{"id":10060,"date":"2012-06-11T09:00:43","date_gmt":"2012-06-11T16:00:43","guid":{"rendered":"https:\/\/magoosh.com\/gre\/?p=10060"},"modified":"2019-07-04T12:22:11","modified_gmt":"2019-07-04T19:22:11","slug":"quantitative-comparison-and-manipulation","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gre\/quantitative-comparison-and-manipulation\/","title":{"rendered":"Quantitative Comparison and Manipulation"},"content":{"rendered":"<p>Many quantitative questions have variables in both columns. While your first instinct may be to work algebraically, this strategy is not always best. Often the fastest way to a solution is by plugging in different values to see which column is greater.<\/p>\n<p>Developing a sense of when to plug in and when to solve algebraically takes practice. Here are a few helpful guidelines when trying to determine which approach to use.<\/p>\n<p>&nbsp;<\/p>\n<h2>\u00a0Work algebraically if the question is a polynomial<\/h2>\n<p>If you are dealing with a polynomial, simplify. Plugging in may require too much calculation. Instead, work with the familiar algebraic forms shown below:<\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_0d8e5d4291377d4a8f9f9debf52b7aff.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(a - b)(a + b) = a^2 - b^2\" title=\"(a - b)(a + b) = a^2 - b^2\"\/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_df7fa4203cb70bffdb6c7e71cd2ee0cb.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(a - b)(a - b) = a^2 - 2ab + b^2\" title=\"(a - b)(a - b) = a^2 - 2ab + b^2\"\/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_1b3e6a4e5fc9d3d8ec84024eb29381b6.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(a - b)^2 = a^2 - 2ab + b^2\" title=\"(a - b)^2 = a^2 - 2ab + b^2\"\/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_9ffd306bb494aacaca4c427f775fdfc9.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(a + b)^2 = a^2 + 2ab + b^2\" title=\"(a + b)^2 = a^2 + 2ab + b^2\"\/><\/p>\n<p>&nbsp;<\/p>\n<p>Now let\u2019s take a look at a question.<\/p>\n<p>&nbsp;<\/p>\n<table width=\"600\">\n<tbody>\n<tr>\n<th><span style=\"text-decoration: underline\">Column A<\/span><\/th>\n<th><span style=\"text-decoration: underline\">Column B<\/span><\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_b52b53055a1236a9d4bcd0c67c7059aa.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(x-2)^2\" title=\"(x-2)^2\"\/><\/td>\n<td style=\"text-align: center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_9661b23035de519a22d5d5d8b2f0a93a.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"x^2 - 4\" title=\"x^2 - 4\"\/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<ol>\n<li>The quantity in Column A is greater<\/li>\n<li>The quantity in Column B is greater<\/li>\n<li>The two quantities are equal<\/li>\n<li>The relationship cannot be determined from the information given<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>In Quantitative Comparison we can make it so each side is equal. Then we can balance the equation, adding and subtracting, multiplying and dividing, where necessary.<\/p>\n<p>&nbsp;<\/p>\n<p>First, note that <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_9661b23035de519a22d5d5d8b2f0a93a.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"x^2 - 4\" title=\"x^2 - 4\"\/> can be factored into <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_287a6eee9bc4a940ef4f2289959fb66e.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(x - 2)(x + 2)\" title=\"(x - 2)(x + 2)\"\/>. Now we can set both columns equal to each other:<\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_1196042ae9df87c839e0ae32dcfd7cce.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(x - 2)(x - 2) = (x - 2)(x + 2)\" title=\"(x - 2)(x - 2) = (x - 2)(x + 2)\"\/><\/p>\n<p>At this point we have to be careful. While algebra tells us to divide both sides by (x &#8211; 2), we need to be aware of the following: if \u00a0x is between 0 and 2, (x-2) yields a negative. However, (x + 2) yields a positive, meaning (x &#8211; 2)(x + 2) gives us a negative. In this case (A) would be bigger.<\/p>\n<p>However, if we divide each side by (x &#8211; 2) and solve we get the following:<\/p>\n<p>DIVIDE EACH SIDE BY (x \u2013 2)<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_6f14b6c71456c829da883e6043041d99.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"x - 2 = x + 2\" title=\"x - 2 = x + 2\"\/><\/p>\n<p>SUBTRACT \u2018X\u2019 FROM BOTH SIDES<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_97c8864ea6783f975333de25250f977b.