The post GMAT Math: How to Divide by a Square Root appeared first on Magoosh GMAT Blog.
]]>1. In the equation above, x =
2. Triangle ABC is an equilateral triangle with an altitude of 6. What is its area?
3. In the equation above, x =
The second one throws in a little geometry. You may want to review the properties of the 30-60-90 Triangle and the Equilateral Triangle if those are unfamiliar. The first one is just straightforward arithmetic. The third is quite hard. For any of these, it may well be that, even if you did all your multiplication and division correctly, you wound up with an answers of the form — something divided by the square root of something — and you are left wondering: why doesn’t this answer even appear among the answer choices? If this has you befuddled, you have found exactly the right post.
When we first met fractions, in our tender prepubescence, both the numerators and denominators were nice easy positive integers. As we now understand, any kind of real number, any number on the entire number line, can appear in the numerator or denominator of a fraction. Among other things, radicals —- that is, square-root expressions —- can appear in either the numerator or denominator. There’s no particular issue if we have the square-root in a numerator. For example,
is a perfectly good fraction. In fact, those of you who ever took trigonometry might even recognize this special fraction. Suppose, though, we have a square root in the denominator: what then? Let’s take the reciprocal of this fraction.
This is no longer a perfectly good fraction. Mathematically, this is a fraction “in poor taste”, because we are dividing by a square-root. This fraction is crying out for some kind of simplification. How do we simplify this?
By standard mathematical convention, a convention the GMAT follows, we don’t leave square-roots in the denominator of a fraction. If a square-root appears in the denominator of a fraction, we follow a procedure called rationalizing the denominator.
We know that any square root times itself equals a positive integer. Thus, if we multiplied a denominator of the square root of 3 by itself, it would be 3, no longer a radical. The trouble is —- we can’t go around multiplying the denominator of fractions by something, leaving the numerator alone, and expect the fraction to maintain its value. BUT, remember the time-honored fraction trick — we can always multiply a fraction by A/A, by something over itself, because the new fraction would equal 1, and multiplying by 1 does not change the value of anything.
Thus, to simplify a fraction with the square root of 3 in the denominator, we multiply by the square root of 3 over the square root of 3!
That last expression is numerically equal to the first expression, but unlike the first, it is now in mathematical “good taste”, because there’s no square root in the denominator. The denominator has been rationalized (that is to say, the fraction is now a rational number).
Sometimes, some canceling occurs between the number in the original numerator and the whole number that results from rationalizing the denominator. Consider the following example:
That pattern of canceling in the simplification process may give you some insight into practice problem #1 above.
This is the next level of complexity when it comes to dividing by square roots. Suppose we are dividing a number by an expression that involves adding or subtracting a square root. For example, consider this fraction:
This is a fraction in need of rationalization. BUT, if we just multiply the denominator by itself, that WILL NOT eliminate the square root — rather, it will simply create a more complicated expression involving a square root. Instead, we use the difference of two squares formula, = (a + b)(a – b). Factors of the form (a + b) and (a – b) are called conjugates of one another. When we have (number + square root) in the denominator, we create the conjugate of the denominator by changing the addition sign to a subtraction sign, and then multiply both the numerator and the denominator by the conjugate of the denominator. In the example above, the denominator is three minus the square root of two. The conjugate of the denominator would be three plus the square root of two. In order to rationalize the denominator, we multiply both the numerator and denominator by this conjugate.
Notice that the multiplication in the denominator resulted in a “differences of two squares” simplification that cleared the square roots from the denominator. That final term is a fully rationalized and fully simplified version of the original.
Having read these posts about dividing by square roots, you may want to give the three practice questions at the top of this article another try, before reading the explanations below. If you have any questions on dividing by square roots or the explanations below, please ask them in the comments sections! And good luck conquering these during your GMAT!
1) To solve for x, we will begin by cross-multiplying. Notice that
because, in general, we can multiply and divide through radicals.
Cross-multiplying, we get
You may well have found this and wondered why it’s not listed as an answer. This is numerically equal to the correct answer, but of course, as this post explains, this form is not rationalized. We need to rationalize the denominator.
Answer = (D)
2) We know the height of ABC and we need to find the base. Well, altitude BD divides triangle ABC into two 30-60-90 triangles. From the proportions in a 30-60-90 triangle, we know:
Now, my predilection would be to rationalize the denominator right away.
Now, AB is simplified. We know AB = AC, because the ABC is equilateral, so we have our base.
Answer = (C)
3) We start by dividing by the expression in parentheses to isolate x.
Of course, this form does not appear among the answer choices. Again, we need to rationalize the denominator, and this case is a little trickier because we have addition in the denominator along with the square root. Here we need to find the conjugate of the denominator —- changing the plus sign to a minus sign — and then multiply the numerator and denominator by this conjugate. This will result in —-
Answer = (A)
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1) Let abcd be a general four-digit number and all the digits are non-zero. How many four-digits numbers abcd exist such that the four digits are all distinct and such that a + b + c = d?
(A) 6
(B) 7
(C) 24
(D) 36
(E) 42
2) Let abcd be a general four-digit number. How many odd four-digits numbers abcd exist such that the four digits are all distinct, no digit is zero, and the product of a and b is the two digit number cd?
(A) 4
(B) 6
(C) 12
(D) 24
(E) 36
3) There are 500 cars on a sales lot, all of which have either two doors or four doors. There are 165 two-door cars on the lot. There are 120 four-door cars that have a back-up camera. Eighteen percent of all the cars with back-up cameras have standard transmission. If 40% of all the cars with both back-up cameras and standard transmission are two-door cars, how many four-door cars have both back-up cameras and standard transmission?
(A) 18
(B) 27
(C) 36
(D) 45
(E) 54
4) At Mnemosyne Middle School, there are 700 students: all the students are boys or girls in the 4^{th} or 5^{th} grade. There are 320 students in the 4^{th} grade, and there are 210 girls in the 5^{th} grade. Fifty percent of the 5^{th} graders and 40% of the 4^{th} graders take Mandarin Chinese. Ninety 5^{th} grade boys do not take Mandarin Chinese. The number of 4^{th} grade girls taking Mandarin Chinese is less than half of the number of 5^{th} grade girls taking Mandarin Chinese. Which of the following could be the number of 4^{th} grade boys in Mandarin Chinese?
(A) 10
(B) 40
(C) 70
(D) 100
(E) 130
5) A hundred identical cubic boxes are currently arranged in four cubes: a single cubic box, a 2 x 2 x 2 cube, a 3 x 3 x 3 cube, and a 4 x 4 x 4 cube. These four are not touching each other. All outward faces are painted and all inward faces are not painted. These four cubes are going to be dismantled and reassembled as a flat 10 x 10 square. The top and all the edges of this 10 x 10 square must be painted, but there is no requirement for paint on the bottom. How many individual faces will have to be painted to accommodate the requirements of this new design?
(A) 0
(B) 5
(C) 9
(D) 16
(E) 27
6) Twelve points are spaced evenly around a circle, lettered from A to L. Let N be the total number of isosceles triangles, including equilateral triangles, that can be constructed from three of these points. A different orientation of the same lengths counts as a different triangle, because a different combination of points form the vertices. What is the value of N?
(A) 48
(B) 52
(C) 60
(D) 72
(E) 120
7) Theresa is a basketball player practicing her free throws. On her first free throw, she has a 60% chance of making the basket. If she has just made a basket on her previous throw, she has a 80% of making the next basket. If she has just failed to make a basket on her previous throw, she has a 40% of making the next basket. What is the probability that, in five throws, she will make at least four baskets?
