{"id":380,"date":"2011-07-08T10:34:52","date_gmt":"2011-07-08T02:34:52","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=380"},"modified":"2024-02-16T21:13:34","modified_gmt":"2024-02-16T13:13:34","slug":"flat-bread-fractions","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2011\/07\/flat-bread-fractions\/","title":{"rendered":"Flat Bread Fractions"},"content":{"rendered":"<p>Some time ago, I posted a fractions puzzle involving flat bread. You can <a title=\"Pratas and Sesame Street\" href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2011\/02\/pratas-and-sesame-street\/\">read about it here<\/a>. Here&#8217;s the question &#8211; if I take a flat piece of bread, and give you half, then you give me back a quarter, then I give you back an eighth, and you give me back 1\/16, and so on ad infinitum, how much of the original flat bread do I have?<\/p>\n<p><!--more-->If you&#8217;ve studied <em>geometric series<\/em>, this is an easy problem &#8211; but what if you haven&#8217;t? Is there any way to find the answer?Can we know what 1 &#8211; 1\/2 + 1\/4 &#8211; 1\/8 + 1\/16 &#8211; &#8230; adds up to?<\/p>\n<p>To quote Bob the Builder and Barack Obama &#8211; <em>yes we can<\/em>!<\/p>\n<p>Here&#8217;s how you do it. Suppose I&#8217;m there, exchanging bits of bread with you.<\/p>\n<ul>\n<li>First, I think &#8220;I start with 1 piece of bread, then I lose 1\/2, then I gain 1\/4, then I lose 1\/8, and so on. You end up with some fraction X of the bread, and I get 1-X&#8221;<\/li>\n<li>Then, I imagine &#8220;What if we swapped roles? Then you end up with 1-X of the bread and I end up with X of the bread. To get my X, I first receive 1\/2 of the bread, then give you 1\/4, then take back 1\/8, then give up 1\/16, and so on&#8221;<\/li>\n<li>Then I imagine &#8220;What if we started with twice as much bread? So you start with 2 pieces of bread, then you give me 1. Then I give up 1\/2, and take 1\/4, then give up 1\/8, and so forth. Since all amounts of bread are doubled, I now have 2X&#8221;<\/li>\n<li>However, as far as I&#8217;m concerned, this is exactly the same situation as I described at the start &#8211; I start with 1 piece of bread, then I lose 1\/2, then gain 1\/4, then lose 1\/8 and so forth. We&#8217;ve just figured out that this gives me 2X of bread, where X is what it gives you.<\/li>\n<\/ul>\n<p>So, this procedure gives me twice as much bread as you. Since there&#8217;s only 1 piece of bread, I must have 2\/3, and you have 1\/3.<\/p>\n<p>If you understand this trick, you are partway towards understanding geometric series generally.<\/p>\n<p>Can you use a similar trick to calculate 1 &#8211; 1\/3 + 1\/9 &#8211; 1\/27 + 1\/81 &#8211; &#8230; ?<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Some time ago, I posted a fractions puzzle involving flat bread. You can read about it here. Here&#8217;s the question &#8211; if I take a flat piece of bread, and give you half, then you give me back a quarter, then I give you back an eighth, and you give me back 1\/16, and so &hellip; <a href=\"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2011\/07\/flat-bread-fractions\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Flat Bread Fractions<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[279,183,217,206,123],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/380"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=380"}],"version-history":[{"count":1,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/380\/revisions"}],"predecessor-version":[{"id":1374,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/380\/revisions\/1374"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=380"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=380"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}