GMAT Quant Topic Practice: 23 Questions with Hidden Solutions

Answer: D. 258

1. The percentage increase applies to the original price: $0.15(240)=36$. The increased price is $240+36=276$.
2. The coupon is a fixed amount, not a percentage. Subtract it after the increase: $276-18=258$.
3. Equivalently, model the entire process as $240(1.15)-18=258$. Subtracting 18 before multiplying would apply the increase to the wrong base.

GMAT takeaway: Write the order of operations from the story before substituting numbers. A fixed discount and a percentage change are different operations.

Review the essential formulas: choosing the right base reference.

Answer: B. $n^2-1$

1. Factor the difference of squares: $n^2-1=(n-1)(n+1)$. Since $n$ is odd, the two factors are consecutive even integers.
2. One of two consecutive even integers is divisible by 4, and the other is divisible by 2. Their product is therefore divisible by $4\cdot2=8$.
3. A single counterexample can eliminate a claim of necessity. When $n=3$, the other choices equal 4, 6, 10, and 12, none of which is divisible by 8.

GMAT takeaway: For a “must be true” question, establish why the correct expression works for every permitted value. Several favorable examples are not a proof.

Review the integers, parity, and divisibility reference.

Answer: A. 10

1. Prime-factorize the number: $360=2^3\cdot3^2\cdot5$. Every prime exponent in a perfect square must be even.
2. The exponent of 3 is already even. Multiply by one factor of 2 and one factor of 5 to make the other exponents even.
3. Thus $k=2\cdot5=10$, and $360(10)=3600=60^2$. Including any additional prime factors would not produce a smaller positive multiplier.

GMAT takeaway: A perfect-square multiplier repairs odd prime exponents. It is not the same as taking the square root of the original number.

Review the prime factors and perfect squares reference.

Answer: C. 15

1. An integer with the required remainder has the form $7q+2$, where $q$ is an integer.
2. Impose the boundaries: $100\le7q+2\le200$. Subtract 2 to obtain $98\le7q\le198$, so the possible integer values of $q$ run from 14 through 28.
3. Count inclusively: $28-14+1=15$. The first qualifying integer is 100 and the last is 198.

GMAT takeaway: Translate a remainder into an equation, then count the permitted quotients. Do not forget the extra 1 when both endpoints are included.

Review the remainders and inclusive counting reference.

Answer: A. $3/8$

1. Convert the decimal: $0.375=3/8$. After the morning shipment, $1-3/8=5/8$ of the original inventory remains.
2. In the afternoon, the warehouse keeps $1-2/5=3/5$ of that remaining inventory.
3. Multiply the remaining fractions: $(5/8)(3/5)=3/8$. The afternoon fraction applies to the remainder, not to the original total.

GMAT takeaway: When a fraction is taken “of what remains,” multiply successive remaining fractions instead of subtracting both fractions from 1.

Review the fractions and decimals reference.

Answer: C. 200

1. Let the original price be $P$. After the discount, the price is $0.80P$. Applying the tax produces $1.10(0.80P)=0.88P$.
2. Set the final amount equal to 176: $0.88P=176$, so $P=200$.
3. Check forward: a 20% discount reduces 200 to 160, and 10% tax adds 16, giving 176. Combining the percentages by subtraction would incorrectly use a 10% net decrease.

GMAT takeaway: Reverse a sequence of percentage changes by dividing by the product of its multipliers. Different percentage bases do not cancel by simple subtraction.

Review the reverse percents and successive changes reference.

Answer: B. 18

1. The two given ratios must use the same value for their shared part, B. Double $A:B=3:5$ to obtain $A:B=6:10$.
2. Now combine the ratios: $A:B:C=6:10:7$. There are $6+10+7=23$ ratio parts in the total.
3. Each part represents $69/23=3$ employees. Department A therefore has $6(3)=18$ employees.

GMAT takeaway: Match the shared quantity before combining ratios. Simply joining 3:5 and 10:7 would treat B as two different amounts.

Review the linked ratios and proportions reference.

Answer: E. 288

1. The radicand is positive, and each prime exponent is even. Taking the principal square root halves those exponents.
2. Thus $\sqrt{2^{10}\cdot3^4}=2^5\cdot3^2$.
3. Evaluate the smaller powers: $2^5=32$ and $3^2=9$, so the value is $32(9)=288$. A square-root symbol denotes the nonnegative root, not both positive and negative values.