8) Suppose a “Secret Pair” number is a four-digit number in which two adjacent digits are equal and the other two digits are not equal to either one of that pair or each other. For example, 2209 and 1600 are “Secret Pair” numbers, but 1333 or 2552 are not. How many “Secret Pair” numbers are there?
(A) 720
(B) 1440
(C) 1800
(D) 1944
(E) 2160
9) In the coordinate plane, a circle with its center on the negative x-axis has a radius of 12 units, and passes through (0, 6) and (0, – 6). What is the area of the part of this circle in the first quadrant?
10) In the coordinate plane, line L passes above the points (50, 70) and (100, 89) but below the point (80, 84). Which of the following could be the slope of line L?
(A) 0
(B) 1/2
(C) 1/4
(D) 2/5
(E) 6/7
11) At the beginning of the year, an item had a price of A. At the end of January, the price was increased by 60%. At the end of February, the new price was decreased by 60%. At the end of March, the new price was increased by 60%. At the end of April, the new price was decreased by 60%. On May 1^{st}, the final price was approximately what percent of A?
(A) 41%
(B) 64%
(C) 100%
(D) 136%
(E) 159%
12) Suppose that, at current exchange rates, $1 (US) is equivalent to Q euros, and 1 euro is equivalent to 7Q Chinese Yuan. Suppose that K kilograms of Chinese steel, worth F Chinese Yuan per kilogram, sold to a German company that paid in euros, can be fashioned into N metal frames for chairs. These then are sold to an American company, where plastic seats & backs will be affixed to these frames. If the German company made a total net profit of P euros on this entire transaction, how much did the US company pay in dollars for each frame?
13) At the Zamenhof Language School, at least 70% of the students take English each year, at least 40% take German each year, and between 30% and 60% take Italian each year. Every student must take at least one of these three languages, and no student is allowed to take more than two languages in the same year. What is the possible percentage range for students taking both English and German in the same year?
(A) 0% to 70%
(B) 0% to 100%
(C) 10% to 70%
(D) 10% to 100%
(E) 40% to 70%
14) On any given day, the probability that Bob will have breakfast is more than 0.6. The probability that Bob will have breakfast and will have a sandwich for lunch is less than 0.5. The probability that Bob will have breakfast or will have a sandwich for lunch equals 0.7. Let P = the probability that, on any given day, Bob will have a sandwich for lunch. If all the statements are true, what possible range can be established for P?
(A) 0 < P < 0.6
(B) 0 ≤ P < 0.6
(C) 0 ≤ P ≤ 0.6
(D) 0 < P < 0.7
(E) 0 ≤ P < 0.7
(A) – 64
(B) – 7
(C) 38
(D) 88
(E) 128
Explanations for this problem are at the end of this article.
Here are twenty-eight other articles on this blog with free GMAT Quant practice questions. Some have easy questions, some have medium, and few have quite challenging questions.
1) GMAT Geometry: Is It a Square?
2) GMAT Shortcut: Adding to the Numerator and Denominator
3) GMAT Quant: Difficult Units Digits Questions
4) GMAT Quant: Coordinate Geometry Practice Questions
5) GMAT Data Sufficiency Practice Questions on Probability
6) GMAT Quant: Practice Problems with Percents
7) GMAT Quant: Arithmetic with Inequalities
8) Difficult GMAT Counting Problems
9) Difficult Numerical Reasoning Questions
10) Challenging Coordinate Geometry Practice Questions
11) GMAT Geometry Practice Problems
12) GMAT Practice Questions with Fractions and Decimals
13) Practice Problems on Powers and Roots
14) GMAT Practice Word Problems
15) GMAT Practice Problems: Sets
16) GMAT Practice Problems: Sequences
17) GMAT Practice Problems on Motion
18) Challenging GMAT Problems with Exponents and Roots
19) GMAT Practice Problems on Coordinate Geometry
20) GMAT Practice Problems: Similar Geometry Figures
20) GMAT Practice Problems: Variables in the Answer Choices
21) Counting Practice Problems for the GMAT
22) GMAT Math: Weighted Averages
23) GMAT Data Sufficiency: More Practice Questions
24) Intro to GMAT Word Problems, Part I
25) GMAT Data Sufficiency Geometry Practice Questions
26) GMAT Data Sufficiency Logic: Tautological Questions
27) GMAT Quant: Rates and Ratios
28) Absolute Value Inequalities
These are hard problems. When you read the solutions, don’t merely read them passively. Study the strategies used, and do what you can to retain them. Learn from your mistakes!
1) We need sets of three distinct integers {a, b, c} that have a sum of one-digit number d. There are seven possibilities:
For each set, the sum-digit has to be in the one’s place, but the other three digits can be permutated in 3! = 6 ways in the other three digits. Thus, for each item on that list, there are six different possible four-digit numbers. The total number of possible four-digit numbers would be 7*6 = 42. Answer = (E)
2) The fact that abcd is odd means that cd must be an odd number and that a & b both must be odd. That limits the choices significantly. We know that neither a nor b can equal 1, because any single digit number times 1 is another single digit number, and we need a two-digit product—there are no zeros in abcd. We also know that neither a nor b can equal 5, because any odd multiple of 5 ends in 5, and we would have a repeated digit: the requirement is that all four digits be distinct.
Therefore, for possible values for a & b, we are limited to three odd digits {3, 7, 9}. We can take three different pairs, and in each pair, we can swap the order of a & b. Possibilities:
Those six are the only possibilities for abcd.
Answer = (B)
3) Total number of cars = 500
2D cars total = 165, so
4D cars total = 335
120 4D cars have BUC
“Eighteen percent of all the cars with back-up cameras have standard transmission.”
18% = 18/100 = 9/50
This means that the number of cars with BUC must be a multiple of 50.
How many 2D cars can we add to 120 4D cars to get a multiple of 50? We could add 30, or 80, or 130, but after that, we would run out of 2D cars. These leaves three possibilities for the total number with BUC:
If a total of 150 have BUC, then 18% or 27 of them also have ST.
If a total of 200 have BUC, then 18% or 36 of them also have ST.
If a total of 250 have BUC, then 18% or 45 of them also have ST.
Then we are told: “40% of all the cars with both back-up cameras and standard transmission are two-door car.”
40% = 40/100 = 2/5
This means that number of cars with both back-up cameras and standard transmission must be divisible by 5. Of the three possibilities we have, only the third words.
Total cars with BUC cams = 250 (120 with 4D and 130 with 2D)
18% or 45 of these also have ST.
40% of that is 18, the number of 2D cars with both BUC and ST.
Thus, the number of 4D cars with both BUC and ST would be
45 – 18 = 27
Answer = (B)
4) 700 student total
4G = total number of fourth graders
5G = total number of fifth graders
We are told 4G = 320, so 5G = 700 – 320 = 380
5GM, 5GF = fifth grade boys and girls, respectively
We are told 5GF = 210, so 5GM = 380 – 210 = 170
4GC, 5GC = total number of 4^{th} or 5^{th} graders, respectively taking Chinese
We are told
5GC = 0.5(5G) = 0.5(380) = 190
4GC = 0.4(4G) = 0.4(320) = 128
4GFM, 4GMC, 5GFC, 5GMC = 4^{th}/5^{th} grade boys & girls taking Chinese
We are told that, of the 170 fifth grade boys, 90 do not take Chinese, so 170 = 90 = 80 do. Thus 5GMC = 80.
5GMC + 5GFC = 5GC
80 + 5GFC = 190
5GFC = 110
We are told:
4GFM < (0.5)(5GFC)
4GFM < (0.5)(100)
4GFM < 55
Thus, 4GFM could be as low as zero or as high as 54.
