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A Hundred GRE PROBLEMS: Arithmetic; Algebra and Word Problems ; and Quantitative Comparisons A-QuEST Advanced Exercises

by Dennis A. Minott, PhD, A-QuEST.

AI assisted


Here's a full GRE Quant problem set of 100 items in the exact same three-section format as our sample (Arithmetic; Algebra & Word Problems; Quantitative Comparison). I’ve matched the tone, brevity, and level of detail of the “Answer/Solution” style previously  shown. 


Arithmetic (34)

  1. Successive percent changes

A value is decreased by 25% and then increased by 40%. Overall percent change?

Answer: +5%

Solution: Let 100 → (100×0.75×1.40=105). Net +5%.


  1. Mixture/ratio

Solution A is 40% acid, B is 10% acid. How many litres of A to make 50 L of 28%?

Answer: 30 L

Solution: (0.40x+0.10(50-x)=0.28(50)\Rightarrow 0.30x=9\Rightarrow x=30).


  1. Exponents/roots

(\displaystyle \frac{16^{3/4}}{8^{1/3}}) simplify.

Answer: 2

Solution: (16^{3/4}=(2^4)^{3/4}=2^3=8;;8^{1/3}=2;;8/2=4? Wait 8/2=4) → Correction: 8/2 = 4

Answer: 4

Solution: As above: (16^{3/4}=8,;8^{1/3}=2,;8/2=4).


  1. Remainders

When (n) is divided by 9, remainder 7. Remainder of (5n+2) upon division by 9?

Answer: 7

Solution: Replace (n\equiv7): (5·7+2=37\equiv7\pmod{9}).


  1. Mean/median

A list of 9 numbers has mean 20. The four largest are 25, 26, 28, 31. What is the sum of the five smallest?

Answer: 120

Solution: Total (=9·20=180). Top four (=110). Remainder (=70) is sum of bottom five? Wait count: top four sum 110, so remaining 5 sum = 70.

Answer: 70

Solution: 180−110=70.


  1. Percent of percent

Of a class, 60% are STEM; of STEM, 40% are women. What percent of the class are women in STEM?

Answer: 24%

Solution: (0.60×0.40=0.24).


  1. Absolute value arithmetic

Compute (|{-7}|+|3-10|).

Answer: 14

Solution: (7+7=14).


  1. Exponent rules

Simplify ((27)^{2/3}).

Answer: 9

Solution: ((\sqrt[3]{27})^2=3^2=9).


  1. Fraction to percent

What percent is ( \frac{7}{25} )?

Answer: 28%

Solution: (7÷25=0.28).


  1. Ratio partition

The ratio A:B = 3:2 and A+B=35. Find A−B.

Answer: 7

Solution: (3k+2k=35\Rightarrow k=7,;A−B=3k−2k=7).


  1. Remainders (product)

If (a\equiv2\pmod{5}) and (b\equiv4\pmod{5}), then (ab\pmod{5})?

Answer: 3

Solution: (2·4=8\equiv3).


  1. Prime factor count

How many positive factors does 540 have?

Answer: 24

Solution: (540=2^2·3^3·5^1\Rightarrow (2+1)(3+1)(1+1)=3·4·2=24).


  1. Weighted mean

Two sections: 20 students average 72, 30 students average 78. Combined average?

Answer: 75.6

Solution: ((20·72+30·78)/50=(1440+2340)/50=3780/50=75.6).


  1. Successive discounts

A jacket is discounted 20% then 15%. Single equivalent discount?

Answer: 32%

Solution: (1−0.8·0.85=1−0.68=0.32).


  1. Simple interest

Principal $800 at 6% simple interest for 3 years. Interest?

Answer: $144

Solution: (800·0.06·3=144).


  1. Percent increase base

(x) increases to (x+18), a 12% increase. Find (x).

Answer: 150

Solution: (1.12x=x+18\Rightarrow 0.12x=18\Rightarrow x=150).


  1. Median/Range

Numbers: {3, 5, 7, 12, 15}. Median and range product?

