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### GMAT Quantitative Problems

## Question 1

## Division & Factoring

When integer y is divided by 2, is the remainder 1?

(1) \((-1)^{(y + 2)} = -1\)

(2) \(y\) is prime.

A) Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

B) Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are not sufficient.

The correct answer is: A)

(1) That \((–1)^{(y + 2)} = -1\), dictates that y + 2 must be odd and therefore y is odd since only an odd exponent can produce a value < 0. Furthermore, if an odd number is divided by 2, the remainder must be 1; SUFFICIENT.

(2) That y is prime does not definitively identify y as definitively even or odd, so the remainder could be either 0 or 1; NOT SUFFICIENT.

The correct answer is A; statement 1 alone is sufficient.

## Question 2

## Fractions and Decimals

Vivek bet \(\frac{1}{4}\) of his available gambling money on baseball, \(\frac{1}{6}\) on football, and \(\frac{1}{3}\) on basketball. If he didn’t bet $45 of his available gambling money, lost on the baseball and football bets, but won double on the basketball bet, which of the following represents Vivek’s current balance compared to his amount of gambling money before the three bets?

A) \(\frac{1}{4}\) more.

B) \(\frac{1}{6}\) more.

C) \(\frac{1}{12}\) more.

D) \(\frac{1}{12}\) less.

E) \(\frac{1}{6}\) less.

The correct answer is: D)

Set Vivek’s available gambling money as \(x\).

According to the problem, he wagered \(\frac{1}{4}x\), \(\frac{1}{6}x\) and \(\frac{1}{3}x\), or, using a common denominator of 12, \(\frac{9}{12}x\). This means that \(\frac{1}{4}\) of his money is $45 and \(x = $180\).

He lost \(\frac{1}{4}x + \frac{1}{6}x = $75\), but still has \( 2 × \frac{1}{3}x = $120\) from his basketball winnings + $45 he kept, so his total after the three bets is $165, which is $15 less than he had originally.

\(\frac {15}{180}=\frac {1}{12}\) that Vivek lost.

## Question 3

## Percentages (Data Sufficiency)

In a vaccination line, 90 people have glasses and 80 people do not have glasses. 40% of the people with glasses have a pre-existing condition and 60% of the people without glasses have a pre-existing condition. If 75% of the people with glasses and pre-existing conditions receive a vaccination, and 50% of the people without glasses and with a pre-existing condition receive a vaccination, what percentage of all the people in line receive a vaccination?

A) 45

B) 40

C) 33

D) 30

E) 25

The correct answer is: D)

40% of the (90) people with glasses have a pre-existing condition. \(90 \times .4 = 36\) people with glasses and a pre-existing condition and 75% of this number receive a vaccination, so \(36 \times .75 = 27\) people with glasses and a pre-existing condition who receive a vaccination.

60% of the (80) people without glasses have pre-existing conditions. \(80 \times .6 = 48\) people without glasses and with pre-existing conditions. 50% of these people receive a vaccination, so \(48 \times .5 = 24\) people without glasses and with a pre-existing condition who receive a vaccination.

The total number of people with a pre-existing condition who receive a vaccination is 27 + 24 = 51. There are 90 + 80 people in line = 170 total, so the percentage of all people in line who receive a vaccination is \(\left(\frac {51}{170} \right) \times 100 = 30\%\).

## Question 4

## Exponents & Radicals (Data Sufficiency)

The amount of bacteria in a culture after some \(t\) hours is given by the function \(f(t)=pe^{kt}\) where \(p\) is a constant. What is the number of bacteria after 8 hours?

(1) There were approximately 275 bacteria after 2 hours

(2) \(k=0.16\)

A) Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

B) Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are not sufficient.

The correct answer is: C)

(1) From the function, \(f(t)=pe^{kt}\), when \(t=2, f(t)=275\). The equation becomes \(pe^{2k}=275\). This is one equation in two unknowns, hence, we cannot solve it. More information is required; NOT SUFFICIENT.

(2) When \(k=0.16\), we have \(f(t)=pe^{0.16t}\). In this case, \(p\) is unknown, hence more information is required; NOT SUFFICIENT.

Considering the two cases, we have \(pe^{2k}=275\) and \(f(t)=pe^{0.16t}\) which reduced to \(pe^{2(0.16)}=275\). We solve the equation

$$\begin{align*} pe^{2(0.16)} &=275\\

pe^{3.2} &=275\\

\ln \left(pe^{3.2}\right) &=\ln 275\\

\ln p + 3.2 &=\ln 275\\

\ln p &=\ln 275 – 3.2 = 2.41677\\

p& =e^{2.41677}=11.21\end{align*}$$.