4GMC = 4GC – 4GFM
If 4GFM = 0, then 4GMC = 128 – 0 = 128
If 4GFM = 54, then 4GMC = 128 – 54 = 74
Thus, fourth grade boys taking Mandarin Chinese could take on any value N, such that 74 ≤ N ≤ 128. Of the answer choices listed, the only one that works is 100.
Answer = (D)
5) The single cube has paint on all six sides. Each of the eight boxes in the 2 x 2 x 2 cube has paint on three sides (8 corner pieces). In the 3 x 3 x 3 cube, there are 8 corner pieces, 12 edge pieces (paint on two sides), 6 face pieces (paint on one side), and one interior piece (no paint). In the 4 x 4 x 4 cube, there are 8 corner pieces, 24 edge pieces, 24 face pieces, and 8 interior pieces. This chart summarizes what we have:
For the 10 x 10 flat square, we will need 4 corner pieces that have paint on three sides, 32 edge pieces that have paint on two sides (top & side), and 64 middle pieces that have paint on one side (the top).
We could use either the single total box or any of the 24 corner boxes for the four corners of the square. That leaves 21 of these, and 35 edge boxes, more than enough to cover the 32 edges of the square. The remaining ones, as well as all 30 face boxes, can be turned paint-side-up to fill in the center. The only boxes that will need to be painted, one side each, are the 9 interior boxes. Thus, we have 9 sides to paint.
Answer = (C)
6) Here’s a diagram.
First, let’s count the equilateral triangles. They are {AEI, BFJ, CGK, DHL}. There are only four of them.
Now, consider all possible isosceles triangles, excluding equilateral triangles, with point A as the vertex. We could have BAL, CAK, DAJ, and FAH. All four of those have a line of symmetry that is vertical (through A and G). Thus, we could make those same four triangles with any other point as the vertex, and we would never repeat the same triangle in the same orientation. That’s 4*12 = 48 of these triangles, plus the 4 equilaterals, is 52 total triangles.
Answer = (B)
7) There are five basic scenarios for this:
Case I: (make)(make)(make)(make)(any)
If she makes the first four, then it doesn’t matter if she makes or misses the fifth!
Case II: (miss)(make)(make)(make)(make)
Case III: (make)(miss)(make)(make)(make)
Case IV: (make)(make)(miss)(make)(make)
Case V: (make)(make)(make)(miss)(make)
Put in the probabilities:
Case I: (0.6)(0.8)(0.8)(0.8)
Case II: (0.4)(0.4)(0.8)(0.8)(0.8)
Case III: (0.6)(0.2)(0.4)(0.8)(0.8)
Case IV: (0.6)(0.8)(0.2)(0.4)(0.8)
Case V: (0.6)(0.8)(0.8)(0.2)(0.4)
Since all the answers are fractions, change all of those to fractions. Multiply the first by (5/5) so it has the same denominator as the other products.
Case I: (3/5)(4/5)(4/5)(4/5)(5/5) = 960/5^5
Case II: (2/5)(2/5)(4/5)(4/5)(4/5) = 256/5^5
Case III: (3/5)(1/5)(2/5)(4/5)(4/5) = 96/5^5
Case IV: (3/5)(4/5)(1/5)(2/5)(4/5) = 96/5^5
Case V: (3/5)(4/5)(4/5)(1/5)(2/5) = 96/5^5
Add the numerators. Since 96 = 100 – 4, 3*96 = 3(100 – 4) = 300 – 12 = 288.
288 + 256 + 960 = 1504
P = 1504/5^5
Answer = (E)
8) There are three cases: AABC, ABBC, and ABCC.
In case I, AABC, there are nine choices for A (because A can’t be zero), then 9 for B, then 8 for C. 9*9*8 = 81*8 = 648.
In case II, ABBC, there are 9 choices for A, 9 for B, and 8 for C. Again, 648.
In case III, ABCC, there are 9 choices for A, 9 for B, and 8 for C. Again, 648.
48*3 = (50 – 2)*3 = 150 – 6 = 144
3*648 = 3(600 + 48) = 1800 + 144 = 1948
Answer = (D)
9)
We know that the distance from A (0,6) to B (0, – 6) is 12, so triangle ABO is equilateral. This means that angle AOB is 60°. The entire circle has an area of
A 60° angle is 1/6 of the circle, so the area of sector AOB (the “slice of pizza” shape) is
The area of an equilateral triangle with side s is
Equilateral triangle AOB has s = 12, so the area is
If we subtract the equilateral triangle from the sector, we get everything to the right of the x-axis.
Again, that’s everything to the right of the x-axis, the parts of the circle that lie in Quadrants I & IV. We just want the part in Quadrant I, which would be exactly half of this.
Answer = (C)
10) One point is (50, 70) and one is (100, 89): the line has to pass above both of those. Well, round the second up to (100, 90)—if the line goes above (100, 90), then it definitely goes about (100, 89)!
What is the slope from (50, 70) to (100, 90)? Well, the rise is 90 – 70 = 20, and the run is 100 – 50 = 50, so the slope is rise/run = 20/50 = 2/5. A line with a slope of 2/5 could pass just above these points.
Now, what about the third point? For the sake of argument, let’s say that the line has a slope of 2/5 and goes through the point (50, 71), so it will pass above both of the first two points. Now, move over 5, up 2: it would go through (55, 73), then (60, 75), then (65, 77), then (70, 79), then (75, 81), then (80, 83). This means it would pass under the third point, (80, 84). A slope of 2/5 works for all three points.
We don’t have to do all the calculations, but none of the other slope values works.
Answer = (D)
11) The trap answer is 100%: a percent increase and percent decrease by the same percent do not cancel out.
Let’s say that the A = $100 at the beginning of the year.
End of January, 60% increase. New price = $160
End of February, 60% decrease: that’s a decrease of 60% of $160, so that only 40% of $160 is left.
10% of $160 = $16
40% of $160 = 4(16) = $64
That’s the price at the end of February.
End of March, a 60% increase: that’s a increase of 60% of $64.
10% of $64 = $6.40
60% of $64 = 6(6 + .40) = 36 + 2.4 = $38.40
Add that to the starting amount, $64:
New price = $64 + $38.40 = $102.40
End of April, 60% decrease: that’s a decrease of 60% of $102.40, so that only 40% of $102.40 is left.
At this point, we are going to approximate a bit. Approximate $102.40 as $100, so 40% of that would be $40. The final price will be slightly more than $40.
Well, what is slightly more than $40, as a percent of the beginning of the year price of $100? That would be slightly more than 40%.
Answer = (A)
12) The K kilograms, worth F Chinese Yuan per kilogram, are worth a total of KF Chinese Yuan. The German company must pay this amount.
Since 1 euro = (7Q) Chinese Yuan, then (1/(7Q)) euro = 1 Chinese Yuan, and (KF/7Q) euros = KF Chinese Yuan. That’s the amount that the Germans pay to the Chinese.
That is the German company’s outlay, in euros. Now, they make N metal chairs, and sell them, making a gross profit of P euros.
That must be the total revenue of the German company, in euros. This comes from the sale to the American company. Since $1 = Q euros, $(1/Q) = 1 euro, so we change that entire revenue expression to euros to dollars, we divide all terms by Q.
That must be the total dollar amount that leaves the American company and goes to the German company. This comes from the sale of N metal frames for chairs, so each one must have been 1/N of that amount.
Answer = (A)
13) First, we will focus on the least, the lowest value. Suppose the minimum of 70% take English, and the minimum of 40% take German. Even if all 30% of the people not taking English take German, that still leaves another 10% of people taking German who also have to be taking English. Thus, 10% is the minimum of this region.
Now, the maximum. Both the German and English percents are “at least” percents, so either could be cranked up to 100%. The trouble is, though, that both can’t be 100%, because some folks have to take Italian, and nobody can take three languages at once. The minimum taking Italian is 30%. Let’s assume all 100% take German, and that everyone not taking Italian is taking English: that’s 70% taking English, all of whom also would be taking German. Thus, 70% is the maximum of this region.