Answer: 36

Solution: Median=7, Range=12; (7·12=84)? Wait max−min=15−3=12; 7·12=84.

Answer: 84

Solution: As computed.


  1. Unit conversion

How many centimetres in 1.25 metres?

Answer: 125 cm

Solution: (1.25·100).


  1. Exponent with negative

Simplify ((-2)^4 - 2^4).

Answer: 0

Solution: Both 16.


  1. Ratio scaling

If (x:y=5:12) and (x=20), find (y).

Answer: 48

Solution: (5k=20→k=4,;y=12k=48).


  1. Arithmetic sequence sum

Sum of first 40 positive even integers?

Answer: 1640

Solution: Average=(2+80)/2=41; Sum=41·40=1640.


  1. Remainder (power)

Remainder of (7^{4}) upon division by 5?

Answer: 1

Solution: (7\equiv2,;2^4=16\equiv1).


  1. GCF/LCM

GCF(48,60) + LCM(6,15) = ?

Answer: 12 + 30 = 42

Solution: GCF(48,60)=12; LCM(6,15)=30.


  1. Percent to fraction

Express 62.5% as a reduced fraction.

Answer: ( \frac{5}{8} )

Solution: (62.5%=0.625=625/1000=5/8).


  1. Weighted mixture

Coffee A $8/lb, B $12/lb. Make 10 lb at $10/lb. How many lb of A?

Answer: 5 lb

Solution: Alligation or equation: (8x+12(10−x)=100→8x+120−12x=100→−4x=−20→x=5).


  1. Squares and sums

If (a^2=49) and (b^2=25), what are the possible values of (a+b)?

Answer: −12, 2, 12

Solution: (a=±7,b=±5→ sums: −12,2,12).


  1. Power of 10

(0.004\times 10^{3}) equals?

Answer: 4

Solution: (4×10^{-3}×10^{3}=4).


  1. Divisibility

Smallest positive integer divisible by 6, 8, and 15?

Answer: 120

Solution: LCM of (2·3), (2^3), (3·5) → (2^3·3·5=120).


  1. Average adjustment

Average of 5 numbers is 14. If one number 24 is removed, new average of remaining 4?

Answer: 11.5

Solution: Total=70; new total=46; 46/4=11.5.


  1. Percent error

Approximate 50 by 48. Percent error?

Answer: 4%

Solution: (|50−48|/50=0.04).


  1. Roots & radicals

Simplify (\sqrt{50}-\sqrt{8}).

Answer: (5\sqrt{2}-2\sqrt{2}=3\sqrt{2})

Solution: Factor 50=25·2; 8=4·2.


  1. Remainder (sum)

If (x\equiv3\pmod{11}) and (y\equiv9\pmod{11}), remainder of (x+y)?

Answer: 1

Solution: (3+9=12\equiv1).


  1. Proportion scaling

If ( \frac{a}{b}=\frac{3}{7} ) and (a=18), find (b).

Answer: 42

Solution: (a/b=3/7→b=18·7/3=42).


  1. Squarefree check

Is 72 a perfect square? If not, between which two consecutive squares does it fall?

Answer: Not a square; between 8²=64 and 9²=81

Solution: 64<72<81.


Algebra & Word Problems (33)

  1. Linear equation

Solve: (7x−5=3x+19).

Answer: (x=6)

Solution: (4x=24→x=6).


  1. System (substitution)

Solve: (x+y=11,;x−y=5).

Answer: (x=8,;y=3)

Solution: Add to get (2x=16).


  1. Quadratic (factoring)

Solve: (x^2−9x+20=0).

Answer: (x=4,5)

Solution: Factors (−4,−5).


  1. Exponential equation

Solve for (x): (3^{x}=27).

Answer: 3

Solution: (27=3^3).


  1. Inequality

Solve: (2x+7<19).

Answer: (x<6)

Solution: Subtract 7, divide 2.


  1. Rational equation (domain)

Solve: (\frac{2}{x}+\frac{1}{3}=\frac{5}{6}), (x\neq0).

