>Key Fraction Rule<\/h3>\r\n\\[ \\frac{xa}{xb} = \\frac a b,\\]\r\n

(at least for positive whole numbers $x$).<\/p>\r\n \r\n\r\nWe say that the fractions $\\frac{xa}{xb}$ and $\\frac a b$ are equivalent<\/strong>.\r\n\r\n<\/div>\r\n \r\n

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Example: Fractions equivalent to 3\/5<\/h3>\r\n

For example,\u00a0\\[\\frac 3 5 \\text{ (sharing three pies among five kids)}\\] yields the same result as<\/p>\r\n

\\[\u00a0\\frac{3 \\cdot 2}{5 \\cdot 2}\\ =\\ \\frac{6}{10} \\text{ (sharing six pies among ten kids)}\\]<\/p>\r\n

and as<\/p>\r\n

\\[\u00a0\\frac{3 \\cdot 100}{5 \\cdot 100}\\ =\\ \\frac{300}{500} \\text{ (sharing 300 pies among 500 kids).}\\]<\/p>\r\n\r\n<\/div>\r\n \r\n

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Think \/ Pair \/ Share<\/h3>\r\n

Write down a lot of equivalent fractions for $\\frac 1 2$, for $\\frac {10}3$, and for 1.<\/p>\r\n\r\n<\/div>\r\n \r\n

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Example: Going Backwards<\/h3>\r\n\\[\\frac{20}{32} \\text{\u00a0(sharing 20 pies among 32 kids) }\\]\r\n

is the same problem as:<\/p>\r\n\\[\u00a0\\frac{5 \\cdot 4}{8 \\cdot 4}\\ = \\ \\frac{5}{8} \\text{\u00a0(sharing five pies among eight kids).} \\]\r\n

Most people say we have cancelled<\/i> or taken a common factor 4 from the numerator and denominator.<\/p>\r\n

Mathematicians call this process reducing<\/i> the fraction to lowest terms. (We have made the numerator and denominator smaller, in fact as small as we can make them!)<\/p>\r\n

Teachers tend to say that we are simplifying<\/i> the fraction. (You have to admit that\u00a0$\\frac 5 8$\u00a0does look simpler than $\\frac {20}{32}$.)<\/p>\r\n\r\n<\/div>\r\n \r\n

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Example: How Low Can You Go?<\/h3>\r\n

As another example,\u00a0$\\frac{280}{350}$\u00a0can certainly be simplified by noticing that there is a common factor of\u00a010\u00a0in both the numerator and the denominator: \\[\u00a0\\frac{280}{350}\r\n\\quad\r\n=\r\n\\quad\r\n\\frac{28 \\cdot 10}{35 \\cdot 10}\r\n\\quad\r\n=\r\n\\quad\r\n\\frac{28}{35}.\\]<\/p>\r\n

We can go further as\u00a028\u00a0and\u00a035\u00a0are both multiples of 7: \\[\u00a0\\frac{28}{35}\r\n\\quad\r\n=\r\n\\quad\r\n\\frac{4 \\cdot 7}{5 \\cdot 7}\r\n\\quad\r\n=\r\n\\quad\r\n\\frac{4}{5}.\\]<\/p>\r\n

Thus, sharing\u00a0280\u00a0pies among\u00a0350\u00a0children gives the same result as sharing\u00a04\u00a0pies among\u00a05\u00a0children! \\[\u00a0\\frac{280}{350}\\ =\\ \\frac{4}{5}. \\]<\/p>\r\n

Since\u00a04\u00a0and\u00a05\u00a0share no common factors, this is as far as we can go with this example (while staying with whole numbers).<\/p>\r\n\r\n<\/div>\r\n \r\n

On Your Own<\/h3>\r\n

Mix and Match: On the top are some fractions that have not been simplified. On the bottom are the simplified answers, but in random order. Which simplified answer goes with which fraction? (Notice that there are fewer answers than questions!)<\/p>\r\n\\[\u00a01. \\ \\ \\frac{10}{20} \\qquad\\\r\n2. \\ \\ \\frac{50}{75} \\qquad\\\r\n3. \\ \\ \\frac{24000}{36000} \\qquad\\\r\n4. \\ \\ \\frac{24}{14} \\qquad\\\r\n5. \\ \\ \\frac{18}{32} \\qquad\\\r\n6. \\ \\ \\frac{1}{40}\\]\r\n\r\n\\[\u00a0a. \\ \\ \\frac{2}{3} \\qquad\\\r\nb. \\ \\ \\frac{9}{16} \\qquad\\\r\nc. \\ \\ \\frac{12}{7} \\qquad\\\r\nd. \\ \\ \\frac{1}{40} \\qquad\\\r\ne. \\ \\ \\frac{1}{2}\\]\r\n\r\n \r\n

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Think \/ Pair \/ Share<\/h3>\r\n

Use the \u201cPies Per Child Model\u201d to explain why<\/strong> the key fraction rule holds. That is, explain why each individual child gets the same amount of pie in these two situations:<\/p>\r\n\r\n