Answer = (C)
14) Let A = Bob eats breakfast, and B = Bob has a sandwich for lunch. The problem tells us that:
P(A) > 0.6
P(A and B) < 0.5
P(A or B) = 0.7
First, let’s establish the minimum value. If Bob never has a sandwich for lunch, P(B) = 0, then it could be that P(A and B) = 0, which is less than 0.5, and it could be that P(A) = 0.7, which is more than 0.6, so that P(A or B) = 0.7. All the requirements can be satisfied if P(B) = 0, so it’s possible to equal that minimum value.
Now, the maximum value. Since P(A or B) = 0.7, both P(A) and P(B) must be contained in this region. See the conceptual diagram.
The top line, 1, is the entire probability space. The second line, P(A or B) = 0.7, fixes the boundaries for A and B. P(A) is the purple arrow, extending from the right. P(B) is the green arrow extending from the left. The bottom line, P(A and B) < 0.5, is the constraint on their possible overlap.
Let’s say that P(A) is just slightly more than 0.6. That means the region outside of P(A), but inside of P(A or B) is slightly less than 1. That’s the part of P(B) that doesn’t overlap with P(A). Then, the overlap has to be less than 0.5. If we add something less than 1 to something less than 5, we get something less than 6. P(B) can’t equal 0.6, but it can any value arbitrarily close to 0.6.
Thus, 0 ≤ P(B) < 0.6.
Answer = (B)
15)
Answer = (E)
The post Challenging GMAT Math Practice Questions appeared first on Magoosh GMAT Blog.
]]>The post GMAT Scores for Top MBA Programs appeared first on Magoosh GMAT Blog.
]]>If you have a strong business school application, you likely won’t need a near-perfect GMAT score for admission into a top MBA program. But how do you know if your GMAT score is up to par with your dream school’s GMAT requirements? Have no fear; we’ve collected GMAT score data from the admissions offices of all the top business schools to bring you the most recent data in average GMAT scores by school.
Special update: We’ve collected the very most recent information for average GMAT scores by school for the top 10 business schools in the United States. See the section immediately below.
Note: This is the most up-to-date information on average GMAT scores by school, GMAT requirements by schools, and other important statistics. All data for Harvard GMAT scores, Stanford GMAT scores, and the rest (including school ranking), comes from U.S. News and Word Report.
Name of MBA Program/Business School | Average GMAT Score | Rank | Enrollment, 2016-2017 |
---|---|---|---|
Harvard Business School | 725 | 1 | 1,872 |
Stanford Graduate School of Business | 733 | 2 (tie) | 824 |
University of Chicago (Booth) | 726 | 2 (tie) | 1,180 |
University of Pennsylvania (Wharton) | 732 | 4 | 1,715 |
Northwestern University (Kellogg) | 724 | 5 (tie) | 1,272 |
Massachusetts Institute of Technology (Sloan) | 716 | 5 (tie) | 806 |
University of California-Berkeley (Haas) | 715 | 7 | 502 |
Yale School of Management | 761 | 8 (tie) | 668 |
Dartmouth (Tuck) | 717 | 8 (tie) | 563 |
Columbia Business School | 715 | 10 | 1,287 |
Of course, there’s a lot more out there than just the top 10. When it comes to finding your fit and researching MBA programs, the ranking numbers don’t tell the whole story.
Scroll down to see average GMAT scores for a wide range of reputable b-schools in the USA.
Note: This information is recent, but is not quite as up-to-date as the date in the table above. Still, these stats should give you a pretty good idea of these schools’ GMAT requirements and expectations. More updates will be coming soon. In the meantime, use this table to get a general idea of where you stand with each school.
(Click the image to open the infographic in a new page and zoom in/out!)
Important to note: officially, the GMAT scale for verbal and quantitative goes up to 60, but in practice, the scale tops out at 51. Nowadays, a verbal subscore of 46 would get you in the 99th GMAT score percentile, while a 51 quant subscore would be in the 97th.
To accurately assess your GMAT score, you must understand the big picture of GMAT admissions, and remember that your GMAT score is just one part of your application.
First, familiarize yourself with GMAT scoring. Then, compare your score to the average GMAT scores by school of admitted students at your target programs. Keep in mind that an average score for a top business school is not the bare minimum you need to get in–approximately half of applicants get into that school with less than that average score. (In other words, not all Wharton students attained a 732 score even though that’s the average Wharton GMAT score). That means you can think about it as just that–an average score.
If your GMAT is good enough for the programs you like (say, for example, you want to go to University of Chicago and your score is a 726, just as Booth’s GMAT score is a 726), then focus your energy on strengthening other aspects of your application. And if your score doesn’t quite make the cut, then consider retaking the GMAT only so you can distinguish yourself from other applicants with a similar application profile to yours.
Ultimately, you have to decide what is a good GMAT score for you. GMAT scores may be paramount to the application process, but even a 720 combined score won’t get you into the best business schools without a strong application to back it up. Your entire profile must honestly and effectively represent your successes, abilities, and potential.
Still … a 720 can’t hurt.
If you’ve checked out an average GMAT score by school and think you need help getting there, then reach out about our Magoosh GMAT Prep! And while you’re at it, leave us a comment below with your thoughts about this infographic.
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]]>The post How Long Should I Study for the GMAT? appeared first on Magoosh GMAT Blog.
]]>1) The first consideration to answer how long you should study for the GMAT is simply: how good are you at the whole standardized-test thing in general? Some people regularly ace standardized tests. Others regularly flub them. This is an estimation—at a gut-level, how comfortable are you, and how successful have you been, with the whole standardized-test thing?
2) How many days you should study depends in part on how many hours a day you can study. Let’s say that 1 hour a day for six months would be very approximately equivalent to six hours a day for one month. The caveat, of course, is most people have real limits concerning how much they can focus. Many also have limitations on how much info they can absorb and assimilate in a single day. Can you put in six hours a day of quality, high-focus study time, day after day, for a month? If so, that’s fantastic. However for most people—not only because of the practical constraints of job and family, but also because of the cognitive constraints on focus and assimilation—the best option would be less-time-per-day over a longer number days studying for the GMAT.
3) Let’s say you have taken a practice test, relatively cold, with little prep, and got some score. We’ll call this a baseline score. What is your target score? How much do you want to improve from this cold-take baseline? Let’s say, with moderate prep, you could improve 50 points over a relatively cold-take. That’s readily do-able. Improving 100 points—that’s more of a challenge. Improving 150 or 200 points or more—that will take exceptionally diligent work. You’ll need to sustain this GMAT study plan over quite some time, and even then, an improvement of this magnitude is not guaranteed.
4) What are your relative strengths? Consider the two big categories—math and verbal. On a 1-10 scale, how would you rank your relative aptitudes in each? This may play into extra time over and above the time you spend studying specifically for the GMAT.
I would say a three month study plan, with 1-2 hours of GMAT study time per weekday and a single 3-4 hour stint on each weekend—that I would call moderate study, probably enough to produce for most people a 50-100 point increase over a relatively cold-take score. Again, this assumes eight hours of sleep a night, a healthy lifestyle, and a normal college-graduate level of learning and remembering.
If you want to improve substantially more than 50-100 points, I would suggest extending your GMAT study time for longer time than three months. In general, the more you can spread your study out over a long period—say, six months—the more time you will have to return a second and even a third time to each topic. This will take advantage of how the brain learns and processes. Repeated exposure helps to encode material into long-term memory.