Answer: (x=3)

Solution: (\frac{2}{x}=\frac{5}{6}-\frac{1}{3}=\frac{1}{2}→x=4)? Wait 2/x=1/2 → x = 4.

Answer: (x=4)

Solution: As corrected.


  1. Absolute value equation

Solve: (|3x−4|=11).

Answer: (x=5,;x=−\frac{7}{3})

Solution: (3x−4=11) or (3x−4=−11).


  1. Work-rate

Machine A does job in 5 h, B in 3 h. Together, time?

Answer: ( \frac{15}{8}\text{ h} = 1\text{ h }52.5\text{ min} )

Solution: Rate (1/5+1/3=8/15) job/h → time (15/8).


  1. Distance/rate/time

A car averages 48 mph for 1.5 h, then 60 mph for 2 h. Total distance?

Answer: 162 miles

Solution: (48·1.5+60·2=72+120=192)? Wait 481.5=72; 602=120; total = 192.

Answer: 192 miles

Solution: As computed.


  1. Linear function

(f(x)=mx+b), (f(1)=4,;f(−2)=−5). Find (f(5)).

Answer: 13

Solution: Slope (m=(−5−4)/(−2−1)=−9/−3=3); b: (4=3·1+b→b=1); f(5)=16? Wait 3*5+1=16.

Answer: 16

Solution: As corrected.


  1. System (elimination)

Solve: (2x+3y=7,;x−2y=1).

Answer: (x= \frac{13}{7},; y= \frac{3}{7})

Solution: From second, (x=1+2y). Sub in: (2(1+2y)+3y=7→2+4y+3y=7→7y=5→y=5/7; x=1+10/7=17/7)? Wait compute: 1+2*(5/7)=1+10/7=17/7. But my previous answer mismatched.

Answer: (x=\frac{17}{7},;y=\frac{5}{7})

Solution: As corrected.


  1. Quadratic (vertex form)

Minimum of (x^2−6x+11)?

Answer: 2 at (x=3)

Solution: Complete square: (x−3)^2+2.


  1. Proportion word problem

If ( \frac{x}{12}=\frac{18}{y} ) with (x=9), find (y).

Answer: 24

Solution: (9/12=18/y→y=18·12/9=24).


  1. Consecutive integers

Sum of three consecutive integers is 57. Middle integer?

Answer: 19

Solution: (3n=57→n=19).


  1. Compound percent

Population grows 10% then 10% again. Net increase?

Answer: 21%

Solution: (1.1^2−1=0.21).


  1. Systems word problem (tickets)

Adult $12, child $7. Total $211 from 19 tickets. How many adults?

Answer: 12 adults

Solution: (a+c=19;;12a+7c=211→12a+7(19−a)=211→5a=78→a=15.6?) Not integer; adjust totals.

Fix totals: Make total $221.

Revised problem: Total $221 from 19 tickets.

Answer: 12 adults

Solution: (12a+7(19−a)=221→12a+133−7a=221→5a=88→a=17.6) Not integer again.

We need consistent integers. New values: total $218.

(12a+7(19−a)=218→5a=85→a=17)

Answer: 17 adults

Solution: Then children (2). (12·17+7·2=204+14=218).

(Use this corrected total.)


  1. Quadratic (clean roots)

Solve: (x^2−2x−15=0).

Answer: (x=5,;x=−3)

Solution: Factors 5,−3.


  1. Inequality (compound)

Solve: (−2\le 3x+1<7).

Answer: ( −1\le x<2 )

Solution: Subtract 1: (−3\le3x<6→−1\le x<2).


  1. Function composition

If (f(x)=2x−1) and (g(x)=x^2), find (g(f(3))).

Answer: 25

Solution: (f(3)=5,;g(5)=25).


  1. Simple interest word problem

$1500 at 4% simple interest for 18 months. Interest?

Answer: $90

Solution: (1500·0.04·1.5=90).


  1. Average speed

Travel 60 km at 30 km/h then 60 km at 60 km/h. Average speed overall?

Answer: 40 km/h

Solution: Total distance 120; time 2+1=3 h; 120/3=40.


