Quantitative Aptitude

Quantitative questions reward a clean model before arithmetic. Write down units, identify the denominator in every percentage, and estimate the scale before calculating. In a plot, read axes and legends first. In geometry and probability, state assumptions; a correct formula used under the wrong conditions gives a wrong answer.

Ratios, percentages, and proportional change

Idea. A ratio compares quantities in the same unit; a percentage is a ratio with denominator 100. If a value changes from $x$ to $y$, percentage change is $100(y-x)/x$ when $x\ne0$. Successive changes multiply factors; they generally do not add.

Solved examples.

1. Question. A solution contains acid and water in ratio 2:7 and has volume 45 L. Find each amount. Solution. (1) Total parts $2+7=9$. (2) One part is $45/9=5$ L. (3) Acid is $2(5)=10$ L and water $7(5)=35$ L. Answer. 10 L acid, 35 L water. Trap. Ratio parts are not percentages until divided by their total.
2. Question. A price rises 20% and then falls 20%. What is the net percentage change? Solution. (1) Take 100 as starting price. (2) Increase gives 120. (3) Decrease is 20% of 120, or 24. (4) Final price 96. Answer. 4% decrease. Shortcut. Factors $1.2\times0.8=0.96$; equal percent rise and fall do not cancel.
3. Question. A class has 36 students; 5/12 are left-handed. How many are right-handed? Solution. (1) Left-handed count is $36(5/12)=15$. (2) Subtract from total: $36-15=21$. Answer. 21. Trap. The requested complement is $7/12$ of the total.
4. Question. An item after 15% discount costs 680. Find its original price. Solution. (1) The paid fraction is $1-0.15=0.85$. (2) Let original price be $P$: $0.85P=680$. (3) $P=680/0.85=800$. Answer. 800. Trap. A 15% discount is applied to original price; do not add 15% of the sale price.
5. Question. Team A completes 3/5 of a task. Team B completes 1/4 of the remainder. What fraction is left? Solution. (1) After A, remainder is $1-3/5=2/5$. (2) B does one fourth of that, $1/10$. (3) Remaining fraction is $2/5-1/10=3/10$. Answer. $3/10$. Trap. “Of the remainder” changes the base.

Numerical computation, powers, logarithms, and sequences

Idea. Use exponent laws only for compatible bases and nonzero denominators: $a^m a^n=a^{m+n}$ and $(a^m)^n=a^{mn}$. For $a>0$, $a\ne1$, $log_a(xy)=\log_a x+\log_a y$ for positive $x,y$. Estimation can reject impossible options before exact arithmetic.

Solved examples.

1. Question. Evaluate $2^5\cdot2^{-2}$. Solution. (1) Same base, so add exponents: $5+(-2)=3$. (2) $2^3=8$. Answer. 8. Trap. A negative exponent means reciprocal, not a negative value.
2. Question. Solve $\log_3 x=4$. Solution. (1) By definition, $x=3^4$. (2) $x=81$, which is positive as required. Answer. 81. Trap. Log arguments must be positive.
3. Question. Approximate $49.8\times 20.1$ without full multiplication. Solution. (1) Round to $50\times20=1000$. (2) Since one factor is 0.2 below 50 and the other 0.1 above 20, the errors nearly offset. Exact value is $49.8(20+0.1)=996+4.98=1000.98$. Answer. About 1001. Trap. Round in a way that preserves scale; state that the result is an estimate.
4. Question. Find the next term: 3, 8, 15, 24, 35, __. Solution. (1) Differences are 5, 7, 9, 11. (2) They increase by 2, so next difference is 13. (3) $35+13=48$. Answer. 48. Trap. Check first differences before guessing a multiplicative pattern.
5. Question. A sequence has first term 4 and each next term is 3 times the previous term minus 2. Find the fourth term. Solution. (1) $t_1=4$. (2) $t_2=3(4)-2=10$. (3) $t_3=28$. (4) $t_4=82$. Answer. 82. Trap. Apply the operation to the preceding term, not to the index.