If, for whatever combination of reasons, you have only a month to prepare for the GMAT, understand that’s not ideal. It will demand both longer stints each day as well as the sustained focus and commitment, in order to get the most out of it. You’re thinking strictly in terms of how many hours to study for the GMAT — not months. For just that one month, be ready to hunker down and work intensely.
If you are planning to take considerably less than a month to prepare for the GMAT—either you are unusually gifted, or you don’t really take the test seriously. How long you study for the GMAT is, to some extent, a statement about how seriously you take the GMAT. If you take the GMAT seriously, then put in the study time to prepare for it. If you don’t take it seriously, then why are you taking it at all? This is your life: it’s not a game, not a stage rehearsal for anything else. Time is precious. Why would you waste significant time and energy and focus and determination on something you don’t take seriously? It’s absolutely necessary to have time doing things that are enjoyable and un-serious in order to refresh and recharge, but why take on something difficult and demanding if you don’t take it seriously? Whoever you are, your time is worth more than that! That’s my 2¢.
This concerns consideration #4 above. If you would rate either of the categories three or below, that’s a red flag. That’s an indication you need extra GMAT study time and thus an extra head start. This is a big curveball in the how-long-do-I-study-for-the-GMAT question!
If you are a math whiz but weak in verbal, and most especially if English is not your first language, then yes, pursue a moderate study schedule, say, a three-month study schedule for folks stronger in math, and in addition to that, READ! Read at least an hour a day—two hours a day would be better. Reading the high-brow material recommended at that blog will accustom your ear to advanced grammatical constructions typical of GMAT Sentence Correction, and will help you practice the analysis skills you will need on both GMAT Critical Reasoning and GMAT Reading Comprehension. Ideally, you will begin this daily reading habit well before the rest of your GMAT studying—a year or more. Where will you get the time to do all this reading? Well, if you sharply reduce TV, video games, and other forms of electronic entertainment, you actually will be doing your brain a favor.
If you are relatively comfortable in verbal, and you haven’t even looked at math since an unfriendly farewell a few years back, then you need to study math, starting pretty much as soon as you finish reading this post. You don’t get a calculator on the GMAT Quant section, so practice mental math—every day, you should add & subtract & multiply & divide in your head. Get remedial books published for high school students, “Algebra Review”, “Geometry Review”, and start reading. Look for every possible application of math in your life. Think areas of rooms, grocery bills, gas mileage, and the like. Do the real world math. Ideally, all this focus on math should begin months before you embark on, say, a three-month study schedule for folks stronger in verbal.
In both cases, this extra focus you give one area or the other should be considered over and above how long you study for the GMAT. These are the extra hours you need to study for the GMAT.
Studying for the GMAT takes a lot of time, regardless of your skill level. Average GMAT students can expect to spend 100-170 hours studying, over the course of 2-3 months. The very top scorers on the GMAT often spend more than 170 hours, with study plans lasting up to 6 months. Keep this in mind when considering how many hours to study for the GMAT.
That’s a general overview of how long to study for the GMAT. The study schedules at the links above will give you more details, and more a sense of what’s required. If you have more questions about your needed GMAT study time situation, please let us know in the comment section below.
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]]>The post GMAT Test Dates | 2016, 2017, 2018 and Beyond! appeared first on Magoosh GMAT Blog.
]]>If you are planning to apply to full-time MBA programs next year to start classes in the next 12-18 months, this is the perfect time to start preparing for your GMAT test date. Even if you aren’t planning to apply for another few years, it’s not too early to take the test! GMAT scores are valid for five years so the sooner you can get this test out of the way, the more time you will have to focus on other aspects of your application, and the less stressed out you will be when deadlines start rolling around.
These timelines will help guide you as you start planning your preparation calendar for the next year. These timelines are based on the most common deadlines for rounds of applications at top MBA programs. Most top schools set MBA application deadlines three times a year, in three rounds. Check with specific schools for exact deadlines for Round 1, Round 2, and Round 3. And check out this article for help figuring out which round you should apply in.
December - February | March | April - May | June | July - August | September - October |
---|---|---|---|---|---|
Study | Take GMAT | Study | Retake GMAT | Essays, etc... | Round 1 due |
March - May | June | July - August | September | October-November | December - January |
---|---|---|---|---|---|
Study | Take GMAT | Study | Retake GMAT | Essays, etc... | Round 2 due |
June - August | September | October - November | December | January - February | March - April |
---|---|---|---|---|---|
Study | Take GMAT | Study | Retake GMAT | Essays, etc... | Round 3 due |
You can register to the test anywhere between six months to 24 hours in advance of your GMAT test date (or GMAT test dates if you are retaking the test; remember you need to allow for a 16-day window between test days!). Unlike the SAT, the GMAT is offered on an ongoing basis, but if you wait too late to register, spots may fill up and you may not get the dates/times you prefer.
Assuming…
Keep in mind that the GMAC recommends that you take the test at least 21 days prior to your application deadline, so that there is ample time for your scores to be processed and sent to your school.
The amount of time you’ll need to study will depend on your strengths and weaknesses, but according to a GMAC survey in 2014, students who scored 700+ prepared for an average of 121 hours. For an idea of what a 700 GMAT score looks like, it may be helpful to check out our GMAT Score Calculator. Factoring in your full-time job and real life, this gives you about 3 months of study time. We have super-detailed study schedules that I would highly recommend you take a look at to help you plan for your GMAT date.
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]]>The post How Much Does the GMAT Cost? A Guide to GMAT Exam Fees appeared first on Magoosh GMAT Blog.
]]>Before we get into any more detail, let’s look at how much the GMAT might cost you.
Scheduling fee | $250 |
Rescheduling fee (more than 7 days out) | $50 |
Rescheduling fee (within 7 days of the exam) | $250 |
Cancellation fee (more than 7 days out) | $170; $80 refund |
Cancellation fee (less than 7 days out) | $250; $0 refund |
Additional score report | $28 |
No matter what country you live in, the GMAT costs $250. Depending on your country, you may also need to pay some taxes on top of this price, though. Paying this $250 fee entitles you to one sitting of the GMAT.
Note, however, that if you choose to pay by phone you will be charged an additional $10. So, if you have easy access to the internet, register online!
The $250 base fee gives you one administration of the exam. GMAC, however, will add to your GMAT cost if you choose to do something like reschedule your exam. Rescheduling costs $50 if you reschedule your exam more than 7 days before your scheduled test. If you cancel, you can also get a refund of $80 (from your initial $250).
If you decide to reschedule or cancel at the very last minute, you’re out of luck. You’ll be charged another full $250. You also will not be entitled to a refund of your initial payment. That makes your GMAT cost essentially double—$500 for one sitting.
You cannot reschedule the GMAT within 24 hours of the test date. Your account history will instead register a “no-show.” Note that this will not be sent to schools in your score report, however.
Besides rescheduling fees, GMAC also charges for score reports. On test day, you are given five (5) free score reports. Any more will cost you $28 each.
Finally, if you cancel your scores and then later decide you want to un-cancel them, the good news is that you. On the other hand, you’ll be charged a $50 reinstatement fee.
You may retake the GMAT once every 16 calendar days, but no more than 5 times in a rolling 12-month period (but let’s be real, taking the GMAT 5 times would not be ideal anyway). However, there are no discounts for taking the test again. Also, don’t forget that the schools you send score reports to will see all of the scores you’ve received in the past five years. Unsure if you should retake the GMAT? You can find out more about GMAT retakes here.
While most of the GMAT cost comes from GMAC directly, you’ll also want to buy some prep materials! Keep in mind that the range of materials is enormous. A full, in-person course will run you thousands of dollars. Something more self-guided, like a GMAT study guide or online test prep could cost you less than $150. You can see some of our Magoosh plans here.