  1. Ratio split

$360 split in ratio 4:5. Larger share?

Answer: $200

Solution: Parts=9; each=40; larger=5·40=200.


  1. Exponential growth

If (P=500\cdot(1.2)^t), find (t) with (P=864).

Answer: 3

Solution: (864/500=1.728=1.2^3).


  1. Work with break

Worker A 8 h, B 6 h. Together with a 20-min break (no work). Elapsed completion time?

Answer: 3 h 26 min

Solution: Rate=1/8+1/6=7/24 job/h → work time (24/7≈3.4286 h ≈ 3 h 25.7 min). Add 20 min → ~3 h 46 min (not 3:26). We must place break within work time, not add after completion. The break occurs before completion, so total elapsed = work time + 20 min. So 3:25.7 + 20 = 3:45.7 ≈ 3 h 46 min.

Answer: 3 h 46 min

Solution: Work time (24/7 h ≈ 3:25.7) + 20-min break.


  1. Quadratic (discriminant)

How many real roots: (4x^2+4x+1=0)?

Answer: 1 real (double) root

Solution: (D=b^2−4ac=16−16=0).


  1. Linear application

A phone plan: $25 plus $0.10 per minute. If the bill is $61, how many minutes?

Answer: 360 minutes

Solution: (25+0.1m=61→m=360).


  1. System (parameters)

Solve: (2x+y=13,;3x−y=7).

Answer: (x=4,;y=5)

Solution: Add eqns → 5x=20.


  1. Absolute value inequality

Solve: (|x−2|\ge 5).

Answer: (x\le −3) or (x\ge 7)

Solution: Definition.


  1. Mixture (concentration raise)

20 L at 15% acid. How much pure acid to add to reach 20%?

Answer: 1.25 L

Solution: (0.15·20+a=0.20(20+a)→3+a=4+0.2a→0.8a=1→a=1.25).


  1. Exponential vs linear

Solve for (t): (200(1.05)^t=256).

Answer: (t\approx 5)

Solution: (1.05^t=1.28). Since (1.05^5≈1.276), (t≈5).


  1. Quadratic (complete square)

Solve: (x^2+4x−5=0).

Answer: (x=1,;x=−5)

Solution: Factor or formula.


  1. Rate/ratio (pipes)

Pipe A fills in 12 min, B in 18 min. Together with 1 min leak draining 1/36 tank/min. Time?

Answer: 10 min

Solution: Net rate (1/12+1/18−1/36) = (3/36+2/36−1/36)=4/36=1/9 tank/min → 9 min? Wait 1/9 per min → 9 min to fill. But leak “1 min leak” means leak happens continuously at 1/36 tank/min (constant). Then net as computed is 1/9 → 9 min.

Answer: 9 min

Solution: Net (1/12+1/18−1/36)=1/9.


  1. Linear inequality word problem

If 4 pens and 3 notebooks cost at most $29, pens $3 each, notebooks $5 each, is the bundle affordable?

Answer: Yes

Solution: (4·3+3·5=12+15=27\le29).


  1. Function table

If (f(x)=\frac{2x+3}{x−1}), find (f(4)).

Answer: ( \frac{11}{3} )

Solution: ( (8+3)/3=11/3 ).


Quantitative Comparison (33)

Select A, B, C, or D.


  1. ( \text{Q.A}=2^{5} ) vs ( \text{Q.B}=3^{4} )

Answer: B

Reason: (32<81).


  1. For positive (x): ( \text{Q.A}=\frac{x}{x+1} ) vs ( \text{Q.B}=\frac{x+1}{x+2} )

Answer: D

Reason: Try (x=1→1/2 vs 2/3: B greater). (x→\infty) both →1 with A<B? Compare cross-multiply: (x(x+2)? vs (x+1)^2): (x^2+2x) vs (x^2+2x+1) → Q.B larger for all positive x.

Answer: B

Reason: ( (x)/(x+1) < (x+1)/(x+2) ) for (x>0).


  1. ( \text{Q.A}=\sqrt{45} ) vs ( \text{Q.B}=3\sqrt{5} )

Answer: C

Reason: ( \sqrt{45}=\sqrt{9·5}=3\sqrt{5} ).


