![\"\"](\"https:\/\/gmatclub.com\/blog\/wp-content\/uploads\/2012\/04\/fs847049-300x200.jpg\" "\\"decimals\\"")
The topic of decimals, and patterns of decimals, seems to be of slightly greater interest to GMAC in the GMAT OG13e than in previous editions.\u00a0 What decimals terminate?\u00a0 What decimals repeat?\u00a0 In this post, we'll take a look at these questions.<\/p>\n
<\/p>\n
Integers are positive and negative whole numbers, including zero.\u00a0 Here are the integers:<\/p>\n
{ \u2026 -3, -2, -1, 0, 1, 2, 3, \u2026}<\/p>\n
When we take a\u00a0ratio\u00a0of two integers, we get a\u00a0rational number.\u00a0 A rational number is any number of the form a\/b, where a & b are integers, and b \u2260 0. Rational numbers are the set of all fractions made with integer ingredients.\u00a0\u00a0 Notice that all integers are included in the set of rational numbers, because, for example, 3\/1 = 3.<\/p>\n
<\/p>\n
When we make a decimal out of a fraction, one of two things happens.\u00a0 It either terminates (comes to an end) or repeats (goes on forever in a pattern).\u00a0 Terminating rational numbers include:<\/p>\n
1\/2 = 0.5<\/p>\n
1\/8 = 0.125<\/p>\n
3\/20 = 0.15<\/p>\n
9\/160 = 0.5625<\/p>\n
<\/p>\n
Repeating rational numbers include:<\/p>\n
1\/3 = 0.333333333333333333333333333333333333\u2026<\/p>\n
1\/7 = 0.142857142857142857142857142857142857\u2026<\/p>\n
1\/11 = 0.090909090909090909090909090909090909\u2026<\/p>\n
1\/15 = 0.066666666666666666666666666666666666\u2026<\/p>\n
<\/p>\n
The GMAT won't give you a complicated fraction like 9\/160 and expect you to figure out what its decimal expression is.\u00a0 BUT, the GMAT could give you a fraction like 9\/160 and ask whether it terminates or not.\u00a0 How do you know?<\/p>\n
Well, first of all, any terminating decimal (like 0.0376) is, essentially, a fraction with a power of ten in the dominator; for example, 0.0376 = 376\/10000 = 47\/1250.\u00a0 Notice we simplified this fraction, by cancelling a factor of 8 in the numerator.\u00a0 Ten has factors of 2 and 5, so any power of ten will have powers of 2 and powers of 5, and some might be canceled by factors in the numerator , but no other factors will be introduced into the denominator.\u00a0 Thus, if the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, then it can be written as something over a power of ten, which means its decimal expression will terminate.<\/p>\n
If the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, the decimal expression terminates.\u00a0 If there is any prime factor in the denominator other than 2 or 5, then the decimal expression repeats.\u00a0 Thus,<\/p>\n
<\/p>\n
1\/24 repeats (there's a factor of 3)<\/p>\n
1\/25 terminates (just powers of 5)<\/p>\n
1\/28 repeats (there's a factor of 7)<\/p>\n
1\/32 terminates (just powers of 2)<\/p>\n
1\/40 terminates (just powers of 2 and 5)<\/p>\n
<\/p>\n
Notice, as long as the fraction is in lowest terms, the numerator doesn't matter at all. Since 1\/40 terminates, then 7\/40, 13\/40, or any other integer over 40 also terminates.Since 1\/28 repeats, then 5\/28 and 15\/28 and 25\/28 all repeat; notice, though that 7\/28 doesn't repeat, because of the cancellation: 7\/28 = 1\/4 = 0.25.<\/p>\n
<\/p>\n
There are certain decimals that are good to know as shortcut, both for fraction-to-decimal conversions and for fraction-to-percent conversions.\u00a0 These are<\/p>\n
<\/p>\n
1\/2 = 0.5<\/p>\n
1\/3 = 0.33333333333333333333333333\u2026<\/p>\n
2\/3 = 0.66666666666666666666666666\u2026<\/p>\n
1\/4 = 0.25<\/p>\n
3\/4 = 0.75<\/p>\n
1\/5 = 0.2 (and times 2, 3, and 4 for other easy decimals)<\/p>\n
1\/6 = 0.166666666666666666666666666\u2026.<\/p>\n
5\/6 = 0.833333333333333333333333333\u2026<\/p>\n
1\/8 = 0.125<\/p>\n
1\/9 = 0.111111111111111111111111111\u2026 (and times other digits for other easy decimals)<\/p>\n
1\/11 = 0.09090909090909090909090909\u2026 (and times other digits for other easy decimals)<\/p>\n
<\/p>\n
There's another category of decimals that don't terminate (they go on forever) and they have no repeating pattern.\u00a0\u00a0 These numbers, the non-terminating non-repeating decimals, are called the irrational numbers.\u00a0 It is impossible to write any one of them as a ratio of two integers.\u00a0 Mr. Pythagoras (c. 570 \u2013 c. 495 bce) was the first to prove a number irrational: he proved that the square-root of 2 \u2014 [pmath]sqrt(2)[\/pmath] \u00a0\u2014 is irrational.\u00a0 We now know: all square-roots of integers that don't come out evenly are irrational.\u00a0 Another famous irrational number is [pmath]pi[\/pmath], or pi, the ratio of a circle's circumference to its diameter.\u00a0 For example,<\/p>\n
[pmath]pi[\/pmath]\u00a0= 3.1415926535897932384626 062862089986280348253421170 028410270193852110555964462 909145648 <\/p>\n That's the first three hundred digits of pi, and the digits never repeat: they go on forever with no repeating pattern.\u00a0 There are infinite many irrational numbers: in fact, the infinity of irrational numbers is infinitely bigger than the infinity of the rational numbers, but that gets into some math (https:\/\/en.wikipedia.org\/ <\/p>\n 1) [pmath]{0.16666...\/0.44444...} = [\/pmath]<\/p>\n (A) 2\/27<\/p>\n (B) 3\/2<\/p>\n (C) 3\/4<\/p>\n (D) 3\/8<\/p>\n (E) 9\/16<\/p>\n <\/p>\n 1) From our shortcuts, we know 0.166666666666\u2026 = 1\/6, and 0.444444444444\u2026 = 1\/9.\u00a0 Therefore\u00a0(1\/6)*(9\/4) = 3\/8.\u00a0 Answer =\u00a0D<\/strong><\/p>\n <\/p>\n This post was written by Mike McGarry, GMAT Expert at\u00a0Magoosh<\/a>, and originally posted\u00a0here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":" The topic of decimals, and patterns of decimals, seems to be of slightly greater interest to GMAC in the GMAT OG13e than in previous editions.\u00a0 What decimals terminate?\u00a0 What decimals…<\/p>\n","protected":false},"author":133,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,783,243,736],"tags":[],"class_list":["post-11189","post","type-post","status-publish","format-standard","hentry","category-gmat","category-magoosh-blog","category-blog","category-quant-gmat","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11189","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/users\/133"}],"replies":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/comments?post=11189"}],"version-history":[{"count":8,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11189\/revisions"}],"predecessor-version":[{"id":58818,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11189\/revisions\/58818"}],"wp:attachment":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/media?parent=11189"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/categories?post=11189"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/tags?post=11189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Practice Question<\/h2>\n
Practice Question Explanation<\/h2>\n