For high-quality free practice, you can also check out:
If you’re hoping to attend a top-tier business school like Harvard, GMAT exam fees will just be a drop in the bucket![/caption]Always remember to put your GMAT cost in perspective. The whole point of taking the GMAT is to get into business school, after all. And that’s very expensive!
Since B-school itself will be pricey, you might want to consider this when budgeting in your GMAT cost. If you’re concerned about expenses, you might consider planning to avoid a rescheduling fee or a retake. You should also be prepared well in advance to send in your score reports to maximize your five free ones. After all, every dollar counts!
This post was updated by Rachel Dale-Kapelke on 3/15/17, with help from Jen Nguyen.
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]]>The post Is Taking the GMAT Hard? appeared first on Magoosh GMAT Blog.
]]>The GMAT exam can help you be competitive in the MBA Admissions process, if you score well. Most North American MBAs and European English-language business schools require GMAT scores. The exam tests your abilities in graduate-level reading, writing, and math. But how hard is it to do well in these skills, and how hard is it to get a competitive score? This brings us to our main question….
Great question! Let’s establish some general parameters for this. Imagine how hard it would be to put a giant squid into a half-Nelson or to climb the Matterhorn wearing rollerblades. Well, the GMAT is considerably easier than those. If you imagine how hard it is to tie your shoes or how hard it is to eat ice cream, well, then the GMAT is harder than those. OK, OK, I am being a little facetious, but a large part of the answer to the question “How hard is the GMAT?” is the frustratingly ambiguous statement “It depends …” Let me explain.
Yes, the GMAT is challenging. It’s supposed to be challenging. It’s supposed to be hard. In mythology, the hero, at the outset of her journey, encounters the “Guardian of the Threshold,” the initial challenge she must face in order to undertake her adventure – what the Tusken Raiders were to Luke Skywalker, or what the first Nazguls were to Frodo & friends at the Prancing Pony. This is precisely what the GMAT is for anyone keen to undertake the adventure of earning an MBA and pursuing a career in the business world. In any context, part of the role of the Guardian of the Threshold is to separate the daring from the lily-livered, the bold & adventurous from those who would prefer to be sheepish followers. The GMAT is hard — in preparing for it and taking it, you will take risks, experience pressure, and feel yourself stretched. If you are the sort of person who doesn’t like risks, doesn’t like pressure, and doesn’t like to feel stretched, then it’s an excellent question why you are pursuing an MBA and a career in management in the first place!
Simply in terms of showing up and taking the test, the GMAT is hard. From the moment you walk into the testing center and they relieve you of any indication of your individuality, until you finally emerge, it will be, at minimum, a little over four hours–four long, difficult hours. Just to maintain concentration and focus during this, you need to be in good physical shape, well-rested, and well-nourished. I would recommend no alcohol for the week leading up to your GMAT. I would recommend not just one, but three or four consecutive nights of 8+ hours of sleep. I would recommend lots of water, healthy snacks, and some stretching during the breaks. During my own GMAT experience, I found myself running out of gas by the end of the test—this may have something to do with the fact that I am old enough to remember Nixon‘s Presidency! If you remember no presidents before Clinton, then your youthful vigor will certainly help you, but, even then, do not underestimate the GMAT’s difficulty — both mentally and physically.
In many ways, this is really the question people are asking when they ask, “how hard is the GMAT?” Sure, any slob can waltz into the GMAT exam with no preparation, do shoddy work, and get an abysmal score without much effort. The GMAT is relatively easy if you simply don’t care how you do. But what if you do care? Then how hard is the GMAT? To answer that question, it helps to know how others score. Only 23% of GMAT takers score over 650, and only 10% cross that magical 700 threshold. Something above 700 is generally what folks have in mind when they consider a “good” GMAT score. The average score on the GMAT (the numerical mean of everyone who takes the test) is 547. That score won’t turn any heads for you. How hard is it to get a GMAT score of a higher caliber?
This is the “it depends …” part. If you regularly score in the 99th percentile of standardized tests, then getting over a 700 on the GMAT shouldn’t be too difficult with moderate preparation. If you regularly flub standardized tests, then acing the GMAT will be that much more difficult. If you remember the percentile of any previous standardized test, the percentile of your SAT score for example, then imagine you score at the same percentile on the GMAT—you can use this official chart to gauge what an equivalent score on the GMAT might be and the Magoosh GMAT score calculator to figure out your GMAT score. You could also take the Magoosh GMAT Diagnostic Test, to give yourself a rough idea of your starting point. Whatever you score cold, on a dry run before any preparation—assume it will not be hard to score this much after preparation on the real test. The question is: how do you improve your score?
Pushing yourself beyond what you already have achieved, pushing yourself toward your own excellence—this is always hard. Improving on the GMAT takes focus, responsibility, dedication, determination, and commitment. Again, if these are qualities you don’t like to exercise, then the whole idea of management in the modern business world might not be for you. If you are ready to do the hard work of improving, then avail yourself of the best GMAT resources. How much you will improve depends very much on how disciplined and how thorough you are willing to be in your preparation. Many folks dream about a spectacular performance but do only moderate preparation. Remember the Great Law of Mediocrity: if you do only what most people do, you will get only what most people get. If you want to stand out, you have to take outstanding action. If you are willing to do outstanding work in your preparation for the GMAT, that’s very hard, but with good material, the results will really pay off.
An ordinary soldier fears his enemy, but a samurai in kensho would experience no separation between self & other, friend & enemy, life & death. While that mindset might seem somewhat extreme, consider that what’s hard about the GMAT—the intellectual challenges, the time pressure, etc.—is not too different from what’s hard about being a manager charged with important decisions in the business world: in other words, what’s “hard” about the GMAT is, in many respects, the same as what’s “hard” about the life & career you are choosing for yourself by pursuing an MBA. If you pursue this life, that level of difficulty and challenge will become, as it were, your “new normal”—get used to this “new normal” now, and what had appeared “hard” about the GMAT will be simply normal. When you routinely expect challenge as a matter of course, nothing is “hard.” That perspective is exactly what I would wish for you as you prepare for the GMAT!
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]]>The post Timing is Everything: The Ideal Time to Apply to Business School appeared first on Magoosh GMAT Blog.
]]>Now is the time to get started! There is still a lot you can do between now and next year’s Round 1 deadlines to improve your chances of admission to b-school. By preparing now you’ll be able to apply to more programs earlier and change your Round 2 strategy if necessary. Also, when you apply early there are more seats and financial aid available. However, your application has to be of the highest quality whenever you apply, so don’t rush!
Here’s what you should be doing now to assure that you have the best possible app at the earliest possible date.
If you haven’t already taken the GMAT, this is the time to prep and take the test, preferably in the spring. Choosing schools without knowing your score leads to unnecessary stress. Taking the test early will give you time to evaluate your score and see if it’s in the range needed for the schools you want to get into. If not, you will still have time to retake the test, reevaluate your target schools, or both. Taking the GMAT early will allow you to focus on the rest of the application process.
Now is the best time to visit schools – when classes are in session. You’ll get to see the professors and students in action and get a feel for the campus. This will help determine your fit with each school. A school may be perfect for you on paper, but if you don’t hit that “fit” factor, then it’s not the best match. During this research phase, you’ll also want to make sure that you’re competitive at the program and that it supports your goals.
Review your record to look for potential weaknesses. Now is the time to take appropriate classes – and ACE them! This will show the adcom that you are able to excel academically.
If you don’t know who to ask, now is the time to consider your various options and possibly raise the subject with people who can write you a strong recommendation. Be sure they see you in positive situations to ensure an amazing letter.
Whether or not you have a formal leadership role in school or work, you can always find ways to become an informal leader. The more the better – you can never have too much leadership in an MBA app. If there’s not enough space to write about it in your essays, be sure to include it in your resume.
