{"id":11670,"date":"2012-05-23T23:35:25","date_gmt":"2012-05-24T06:35:25","guid":{"rendered":"https:\/\/gmatclub.com\/blog\/?p=11670"},"modified":"2012-05-23T10:54:54","modified_gmt":"2012-05-23T17:54:54","slug":"11670","status":"publish","type":"post","link":"https:\/\/gmatclub.com\/blog\/11670\/","title":{"rendered":"Rational Irrationality on the GMAT"},"content":{"rendered":"

\"\"<\/a>\u00a0Rational Irrationality on the GMAT<\/strong><\/p>\n

By: Eli Meyer<\/strong><\/p>\n

Despite the name, irrational numbers aren\u2019t crazy. You might think they are, and you wouldn\u2019t be alone. The followers of Pythagoras (yes, the right triangle guy!) believed that irrational numbers were heretical; supposedly, they threw the first person to prove irrationals existed into the ocean! But in fact, the word \u2018irrational\u2019 refers to the mathematical concept of the \u2018ratio.\u2019 Irrational numbers are numbers that cannot be expressed at the ratio of two integers, a\/b.<\/em><\/p>\n

There are an infinite number of irrational numbers, though far fewer are of mathematical note. If you\u2019ve ever studied statistics or calculus, you\u2019ve probably run across the natural number e, <\/em>and ancient architects had a centuries-long love affair with \u03c6, the golden ratio. But on the GMAT, there are only two types of irrational numbers that will be relevant to you: \u03c0 and radicals.<\/p>\n

Both of these irrationals will appear most often in geometry questions, and both of them have the same basic rule: you don\u2019t need an exact numerical value. Because they can\u2019t be reduced to a ratio (assuming, in the case of a radical, that the number underneath isn\u2019t a perfect square), their values can\u2019t be effectively determined with pen and paper. So generally, they will simply appear as-is in the answer choices. An answer choice to a circle problem will take the form of 3\u03c0, not 9.42.<\/p>\n

As a result, we can treat these irrational terms just like we would a variable like an x<\/em> or y. <\/em>So on complex algebraic statements that feature irrational numbers, we can simplify by adding coefficients to irrationals and combining like terms. In other words, if we are given:<\/p>\n

(3 X\u00a0\u221a2) +\u00a0\u221a2<\/strong><\/p>\n

we can express the first term with an algebraic coefficient:<\/p>\n

3 \u221a2 +\u00a0\u221a2<\/strong><\/p>\n

Then, we can combine like terms, getting:<\/p>\n

4\u221a2<\/strong><\/p>\n

Not only is this helpful and efficient, it\u2019s necessary; GMAT answer choices will almost invariably be written in the simplest terms possible.<\/p>\n

Similarly, Irrationals can (and sometimes must) be factored out as well. If the GMAT gives us a multi-irrational tangle like this one:<\/p>\n

3\u03c0+\u03c0\u221a2<\/strong><\/p>\n

Then anything we can do to make it more manageable will help us find the correct answer choice. In this case, we are adding two terms that are both multiplied by \u03c0. That means we can pull the \u03c0 out. The result is a little simpler:<\/p>\n

\u03c0\u00a0(3+\u221a2)<\/strong><\/p>\n

This could help us find the answer to a tough problem with circles and triangle. So get comfortable manipulating these indivisible terms, and you\u2019re one step closer to acing the GMAT quantitative section.<\/p>\n

Problem<\/strong><\/p>\n

A right circular cylinder has a height of 20 and a radius of 5. A rectangular solid with a height of 15, and a square base is placed in the cylinder such that each of the corners of the solid is tangent to the cylinder wall. Liquid is then poured into the cylinder such that it reaches the rim. What is the volume of the liquid?<\/p>\n

(A) 500(\u03c0 \u2013 3)<\/p>\n

(B) 500(\u03c0 \u2013 2.5)<\/p>\n

(C) 500(\u03c0 -2)<\/p>\n

(D) 500(\u03c0 \u2013 1.5)<\/p>\n

(E) 500(\u03c0- 1)<\/p>\n

 <\/p>\n

Solution<\/strong><\/p>\n

Step 1: Analyze the Question<\/strong><\/p>\n

For any Geometry question without a figure, our first step must always be to draw a quick sketch. This is particularly important for solids and complex word problems such as this question. The challenging element of this question is that a square solid is inscribed inside the right circular cylinder. Our sketch shows this:<\/p>\n

\"\"<\/a><\/p>\n

Notice that we drew the radius of the circle to end at the corner where the circle and square meet. When working with multiple shapes, look for points or sides where the two shapes overlap, since you can use the common side or point to connect the equations of the shapes.<\/p>\n

Step 2: State the Task<\/strong><\/p>\n

We must determine the volume of the liquid that has filled the cylinder. The volume of the liquid will be equal to the volume of the cylinder minus the volume of the rectangular solid. So we must calculate the volume of the cylinder and the volume of the rectangular solid.<\/p>\n

Step 3: Approach Strategically<\/strong><\/p>\n

The volume of the cylinder is<\/p>\n

To calculate the volume of the rectangular solid, we\u2019ll need to determine the lengths of the sides of the square. We can draw a right triangle to help.<\/p>\n

\"\"<\/a><\/p>\n

Recognizing a 45-45-90 triangle (always true of a bisected square), we know that the ratio of the side lengths of the triangle and can solve for the value of x because we know the hypotenuse is 10.<\/p>\n

\"\"<\/a><\/p>\n

We can now calculate the volume of the liquid: Volume of Liquid = Volume of the Cylinder \u2013 Volume of the Rectangular Solid<\/p>\n

Volume of the Liquid =\u00a0\"\"<\/a><\/p>\n

Step 4: Confirm Your Answer<\/strong><\/p>\n

Reread the question, making sure that you did not misread any important information.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u00a0Rational Irrationality on the GMAT By: Eli Meyer Despite the name, irrational numbers aren\u2019t crazy. You might think they are, and you wouldn\u2019t be alone. The followers of Pythagoras (yes,…<\/p>\n","protected":false},"author":120,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-11670","post","type-post","status-publish","format-standard","hentry","category-uncategorized","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11670","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\/120"}],"replies":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/comments?post=11670"}],"version-history":[{"count":8,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11670\/revisions"}],"predecessor-version":[{"id":11678,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11670\/revisions\/11678"}],"wp:attachment":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/media?parent=11670"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/categories?post=11670"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/tags?post=11670"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}