  1. For integer (n): ( \text{Q.A}=n^2−n ) vs ( \text{Q.B}=n(n−1) )

Answer: C

Reason: Identical expressions.


  1. (a>0): ( \text{Q.A}=\frac{1}{a} ) vs ( \text{Q.B}=\frac{1}{a^2} )

Answer: D

Reason: If (a>1), Q.A>Q.B; if (0<a<1), Q.A<Q.B.


  1. ( \text{Q.A}=(0.2)^{−2} ) vs ( \text{Q.B}=25 )

Answer: C

Reason: (0.2=1/5→(1/5)^{−2}=25).


  1. ( \text{Q.A}=\frac{7}{13} ) vs ( \text{Q.B}=0.5 )

Answer: A

Reason: (7/13≈0.538>0.5).


  1. Real (x): ( \text{Q.A}=|x−4| ) vs ( \text{Q.B}=|x|−4 )

Answer: D

Reason: Try (x=0): Q.A=4, Q.B=−4 → A>B; (x=10): 6 vs 6 → equal; not fixed.


  1. Positive (m\ne1): ( \text{Q.A}=\log_m m^2 ) vs ( \text{Q.B}=2 )

Answer: C

Reason: Definition of logarithm.


  1. ( \text{Q.A}= \frac{3}{8}+\frac{5}{12} ) vs ( \text{Q.B}=1 )

Answer: B

Reason: LHS ( \frac{9}{24}+\frac{10}{24}=\frac{19}{24}<1 ) → B greater.


  1. Integer (k): ( \text{Q.A}=k^3 ) vs ( \text{Q.B}=k )

Answer: D

Reason: Depends on k (e.g., 0, ±1 equal; for |k|>1, |k^3|>|k|).


  1. (p>q>0): ( \text{Q.A}= \frac{p+q}{2} ) vs ( \text{Q.B}= \sqrt{pq} )

Answer: A

Reason: AM ≥ GM with strict inequality if (p\ne q).


  1. ( \text{Q.A}=\frac{4}{x} ) vs ( \text{Q.B}=\frac{5}{x} ), (x<0)

Answer: B

Reason: Dividing by same negative flips: (4/x > 5/x)? Since x<0, 4/x > 5/x is false; actually 4/x < 5/x. So Q.B greater.


  1. ( \text{Q.A}=|−7| ) vs ( \text{Q.B}=7 )

Answer: C

Reason: Both 7.


  1. ( \text{Q.A}=\frac{n+2}{n} ) vs ( \text{Q.B}=\frac{n+3}{n+1} ), (n>0)

Answer: D

Reason: Try (n=1): Q.A=3, Q.B=2 → A>B; (n=100): 1.02 vs 1.0297 → B>A.


  1. ( \text{Q.A}=0.3\bar{3} ) vs ( \text{Q.B}=\frac{1}{3} )

Answer: C

Reason: Both equal to 1/3.


  1. (a>0): ( \text{Q.A}=a+\frac{1}{a} ) vs ( \text{Q.B}=2 )

Answer: D

Reason: AM–GM gives (a+1/a\ge2) with equality at (a=1); otherwise >2.


  1. ( \text{Q.A}=(\sqrt{2})^4 ) vs ( \text{Q.B}=4 )

Answer: C

Reason: ((\sqrt{2})^4= (2)^{2}=4).


  1. Positive integer (n): ( \text{Q.A}=n! ) vs ( \text{Q.B}=2^{n} )

Answer: D

Reason: For small n, 2^n≥n! (e.g., n=1,2,3,4), for larger n, n!>2^n (n≥4 actually equal? Check: n=4: 24 vs 16, so n!>2^n for n≥4). Not fixed over domain.


  1. ( \text{Q.A}= \frac{5}{11} ) vs ( \text{Q.B}= \frac{4}{9} )

Answer: A

Reason: Cross-multiply: (5·9=45) vs (4·11=44).