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"0 = 4\" title=\"0 = 4\"\/><\/p>\n<p>&nbsp;<\/p>\n<p>This hardly looks like a solution (in fact it looks like I forgot to go to grade school!). However, what this yields is the important insight: Column B is now 4 greater than Column A. Therefore, the answer is (D). Alternatively you could pick numbers, such as &#8216;0&#8217; and &#8216;4&#8217;. Each gives us different answers, leading to the same conclusion: Answer (D).<\/p>\n<p>&nbsp;<\/p>\n<h2>\u00a0If the question has variables, but there no polynomials, plug in values.<\/h2>\n<p>This advice pertains to variables that are not in polynomial form, as seen above. Here come up with easy numbers to plug in to see which values the columns yield.<\/p>\n<p>&nbsp;<\/p>\n<p>0 &gt; x &gt; y &gt; z &gt; -1<\/p>\n<p>&nbsp;<\/p>\n<table width=\"600\">\n<tbody>\n<tr>\n<th><span style=\"text-decoration: underline\">Column A<\/span><\/th>\n<th><span style=\"text-decoration: underline\">Column B<\/span><\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_1d72a5972bafc1b930aef34b0e3dccb8.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"x\/y\" title=\"x\/y\"\/><\/td>\n<td style=\"text-align: center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_bfea2d559b2af1bd1be87a7ebe041aeb.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"y\/z\" title=\"y\/z\"\/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<ol>\n<li>The quantity in Column A is greater<\/li>\n<li>The quantity in Column B is greater<\/li>\n<li>The two quantities are equal<\/li>\n<li>The relationship cannot be determined from the information given<\/li>\n<\/ol>\n<p><span style=\"text-decoration: underline\"><br \/>\n<\/span><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_965_850e371691790b510dc43d7433d0fc39.png\" style=\"vertical-align:-35px; display: inline-block ;\" alt=\"{-1\/4}\/ {-1\/2} = 1\/2\" title=\"{-1\/4}\/ {-1\/2} = 1\/2\"\/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_965_48ab4557ad65470e7a053d8c7e3048a2.png\" style=\"vertical-align:-35px; display: inline-block ;\" alt=\"{-1\/2}\/ {-3\/4} = 2\/3\" title=\"{-1\/2}\/ {-3\/4} = 2\/3\"\/>.<\/p>\n<p>&nbsp;<\/p>\n<p>It would be very nice if we could just stop here and choose answer (B). However, things are not so simple. When plugging in one set of numbers we will inevitably come up with one outcome. Once we\u2019ve plugged in and come up with one answer, whether it is (A), (B), or (C), our job is to disprove that answer.<\/p>\n<p>&nbsp;<\/p>\n<p>Can we make Column A larger than Column B? Well what if we plug in a values for x and y that are very close to one another. <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_965_62263431378433ce80bed05896c04aa7.png\" style=\"vertical-align:-35px; display: inline-block ;\" alt=\"{-1\/10}\/ {-1\/9} = 9\/10\" title=\"{-1\/10}\/ {-1\/9} = 9\/10\"\/>. Now, for z we can plug in a value close to -1, say -8\/9. This gives us <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gre\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_965_c6f089e726f91f25d53ef754601096a2.png\" style=\"vertical-align:-35px; display: inline-block ;\" alt=\"{-1\/9}\/{-8\/9} = 1\/8\" title=\"{-1\/9}\/{-8\/9} = 1\/8\"\/>. Now you can see (B) is much smaller. Therefore the answer is (D).<\/p>\n<p>&nbsp;<\/p>\n<h2><strong>Takeaway<\/strong><\/h2>\n<p>You should be adept at both algebra and plugging-in to efficiently\u2014and accurately\u2014answer a quantitative comparison question that contains variables. Typically it is best to simply check for polynomials before plugging in. To test out your skills, try these <a href=\"https:\/\/magoosh.com\/gre\/gre-math-practice-questions\/\">GRE math questions for practice<\/a>!<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Many quantitative questions have variables in both columns. While your first instinct may be to work algebraically, this strategy is not always best. Often the fastest way to a solution is by plugging in different values to see which column is greater. Developing a sense of when to plug in and when to solve algebraically [&hellip;]<\/p>\n","protected":false},"author":10,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25,54],"tags":[],"ppma_author":[12264],"class_list":["post-10060","post","type-post","status-publish","format-standard","hentry","category-math","category-math-question-types"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Quantitative Comparison and Manipulation - Magoosh Blog \u2014 GRE\u00ae Test<\/title>\n<meta name=\"description\" content=\"Developing a sense of when to plug in and when to solve algebraically takes practice. 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