Quants 1 — BSSC Quantitative Aptitude

• Key Fact
Speed of P = $27 \times \frac{5}{18} = 7.5$ m/s. Speed of Q = $36 \times \frac{5}{18} = 10$ m/s. Time taken by P to complete one round = $\frac{600}{7.5} = 80$ seconds. Time taken by Q = $\frac{600}{10} = 60$ seconds. They will meet at the starting point at LCM(80, 60) = 240 seconds.

• Related Formula(s)
$T = \frac{D}{S}$, $\text{Meeting time at start} = \text{LCM}(T_1, T_2)$

• Shortcut / Trick
Instead of full LCM calculation, just observe the times 80s and 60s. The lowest common multiple of 8 and 6 is 24, so for 80 and 60 it is 240 instantly without writing anything down.

• Why wrong options are wrong
120 seconds: LCM of 60 and 40 (calculation slip in speed conversion).
300 seconds: Arbitrary multiple, fails divisibility by 80.
480 seconds: A common multiple but not the least common multiple (LCM).

• Time-Saving Tip
Always remember that meeting at the starting point requires taking the LCM of their individual round times, whereas meeting anywhere on the track requires dividing track length by relative speed.

• Additional Info
If asked when they will meet for the first time anywhere on the track, it would be $\frac{600}{(10 - 7.5)} = \frac{600}{2.5} = 240$ seconds. In this specific case, the first meeting anywhere coincides with the first meeting at the start.

• Key Fact
For every 1 kg (1000g) sold, he saves 200g (0.2 kg). Total transactions made out of 500 kg = 500. Total wheat saved = $500 \times 0.2 = 100$ kg. Selling 100 kg in the black market yields Rs. 2500, so the black market SP = $\frac{2500}{100} = 25$ Rs/kg. The ratio of black market SP to original CP = 25 : 20 = 5:4.

• Related Formula(s)
$\text{Profit} = \text{Selling Price} - \text{Cost Price}$

• Shortcut / Trick
He saves $\frac{1}{5}$th of a kg per 1 kg transaction. Total saved = $\frac{500}{5} = 100$ kg. Mentally divide 2500 by 100 to get Rs. 25. Ratio 25:20 simplifies to 5:4 instantly.

• Why wrong options are wrong
4:5: Inverse ratio, obtained if you place CP in the numerator instead of SP.
5:6: Calculation error from assuming he saves 100g instead of 200g per kg.
6:5: Arbitrary error from using 1000g instead of 800g as the base for transactions.

• Time-Saving Tip
In such "saved inventory" problems, always find the amount saved per transaction first. Multiply this by the total number of assumed 1kg transactions to get the total stolen/saved quantity.

• Additional Info
His profit percentage on regular customers is based on the weight given, which is $\frac{200}{800} \times 100 = 25\%$.

• Key Fact
Let the final quotient be 0 for the least number. Reverse work: $N_3 = 6 \times 0 + 4 = 4$. $N_2 = 5 \times 4 + 3 = 23$. $N_1 = 4 \times 23 + 2 = 94$. The least number is 94. Successive division of 94 by 6, 5, 4: $94 / 6$ gives quotient 15, remainder 4. $15 / 5$ gives quotient 3, remainder 0. $3 / 4$ gives quotient 0, remainder 3. The new remainders are 4, 0, 3. Sum = $4 + 0 + 3 = 7$.

• Related Formula(s)
$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$

• Shortcut / Trick
Use the ladder method: write divisors (4,5,6) and remainders (2,3,4) below them. Start with 0 at the end. $(6 \times 0)+4=4$, $(5 \times 4)+3=23$, $(4 \times 23)+2=94$. Then apply the same ladder forwards using 6,5,4 on 94 to fetch the new remainders rapidly.

• Why wrong options are wrong
5: Obtained if you add the quotients instead of remainders.
9: The sum of the original remainders (2+3+4), a trap for students who assume reversing divisors keeps the remainders same.
12: A random calculation slip during the multiplier step.

• Time-Saving Tip
Always take the final quotient as 0 to find the absolute least number for successive division problems, which keeps calculations small and manageable.

• Additional Info
Successive division means dividing the quotient of the previous step by the new divisor, not dividing the original number each time.

• Key Fact
Total Work = LCM(20, 30) = 60 units. Efficiency of P = $60/20 = 3$ units/day. Efficiency of Q = $60/30 = 2$ units/day. In 7 days (P, Q, P, Q, P, Q, P), P works for 4 days and Q works for 3 days. Work done = $4(3) + 3(2) = 12 + 6 = 18$ units. Remaining work = $60 - 18 = 42$ units. Time taken by Q to finish = $\frac{42}{2} = 21$ days.

• Related Formula(s)
$\text{Total Work} = \text{Efficiency} \times \text{Time}$

• Shortcut / Trick
A 2-day cycle produces $3+2=5$ units. 7 days means 3 full cycles plus 1 extra day (P's turn). Work done = $3 \times 5 + 3 = 18$ units. Remaining = 42. Since Q's speed is 2, just halve 42 to get 21 days instantly.

• Why wrong options are wrong
14 days: Error from mistakenly dividing remaining work by combined efficiency (42/3) or P's efficiency.
19 days: Subtracted wrong units from total work.
22 days: Miscounted the days P and Q worked in the 7-day period (e.g., assuming 3 days P, 4 days Q).