What can you say about your goals – your planned industry, company function – that is interesting? Now is the time to read books, journals, and company reports. Talk to people. In less than 10 minutes, with good questions, you can get informative, instructive information that will make your essay stand out from the others.
Now is the time to get 95% of your resume done. You can adjust it for new developments along the way. It’s good to have this ready if you have the opportunity to visit a school or meet with an adcom member earlier than you’d planned.
Need more help determining where you are, where you’re going and how to stay on track? Download our free guide, MBA Action Plan: 6 Steps for the 6 Months Before You Apply.
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]]>The post Matemática GMAT: Decimais Exatos e Periódicos appeared first on Magoosh GMAT Blog.
]]>Inteiros são números positivos e negativos reais, incluindo o zero. Aqui estão alguns inteiros:
{ … -3, -2, -1, 0, 1, 2, 3, …}
Quando fazemos uma razão entre dois inteiros, nós temos um número racional. Um número racional é qualquer número no formato a/b, onde a e b são inteiros e b ≠ 0. Números racionais são o conjunto de todas frações feitas com fatores inteiros. Perceba que todos os inteiros estão incluídos no conjunto de números racionais, pois, por exemplo, 3/1 = 3.
Quando fazemos um decimal de uma fração, uma das duas coisas acontece. Ou termina (decimal exato) ou repete (continua para sempre em um padrão, chamados de decimais periódicos). Números racionais exatos incluem:
1/2 = 0.5
1/8 = 0.125
3/20 = 0.15
9/160 = 0.05625
Números racionais periódicos incluem:
1/3 = 0.333333333333333333333333333333333333…
1/7 = 0.142857142857142857142857142857142857…
1/11 = 0.090909090909090909090909090909090909…
1/15 = 0.066666666666666666666666666666666666…
O GMAT não dará uma fração complicada como 9/160 e esperará que você descubra qual sua expressão decimal. MAS, o GMAT pode fornecer uma fração como 9/160 e perguntar se é exato ou não. Mas como saber?
Bem, primeiro de tudo, qualquer terminação decimal (como 0.0376) é, essencialmente, uma fração com uma potência de dez no denominador. Por exemplo, 0.0376 = 376/10000 = 47/1250. Note que simplificamos esta fração, dividindo o numerador por 8. O dez é múltiplo de 2 e 5, então qualquer potência de dez irá ser potência de 2 e de 5, e algumas podem ser canceladas por fatores no numerador, mas nenhum outro fator será introduzido no denominador. Então, se a fatoração primária do denominador de uma fração possui apenas múltiplos de 2 e múltiplos de 5, então pode ser escrita como algo com potência de 10, o que significa que sua expressão decimal será exata.
Se a fatoração primária do denominador de uma fração possui apenas múltiplos de 2 e de 5, as expressões decimais são exatas. Se há algum fator primário no denominador que não seja 2 ou 5, então a expressão decimal é periódica. Deste modo,
1/24 é periódica (há um múltiplo de 3)
1/25 é exata (apenas múltiplos de 5)
1/28 é periódica (há um múltiplo de 7)
1/32 é exata (apenas múltiplos de 2)
1/40 é exata (apenas múltiplos de 2 e 5)
Note que, contanto que a fração esteja nos seus menores termos, o numerador não importa. Já que 1/40 é exata, então 7/40, 13/40 ou qualquer outro inteiro sobre 40 também é.
Já que 1/28 é periódica, então 5/28 e 15/28 e 25/28 também são. Note, entretanto, que 7/28 não é periódica por causa do cancelamento: 7/28 = 1/4 = 0.25.
Há alguns decimais que são úteis como atalhos, tanto para conversões de fração-para-decimal quanto para conversões de fração-para-porcentagem. Esses são
1/2 = 0.5
1/3 = 0.33333333333333333333333333…
2/3 = 0.66666666666666666666666666…
1/4 = 0.25
3/4 = 0.75
1/5 = 0.2 (e vezes 2, 3 e 4 para decimais fáceis)
1/6 = 0.166666666666666666666666666….
5/6 = 0.833333333333333333333333333…
1/8 = 0.125
1/9 = 0.111111111111111111111111111… (e vezes outros dígitos para outros decimais fáceis)
1/11 = 0.09090909090909090909090909… (e vezes outros dígitos para outros decimais fáceis)
Há outra categoria de decimais periódicos (que continuam para sempre) e eles não têm padrões de repetição. Esses números, os decimais periódicos não repetitivos, são chamados de números irracionais. É impossível escrever qualquer forma deles como razão de dois inteiros. O Sr. Pitágoras (c. 570 – c. 495 aC) foi o primeiro a provar um número irracional: ele provou que a raiz quadrada de 2 — — é irracional. Nós todos sabemos: toda raíz quadrada de um inteiro cuja solução não é outro número inteiro é irracional Outro número racional famoso é o , ou pi, a razão da circunferência de um círculo pelo seu diâmetro. Por exemplo,
= 3.1415926535897932384626433832795028841971693993751058209749445923078164
0628620899862803482534211706798214808651328230664709384460955058223172535940812848111745
0284102701938521105559644622948954930381964428810975665933446128475648233786783165271201
909145648566923460348610454326648213393072602491412737…
Esses são os primeiros trezentos dígitos do pi, e eles nunca se repetem: continuam para sempre sem padrões repetitivos. Há uma infinidade de outros números irracionais: na verdade, a infinidade de números irracionais é infinitamente maior que a infinidade de números racionais, mas isto leva a uma matemática (http://en.wikipedia.org/wiki/Aleph_number) que é muito mais avançada que a do GMAT.
1)
(A) 2/27
(B) 3/2
(C) 3/4
(D) 3/8
(E) 9/16
1) A partir dos nossos atalhos, nós sabemos que 0.166666666666… = 1/6, e 0.444444444444… = 4/9. Portanto (1/6)*(9/4) = 3/8. Resposta = D
Esta postagem apareceu originalmente em inglês no Magoosh blog e foi traduzida por Jonas Lomonaco.
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]]>The post Matemática GMAT: a Questão de “Pelo Menos” de Probabilidade appeared first on Magoosh GMAT Blog.
]]>2) Suponha que você jogue uma moeda não-viciada seis vezes. Qual é a probabilidade de, em seis jogadas, sair pelo menos uma cara?
3) Em um certo jogo, você escolhe uma carta de um baralho padrão de 52 cartas. Se a carta é de copas, você ganha. Se não for de copas, a pessoa a repõe no bolo de cartas, embaralha e puxa novamente. Esse processo é repetido até sair uma de copas, e a questão é calcular: quantas vezes a pessoa precisa puxar antes que consiga uma carta de copas e ganhe? Qual é a probabilidade de puxar pelo menos duas cartas que não sejam de copas nas primeiras duas jogadas, e só pegar a primeira carta de copas pelo menos na terceira jogada?
Há uma regra muito simples e importante relacionando P(A) e P(A’), conectando a probabilidade de qualquer evento acontecer com a probabilidade daquele mesmo evento não acontecer. Para qualquer evento bem definido, é 100% verdade que este pode ou não acontecer. O GMAT não irá fazer perguntas de probabilidade sobre eventos bizarros dos quais, por exemplo, você não saberia dizer se aconteceu ou não, ou eventos complexos que podem, de alguma forma, tanto acontecer quanto não acontecer. Para qualquer evento A em uma questão de probabilidade no GMAT, os dois cenários “A acontece” e “A não acontece” esgotam as possibilidades que podem acontecer. Com certeza, nós podemos falar: um dos dois irá ocorrer. Em outras palavras
P(A OU A’) = 1
Ter a probabilidade de 1 significa certeza garantida. Obviamente, por uma variedade de razões lógicas profundas, os eventos “A” e “não A (ou A’)” são desconexos e não têm sobreposição. A regra do OU, discutida no último post, implica:
P(A) + P(A’) = 1
Subtraia o primeiro termo para isolar P(A’).