We now solve the equation

When \(t=8\), we have

$$f(8)=11.21e^{0.16(8)} =11.21e^{1.28}=40.32$$

The correct answer is C; BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

## Question 5

## Geometry (Data Sufficiency)

A cylindrical layer cake has two vanilla layers surrounding a marzipan layer, represented by the shaded region, as shown below. Is the cake split into three layers of equal volume?

(1) The height of the marzipan layer is equal to the radius of the whole cake.

(2) The volume of the entire cake is 81π cubic inches.

A) Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

B) Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are not sufficient.

The correct answer is: E)

Note that the volume for a cylinder can be found from the formula \(πr^2 × height\) where r = radius and the height is perpendicular to the radius.

(1) That the height for the marzipan layer is equal to the radius of the cake means that the volume of the marzipan layer can be found from the simplified formula \(πr^3\), but the total cake volume remains dependent on the actual height of the cake; NOT sufficient.

(2) Represent the information algebraically as \(πr^2 × height = 81\), but no information is provided about the height in relation to the radius; NOT sufficient.

(Together) The information together still does not relate the height to the radius.

For instance, the radius could be 3 and the ratio of the marzipan layer to the others could be equal as each would have a volume of 27π.

Alternatively, the radius could = 1 and the marzipan layer would be significantly smaller than the other two layers.

The correct answer is E; both statements together are still not sufficient.

## Question 6

## Inequalities

At Rounders Grocer, orange slices cost $2 a pound, pineapple chunks cost $3 a pound, and cut watermelon cost $5 a pound. If Sally buys enough of these three fruits from Rounders to make five pounds of fruit salad, and at least one pound of each, which fruit did she buy the most of by weight?

(1) Sally spent less than $5 on orange slices.

(2) Sally spent more than $18 on her fruit salad.

A) Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

B) Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are not sufficient.

The correct answer is: B)

First note that Sally must buy at least one pound of each of the fruits, which would result in a sum fixed cost of $10.

(1) If Sally spent less than $5 on orange slices, then she spent must have spent more on one of the other two fruits, but it is unclear which; NOT sufficient

(2) If Sally spent more than $18 then she must spend more than $8 on the two-variable pounds of fruit, which could only be accomplished by buying two more pounds of cut watermelon; Sufficient

The correct answer is B; statement 2 alone is sufficient.

## Question 7

## Verbal – Sentence Correction

In the wake of his passing, one of the most heartwarming things to come out is the volume of people who have shared their own letters from Stephen Sondheim, revealing his astounding generosity of spirit that has certainly resulted in countless ripples throughout the musical theater field.

A) In the wake of his passing, one of the most heartwarming things to come out is the volume of people who have shared their own letters from Stephen Sondheim, revealing his astounding generosity of spirit that has certainly resulted in countless ripples throughout the musical theater field.

B) One of the most heartwarming things to come out in the wake of his passing, revealing his astounding generosity of spirit that has certainly resulted in countless ripples throughout the musical theater field was the volume of people sharing their own letters from Stephen Sondheim.

C) Revealing his astounding generosity of spirit that has certainly resulted in countless ripples throughout the musical theater field, in the wake of his passing, one of the most heartwarming things to come out is the volume of people who have shared their own letters from Stephen Sondheim.

D) Stephen Sondheim, in the wake of his passing, revealed his astounding generosity of spirit that has certainly resulted in countless ripples throughout the musical theater field as one of the most heartwarming things to come out is the volume of people who have shared their own letters from him.

E) In the wake of his passing, Stephen Sondheim, revealing his astounding generosity of spirit that has certainly resulted in countless ripples throughout the musical theater field, shared one of the most heartwarming things which is the volume of people who have their own letters from him.

The correct answer is: A)

A. Yes. Each instance of the pronoun “his” is properly used as a possessive for “Stephen Sondheim”, and each of the subordinate clauses properly refers to Sondheim as well.

B. No. The verb tenses “was” and “sharing” used in this choice are non-parallel.

C. No. This choice incorrectly changes the intended meaning of the original sentence by indicating that the “countless ripples throughout the musical theater field” occurred “in the wake of his passing”.

D. No. This choice illogically indicates that “Stephen Sondheim” “revealed his astounding generosity of spirit” “in the wake of his passing”, which could not happen because he would have already been dead.