GMAT takeaway: Simplify exponents before computing a large radicand. Keep the principal square root distinct from solving an equation such as x squared equals a positive number.

Review the exponents and roots reference.

Answer: C. 7

1. Factor each numerator as a difference of squares: $x^2-16=(x-4)(x+4)$ and $x^2-9=(x-3)(x+3)$.
2. The stated restrictions permit cancellation of the denominator factors. The expression becomes $(x+4)-(x-3)$.
3. Distribute the subtraction: $x+4-x+3=7$. The restrictions remain part of the original problem even after the expression simplifies.

GMAT takeaway: Cancel factors, not separate terms. When subtracting a simplified expression, distribute the negative sign across every term.

Review the algebraic identities and rational expressions reference.

Answer: D. 97

1. Use the identity $(x+y)^2=x^2+2xy+y^2$. Rearrange it to isolate the requested expression: $x^2+y^2=(x+y)^2-2xy$.
2. Substitute the given sum and product: $13^2-2(36)=169-72=97$.
3. You do not need to solve separately for x and y. As a check, 4 and 9 have sum 13 and product 36; their squares sum to $16+81=97$.

GMAT takeaway: Solve for the expression the question asks for. Finding every underlying variable can add work without adding useful information.

Review the equations and quadratic identities reference.

Answer: B. 6

1. An absolute value less than 7 means the expression lies strictly between -7 and 7: $-7<2x-5<7$.
2. Add 5 throughout to obtain $-2<2x<12$, then divide by the positive number 2: $-1<x<6$.
3. The permitted integers are 0, 1, 2, 3, 4, and 5, for a total of 6. The endpoints -1 and 6 are excluded because they make the absolute value equal to 7.

GMAT takeaway: Translate absolute value into an interval and preserve strict endpoints. Count only integers after solving the real-number inequality.

Review the inequalities and absolute value reference.

Answer: B. 15

1. First evaluate the newly defined operation. Since $f(4)=2(4)-3=5$, we have $4\diamond2=5+2(2)=9$.
2. The question then asks for the function applied to that result: $f(9)=2(9)-3=15$.
3. The intermediate result 9 is not the answer. There are two layers: evaluate the diamond operation, then apply f to its output.

GMAT takeaway: Follow the definition of a new symbol exactly. For nested operations, work from the innermost input outward.

Review the functions and newly defined operations reference.

Answer: E. 2.5

1. Convert the volume: $1.8\text{ m}^3=1800\text{ liters}$. Convert the time: $12\text{ minutes}=720\text{ seconds}$.
2. Divide volume by time: $1800/720=180/72=5/2=2.5$ liters per second.
3. As a check, 2.5 liters per second is 150 liters per minute; over 12 minutes, that is 1,800 liters.

GMAT takeaway: Convert both numerator and denominator to the requested units. A correct division using minutes instead of seconds answers a different question.

Review the word problems and unit conversions reference.

Answer: C. 48

1. Average speed is total distance divided by total time. The total distance is $90+90=180$ kilometers.
2. The times are $90/40=9/4$ hours and $90/60=3/2$ hours. Total time is $9/4+6/4=15/4$ hours.
3. Therefore the average speed is $180\div(15/4)=180(4/15)=48$ kilometers per hour. The arithmetic mean of 40 and 60 is not appropriate because the driver spends unequal times at the two speeds.

GMAT takeaway: Use the definition of average speed. Equal distances produce unequal time weights unless the speeds are equal.

Review the distance, speed, and average rates reference.

Answer: D. 8

1. The combined rate is $1/10+1/15=3/30+2/30=1/6$ job per hour.
2. In the first 3 hours, the machines complete $3(1/6)=1/2$ of the job. The remaining half is completed by A alone at $1/10$ job per hour.
3. A needs $(1/2)\div(1/10)=5$ additional hours. Total elapsed time is $3+5=8$ hours, not just the final 5 hours.

GMAT takeaway: Split a work problem whenever the contributing team changes. Add completed work across intervals, then add elapsed times.

Review the work rates with a changing team reference.