P(A’) = 1 – P(A)
Isto é conhecido em probabilidade como regra de um evento complementar, porque a região probabilística na qual um evento não acontece complementa a região na qual ele ocorre. Esta é a ideia crucial no geral, para todas as questões de probabilidade do GMAT, que será muito importante para resolver questões de “pelo menos” em particular.
Suponha que um evento A é uma afirmação envolvendo palavras de “pelo menos” – o que afirmaria os constituintes de “não A”? Em outras palavras, como negamos uma afirmação “pelo menos”? Vamos ser concretos. Suponha que há alguns eventos que envolvam apenas dois resultados: sucesso e fracasso. O evento pode ser, por exemplo, fazer um arremesso de basquete ou jogar uma moeda e tirar cara. Agora, suponha que temos uma “disputa” envolvendo dez desses eventos consecutivos, e nós estamos contando o número de sucessos dessas dez tentativas. O evento A será definido como: A = “há pelo menos 4 sucessos nessas dez tentativas.” Que resultados iriam constituir “não A”? Bem, vamos pensar sobre isso. Em dez tentativas, uma pode ser de nenhum sucesso, exatamente um sucesso, exatamente dois sucessos, até dez sucessos. Há onze resultados possíveis, os números de 0 – 10, para o número de sucessos que pode ocorrer em 10 tentativas. Considere o seguinte diagrama de números de sucessos possíveis em dez tentativas.
Os números em roxo são membros de A, “pelo menos 4 sucessos” em dez tentativas. Portanto, os números verdes são os espaços complementares, a região de “não A”. Em palavras, como iríamos descrever as condições que lhe colocariam na região verde? Nós podemos dizer: “não A” = “três sucessos ou menos” em dez tentativas. A negação, o oposto, de “pelo menos quatro” é “três ou menos”.
Abstraindo disso, a negação ou oposto de “pelo menos n” é a condição “(n – 1) ou menos”. Um caso particularmente interessante é n = 1: a negação ou o oposto de “pelo menos um” é “nenhum.” Esta última afirmação é uma ideia extremamente importante, indiscutivelmente a chave para resolver a maior parte das questões de “pelo menos” que você encontrará no GMAT.
A maior ideia das questões de “pelo menos” no GMAT é: sempre é mais fácil descobrir a probabilidade do complementar. Por exemplo, no cenário acima das dez tentativas de alguma coisa, calcular “pelo menos 4” diretamente iria envolver sete cálculos diferentes (para os casos de 4 a 10), enquanto calcular “três ou menos” iria envolver apenas quatro cálculos separados (para os casos de 0 a 3). No extremo – e extremamente comum – caso de “ao menos um”, a abordagem direta iria envolver o cálculo de um quase caso, mas o cálculo do complementar envolve simplesmente calcular a probabilidade para o caso de “nenhum” e então subtraí-lo de um.
P(A’) = 1 – P(A)
P(pelo menos um sucesso) = 1 – P(nenhum sucesso)
Este é um dos atalhos mais poderosos e que economizam o seu tempo em todo o GMAT.
Considere a simples questão a seguir.
4) Dois dados são arremessados. Qual é a probabilidade de se tirar um 6 em pelo menos um deles?
Acontece que calcular isso diretamente iria envolver um cálculo relativamente longo – a probabilidade de tirar exatamente um 6, em qualquer dado, e a rara probabilidade de ambos darem 6. Este cálculo poderia facilmente levar muitos minutos para ser concluído.
Em vez disso, nós iremos usar o atalho definido acima:
P(A’) = 1 – P(A)
P(pelo menos um 6) = 1 – P(nenhum 6)
Qual a probabilidade de ambos os dados darem nenhum 6? Bem, primeiro, vamos considerar apenas um dado. A probabilidade de arremessar um 6 é de 1/6, então a probabilidade de arremessar algo diferente de 6 (não 6) é 5/6.
P(dois dados, nenhum 6) = P(“não 6” no dado nº 1 E “não 6” no dado nº 2)
Como vimos no último post, a palavra E significa multiplicação. (Claramente, o resultado de cada dado é independente do outro). Então:
P(dois dados, nenhum 6) =(5/6)*(5/6) = 25/36
P(pelo menos um 6) = 1 – P(nenhum 6) = 1 – 25/36 = 11/36
O que poderia ser um cálculo bem longo tornou-seincrivelmente simples com esse atalho. Isto pode ser um enorme quebra-galho para economizar o seu tempo no GMAT!
Após ler este post, tente resolver novamente as três questões práticas acima antes de ler suas respostas abaixo. E mais, aqui está uma questão grátis, com vídeo explicativo, sobre esse mesmo tema:
5) http://gmat.magoosh.com/questions/839
O próximo artigo da série irá explorar as questões de probabilidade que envolvem técnicas de contagem.
1) P(pelo menos uma vogal) = 1 – P(nenhuma vogal)
A probabilidade de pegar uma letra que não seja uma vogal no primeiro conjunto é de 3/5. A probabilidade de não pegar nenhuma vogal no segundo conjunto é 5/6. Para não pegar vogal alguma, não podemos pegar nenhuma no primeiro conjunto E nenhuma no segundo conjunto. De acordo com a regra E, nós multiplicamos as probabilidades.
P(nenhuma vogal) = (3/5)*(5/6) = 1/2
P(pelo menos uma vogal) = 1 – P(nenhuma vogal) = 1 – 1/2 = 1/2
Resposta = C
2) P(pelo menos um H) = 1 – P(nenhum H)
Em uma jogada, P(“não H”) = P(T) = 1/2. Nós precisaríamos que isso acontecesse seis vezes – isto é, seis eventos independentes unidos pela palavra E, o qual significa que são multiplicados entre si.
Resposta = E
3) Um baralho completo de 52 cartas contém 13 cartas de cada um dos quatro naipes. A probabilidade de puxar uma carta de copas em um baralho completo é 1/4. Portanto, a probabilidade de tirar uma carta que “não seja copas” é 3/4.
P(pelo menos três jogadas para ganhar) = 1 – P(ganhar em duas ou menos jogadas)
Além disso,
P(ganhar em duas ou menos jogadas) = P(ganhar em uma jogada OU ganhar em duas jogadas)
= P(ganhar em uma jogada) + P(ganhar em duas jogadas)
Ganhar em uma jogada significa: eu seleciono uma carta do baralho e acontece de ser de copas. Acima, nós já falamos: a probabilidade disto acontecer é de 1/4.
P(ganhar em uma jogada) = 1/4
Ganhar em duas jogadas significa: minha primeira jogada era uma carta “não copas”, P = 3/4, E a segunda jogada é de copas, P = 1/4. Porque nós repomos e embaralhamos, as jogadas são independentes, então o E significa multiplicação.
P(ganhar em duas jogadas) =(3/4)*(1/4) = 3/16
P(ganhar em duas ou menos jogadas) =P(ganhar em uma jogada) + P(ganhar em duas jogadas)
= 1/4 + 3/16 = 7/16
P(pelo menos três jogadas para ganhar) = 1 – P(ganhar em duas ou menos jogadas)
= 1 – 7/16 = 9/16
Resposta = B
Esta postagem apareceu originalmente em inglês no Magoosh blog e foi traduzida por Jonas Lomonaco.
The post Matemática GMAT: a Questão de “Pelo Menos” de Probabilidade appeared first on Magoosh GMAT Blog.
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