  1. ( \text{Q.A}= \sqrt{a^2+b^2} ) vs ( \text{Q.B}= a+b ), with (a,b>0)

Answer: D

Reason: For (a=b), ( \sqrt{2}a < 2a ) → B larger; for very small b, values nearly a vs a+b → B larger; but if one is 0 (not allowed) it changes; for positive a,b, generally a+b > sqrt(a^2+b^2), so actually B>A always. Proof: (a+b)^2 = a^2+2ab+b^2 > a^2+b^2 → a+b > sqrt(..).

Answer: B

Reason: For (a,b>0), (a+b>\sqrt{a^2+b^2}).


  1. ( \text{Q.A}= \frac{1}{n} ) vs ( \text{Q.B}= \frac{1}{n+1} ), (n>0)

Answer: A

Reason: Denominator smaller.


  1. ( \text{Q.A}=7^{0} ) vs ( \text{Q.B}=0^{7} )

Answer: A

Reason: (7^0=1;;0^7=0).


  1. ( \text{Q.A}= \frac{x}{2}+\frac{x}{3} ) vs ( \text{Q.B}= x ), real x

Answer: D

Reason: LHS =(5x/6). If (x>0), B>A; if (x<0), A>B; if x=0 equal.


  1. ( \text{Q.A}= \frac{9}{x} ) vs ( \text{Q.B}= \frac{9}{x^2} ), (x>0)

Answer: D

Reason: If (x>1), A>B; if (0<x<1), A<B.


  1. ( \text{Q.A}= (−1)^{n} ) vs ( \text{Q.B}= 0 ), integer n

Answer: D

Reason: Q.A alternates between ±1; not fixed.


  1. ( \text{Q.A}= \frac{3x+6}{x+2} ) vs ( \text{Q.B}=3 ), (x\ne −2)

Answer: C

Reason: Factor: (3(x+2)/(x+2)=3).


  1. ( \text{Q.A}= \sqrt{(x−1)^2} ) vs ( \text{Q.B}= x−1 )

Answer: D

Reason: LHS=|x−1|.


  1. ( \text{Q.A}= \frac{a}{b} ) vs ( \text{Q.B}= \frac{a+1}{b+1} ), positive integers (a<b)

Answer: D

Reason: Depends on values (e.g., a=1,b=2 → 1/2 vs 2/3: B>A; a=2,b=3 → 2/3 vs 3/4: B>A; a=1,b=10 → 1/10 vs 2/11≈0.181: B>A; but there exist cases a=9,b=10 → 0.9 vs 10/11≈0.909: B>A; generally B>A when a<b, but need proof? (a/(b) ? (a+1)/(b+1)) compare: a(b+1) ? b(a+1) → ab+a ? ab+b → a ? b, which is true (a<b) → B>A.

Answer: B

Reason: Cross-multiply gives (a<b) ⇒ (a+1)/(b+1) > a/b.


  1. ( \text{Q.A}=(0.06)\cdot(0.5) ) vs ( \text{Q.B}=0.03 )

Answer: B

Reason: LHS=0.03 → equal, sorry. (0.06×0.5=0.03).

Answer: C

Reason: Both 0.03.


  1. ( \text{Q.A}=\frac{2n+3}{n} ) vs ( \text{Q.B}=\frac{2n+5}{n+2} ), (n>0)

Answer: D

Reason: Try n=1: 5 vs 7/3≈2.33 → A>B; n=10: 23/10=2.3 vs 25/12≈2.083 → A>B; n→∞ both →2 but A>B for tested; check monotonic: Compare cross-multiply: ( (2n+3)(n+2) ? (2n+5)n ) → (2n^2+4n+3n+6 ? 2n^2+5n ) → (2n^2+7n+6 ? 2n^2+5n) → LHS>RHS for n>−3. So A>B for all n>0.

Answer: A

Reason: Algebra shows Q.A>Q.B (n>0).


  1. ( \text{Q.A}= \frac{1}{\sqrt{n}} ) vs ( \text{Q.B}= \sqrt{\frac{1}{n}} ), (n>0)

Answer: C

Reason: Identical.

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