Permutations, combinations, and numerical relations

Idea. If order matters when choosing $r$ objects from $n$ distinct objects, use $P(n,r)=n!/(n-r)!$. If order does not matter, use $\binom nr=n!/[r!(n-r)!]$. These assume distinct objects and valid integer ranges $0\le r\le n$.

Solved examples.

1. Question. How many ordered pairs of different digits can be chosen from 1 through 6? Solution. (1) Choose the first digit in 6 ways. (2) Choose a different second digit in 5 ways. (3) $6\cdot5=30$. Answer. 30. Trap. Pair order matters.
2. Question. A committee of 2 is chosen from 7 people. How many committees? Solution. (1) Order does not matter. (2) $\binom72=7\cdot6/2=21$. Answer. 21. Trap. Counting AB and BA separately would double count.
3. Question. How many distinct arrangements of the letters in LEVEL are there? Solution. (1) There are 5 letters; L repeats twice and E repeats twice. (2) Divide $5!$ by $2!2!$. (3) $120/4=30$. Answer. 30. Trap. Divide by each repeated-letter factorial.
4. Question. A code has two different letters selected from A, B, C, D followed by one digit from 0–9. How many codes? Solution. (1) Ordered letter choices: $4\cdot3=12$. (2) Ten digit choices, including zero. (3) Product rule gives $12\cdot10=120$. Answer. 120. Trap. A code is not a number, so a leading zero in its digit position is permitted.
5. Question. A bag has 5 red and 3 blue distinct tokens. In how many ways can 2 red tokens be selected? Solution. (1) Only the 5 red tokens are eligible. (2) Selection order is irrelevant. (3) $\binom52=10$. Answer. 10. Trap. Do not include blue tokens when the color is specified.

Data interpretation: tables, plots, maps, and estimates

Idea. Read the title, units, axes, legend, and time period before calculating. For a bar chart, bar height is the quantity; for a pie chart, a sector's fraction is its angle divided by $360^\circ$. A map needs its scale and direction. In a 3-D plot, identify which variable each axis encodes.

Solved examples.

1. Question. A table reports output (units): Mon 24, Tue 30, Wed 36. What is the percentage rise from Monday to Wednesday? Solution. (1) Change is $36-24=12$. (2) Base is Monday's 24. (3) $12/24\times100=50\%$. Answer. 50%. Trap. Use the starting day's value as denominator.
2. Question. A pie chart shows four categories at $90^\circ, 120^\circ, 60^\circ, 90^\circ$. If total is 720, find the amount in the second category. Solution. (1) Its share is $120/360=1/3$. (2) $720/3=240$. Answer. 240. Trap. Sector angle is a fraction of 360 degrees, not 100 degrees.
3. Question. A line plot gives readings at x=0,1,2,3 as 4, 7, 7, 10. What is the mean reading? Solution. (1) Sum is $4+7+7+10=28$. (2) There are four readings. (3) Mean $28/4=7$. Answer. 7. Trap. Count observations, not intervals.
4. Question. A map scale is 1 cm to 8 km. Two sites are 3.5 cm apart on the map. Find straight-line distance. Solution. (1) Multiply map distance by scale: $3.5\times8=28$. (2) Scale refers to straight-line map distance. Answer. 28 km. Trap. Road distance can be longer and is not inferred from a straight map segment.
5. Question. In a 3-D chart, height is production, x-axis is month, and y-axis is factory. Factory A produces 40 in month 1 and 52 in month 2. Find its increase. Solution. (1) Hold factory fixed at A. (2) Compare month 2 with month 1. (3) Difference $52-40=12$; relative rise $12/40=30\%$. Answer. Increase 12 units, or 30%. Trap. Do not compare bars belonging to different factories.

Geometry, mensuration, statistics, and probability

Idea. Perimeter measures boundary length; area measures a two-dimensional region; volume measures three-dimensional space. Use consistent units. For equally likely outcomes, probability is favorable outcomes divided by total outcomes; this ratio assumes every outcome has equal chance.