Answer: B. $26\frac23\%$

1. Initially, the tank contains $0.40(30)=12$ liters of concentrate.
2. The removed 10 liters have the same 40% concentration, so they contain $0.40(10)=4$ liters of concentrate. That leaves $12-4=8$ liters of concentrate.
3. Adding pure water restores the total volume to 30 liters without adding concentrate. The final concentration is $8/30=4/15$, or $26\frac23\%$.

GMAT takeaway: Removing a well-mixed sample removes every ingredient proportionally. Track the ingredient amount separately from the restored total volume.

Review the mixtures and replacement reference.

Answer: A. 110.40

1. For the first account, simple interest is $1500(0.08)(3)=360$, giving a final balance of $1500+360=1860$.
2. For the second account, the first year's interest is 120, so the balance becomes 1620. The second year's interest is $1620(0.08)=129.60$, giving a final balance of 1749.60.
3. Subtract the two balances: $1860-1749.60=110.40$. Compound interest applies to the growing balance, but this account also has a shorter investment period.

GMAT takeaway: Keep the interest model and the time period separate. A higher-growth model does not automatically yield the larger final amount when durations differ.

Review the simple and compound interest reference.

Answer: C. 76

1. Reconstruct each group's total score. The first total is $18(70)=1260$, and the second is $12(85)=1020$.
2. The combined total is $1260+1020=2280$. Divide by the combined number of students: $2280/30=76$.
3. The result is closer to 70 because more students belong to that group. Averaging 70 and 85 directly would incorrectly give the two groups equal weight.

GMAT takeaway: Combine totals and counts, not just averages. Group sizes determine the weights.

Review the statistics and weighted means reference.

Answer: B. 61

1. The number speaking at least one language is $80-12=68$.
2. Use inclusion-exclusion to find the overlap: $40+35-\text{both}=68$, so 7 employees speak both languages.
3. Exactly one excludes those 7 employees from the union: $68-7=61$. Equivalently, add French only and Spanish only: $(40-7)+(35-7)=61$.

GMAT takeaway: “At least one” includes the overlap, while “exactly one” excludes it. Identify which population the question requests before applying a set formula.

Review the overlapping sets reference.

Answer: E. 105

1. Choose the 2 women without regard to order: $\binom72=(7\cdot6)/2=21$.
2. Choose the 1 man in 5 ways. Every selection of women can be paired with every selection of the man.
3. Multiply the independent choice counts: $21(5)=105$. Using $7\cdot6$ for the women would count each unordered pair twice.

GMAT takeaway: Ask whether roles make order matter. An unranked committee uses combinations within each required category.

Review the counting and combinations reference.

Answer: C. $1/3$

1. The condition changes the sample space. List the ordered outcomes with sums greater than 9: $(4,6),(5,5),(6,4),(5,6),(6,5),(6,6)$.
2. There are 6 such outcomes, all equally likely because the dice are fair and independent. Only $(5,5)$ and $(6,6)$ show the same number.
3. The conditional probability is therefore $2/6=1/3$. Dividing by 36 would ignore the information that the sum is already known to exceed 9.

GMAT takeaway: For conditional probability, restrict the denominator to outcomes consistent with the given condition. Treat the two dice as distinguishable when enumerating ordered outcomes.

Review the conditional probability reference.

Answer: A. 288

1. There are 6 equal increments from the fourth term to the tenth term. The common difference is $(38-14)/(10-4)=24/6=4$.
2. The first term is $14-3(4)=2$. The twelfth term is $2+11(4)=46$.
3. The sum of 12 equally spaced terms is $12(2+46)/2=6(48)=288$. Use 11 increments, not 12, to move from the first term to the twelfth.

GMAT takeaway: Count gaps between term positions carefully. Once the endpoints are known, an arithmetic sum is the number of terms times their average.

Review the sequences and partial sums reference.

Answer: D. 48

1. For a fixed sum, the product is largest when the two numbers are as close together as the restrictions permit. Equal values would give $a=b=7$, but the problem says they are distinct.
2. The closest permitted positive integers are therefore 6 and 8, with product $6(8)=48$.
3. To verify the maximum algebraically, write $b=14-a$. Then $ab=a(14-a)=49-(a-7)^2$. Since a is an integer and cannot equal 7, the smallest possible value of $(a-7)^2$ is 1, giving a maximum of 48.

GMAT takeaway: Check every restriction before selecting the numerically largest result. An unconstrained maximum may be forbidden by “integer,” “distinct,” or another condition.

Review the mixed application: checking restrictions reference.