E. No. This choice illogically indicates that “Stephen Sondheim” “shared one of the most heartwarming things”, which could not happen because he would have already been dead.

## Question 8

## Verbal – Sentence Correction

Satire was __used by writers like Jonathan Swift and artists like Honoré Daumier, both of whom__ employed scathing commentary in their work to call attention to the ills of society in the hope of correcting them.

A) used by writers like Jonathan Swift and artists like Honoré Daumier, both of whom

B) used by writers such as Jonathan Swift and artists such as Honoré Daumier, both of which

C) used by writers and artists such as Jonathan Swift and Honoré Daumier, both of them

D) what writers like Jonathan Swift and artists like Honoré Daumier used, and both of them

E) what writers and artists like Jonathan Swift and Honoré Daumier used, and both of them

The correct answer is: A)

A. Yes. A proper comparison is analogizing “writers like Jonathan Swift and artists like Honoré Daumier” and the modifying phrase “both of whom employed” properly describes Swift and Daumier.

B. No. The pronoun “which” incorrectly refers to the two people, “Jonathan Swift” and “Honoré Daumier”.

C. No. The phrase “writers and artists such as Jonathan Swift and Honoré Daumier” changes the intended meaning of the original sentence.

D. No. The pronoun “what” introduces an ambiguity in the sentence and changes the intended meaning of the original sentence.

E. No. The pronoun “what” introduces an ambiguity in the sentence and changes the intended meaning of the original sentence.

## Question 9

## Verbal – Reading Comprehension

If King Tut were around today, could he send a text in Egyptian hieroglyphics? Yes, with the right font and keyboard. That’s because the writing system of the pharaohs has already been included in the Unicode Standard, meaning that a character like the Eye of Horus has a code point, 13080, that will render the same way on a tablet in Cairo and a smartphone in Beijing. Because Mayan hieroglyphs have yet to be encoded, the ancient Mayan emperor K’inich Janaab’ Pakal would have to stick to emoji—but that’s about to change.

Unicode is the international encoding standard that makes it possible for users to read, write, and search in a wide range of written languages on all manner of devices without technical miscommunication. Made up of a mix of academics, stakeholders, and interested volunteers, the Unicode Consortium has encoded 139 of the writing systems, technically known as scripts, ever to have existed. Given that alphabets like Cyrillic, Arabic, and Devanagari serve more than 60 languages each and that 500 languages use the Latin alphabet, Unicode makes electronic communication possible in almost a thousand languages. But there are more than a hundred writing systems to go.

In June 2017, the Unicode Consortium rolled out its tenth version in 26 years, which included four scripts as well as the Bitcoin sign and 56 new emoji. The scripts introduced this year include Nüshu, a writing system that was developed by women in the Hunan Province of nineteenth-century China as a workaround when they were denied formal education. Also newly available is Zanabazar Square, created by a Mongolian monk in the seventeenth century to write spiritual texts in Mongolian, Tibetan, and Sanskrit. Crucial as these steps toward cultural empowerment may be, it is the textable faces, socks, mermen, and the like that have brought this global standard into the limelight.

As Michael Erard, writer-in-residence at the Max Planck Institute for Psycholinguistics, recently pointed out in the *New York Times*, the incorporation of emoji into Unicode is a double-edged sword for those interested in preserving ancient and minority languages. While, on the one hand, emoji eat up the time of the already strapped volunteers who could be working on other scripts, customer demand for emoji also incentivizes voting members of the Unicode Consortium like Microsoft, Apple, and Google’s Alphabet to update their systems with each new version that Unicode releases.

**Question 1 –**

What function does the phrase: “…a character like The Eye of Horus has a code point, 13080…” serve in the passage?

A) It provides an example of Unicode’s standardization across languages, countries, and platforms.

B) It indicates that an analogous Mayan hieroglyph would have the same code point.

C) It describes the order in which new languages are incorporated into the Unicode standard.

D) It describes the manner in which a particular character might be encoded into Unicode.

E) It provides an indication of the size of the Unicode character set.

The correct answer is: A)

A. Yes. This answer choice is stated explicitly in the first paragraph.

B. No. This answer choice is unsupported by the passage since no evidence is provided that analogous characters would have the same code point.

C. No. This answer choice is unsupported by the passage since no information is provided concerning the order in which new languages are incorporated.

D. No. This answer choice is unsupported by the passage since no information is provided concerning the manner in which languages are encoded.

E. No. This answer choice is unsupported by the passage since no information is provided concerning the size of the character set.

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