• Time-Saving Tip
For alternate days, always bundle the work into cycles (e.g., 2 days = 5 units). It prevents manually counting out day 1, day 2, day 3, etc., saving massive time on larger numbers.

• Additional Info
If Q had started the work, in 7 days Q would work 4 days and P would work 3 days, altering the remaining work amount.

• Key Fact
When a 3D figure is melted into another, their volumes remain equal. Volume of Cylinder = $\pi r^2h = \pi \times 16^2 \times 24 = \pi \times 256 \times 24 = 6144\pi$. Volume of one Sphere = $\frac{4}{3}\pi r^3 = \frac{4}{3}\pi(2)^3 = \frac{32\pi}{3}$. Number of spheres = $\frac{\text{Volume of Cylinder}}{\text{Volume of Sphere}} = \frac{6144\pi}{\frac{32\pi}{3}} = 6144 \times \frac{3}{32} = 192 \times 3 = 576$.

• Related Formula(s)
$V_{\text{cylinder}} = \pi r^2 h$
$V_{\text{sphere}} = \frac{4}{3} \pi r^3$

• Shortcut / Trick
Set the equations directly: $\pi \times 16 \times 16 \times 24 = n \times \frac{4}{3} \pi \times 2 \times 2 \times 2$. Cancel $\pi$. Simplify LHS vs RHS: $2 \times 2 \times 2 = 8$. $4 \times 8 = 32$. $16 \times 16 \times 24 = n \times \frac{32}{3}$. Cancel 32 with $16 \times 2$ to get $8 \times 24 = n \times \frac{1}{3}$. $192 \times 3 = 576$.

• Why wrong options are wrong
144: Forgot to multiply by the 4 from the sphere formula or inverted the fraction.
384: Forgot the $\frac{1}{3}$ term in the sphere's volume formula.
768: Used $2^2$ instead of $2^3$ for the sphere's radius calculation.

• Time-Saving Tip
Don't calculate full volumes like $6144\pi$. Keep the terms as un-multiplied factors ($16 \times 16 \times 24$) on both sides of the equal sign and cancel them out. It heavily reduces calculation errors.

• Additional Info
If the material suffered a 10% wastage during melting, you would multiply the original volume by 0.9 before dividing.

• Key Fact
Let the percentage score in Paper A be $x\%$. The score in Paper B is $\frac{8}{10} = 80\%$. The ratio of weights for A and B is 2:1. The overall percentage is the weighted average. $\frac{2x + 1(80)}{2 + 1} = 70$. Solving this: $2x + 80 = 210 \implies 2x = 130 \implies x = 65\%$. The marks obtained in Paper A = 65% of 10 = 6.5.

• Related Formula(s)
$\text{Weighted Average} = \frac{w_1x_1 + w_2x_2}{w_1 + w_2}$

• Shortcut / Trick
Use alligation on percentages. Overall is 70%. B is 80%. Difference between B and Overall is $80 - 70 = 10$. This corresponds to A's weight of 2 parts. So 2 parts = 10, meaning 1 part = 5. A must be lower than the average by 5. A = $70 - 5 = 65\%$. 65% of 10 marks is 6.5.

• Why wrong options are wrong
5.5: Arithmetic error during cross multiplication.
6.0: Assumed a simple average instead of weighted average (if simple average, $80+x=140 \implies x=60$, so 6.0 marks).
7.0: Assigned the 2:1 ratio in the reverse order for the subjects.

• Time-Saving Tip
Alligation is always faster than the weighted average equation when you are given the final average and the weights ratio.

• Additional Info
Even if the total marks were different, calculating everything in percentages first keeps the numbers small and avoids confusion with the weightage.

• Key Fact
Prime factorization of $14400 = 144 \times 100 = (2^4 \times 3^2) \times (2^2 \times 5^2) = 2^6 \times 3^2 \times 5^2$. Factors divisible by $18 (2^1 \times 3^2)$: Take out $2^1 \times 3^2$, remaining is $2^5 \times 5^2$. Number of such factors = $(5+1)(2+1) = 6 \times 3 = 18$. Factors divisible by $36 (2^2 \times 3^2)$: Take out $2^2 \times 3^2$, remaining is $2^4 \times 5^2$. Number of such factors = $(4+1)(2+1) = 5 \times 3 = 15$. Factors divisible by 18 but not 36 = $18 - 15 = 3$.

• Related Formula(s)
$\text{Number of factors of } p^a q^b r^c = (a+1)(b+1)(c+1)$

• Shortcut / Trick
To be divisible by 18 ($2^1 \times 3^2$) but NOT 36 ($2^2 \times 3^2$), the factor must have exactly $2^1$ in its prime factorization, not $2^2$ or higher. It must also have $3^2$ or higher (max is $3^2$), and any power of 5 (up to $5^2$). So choices for powers of 2 is exactly 1 option (which is $2^1$). Choices for 3 is 1 option ($3^2$). Choices for 5 are 3 options ($5^0, 5^1, 5^2$). Total combinations = $1 \times 1 \times 3 = 3$ factors instantly.

• Why wrong options are wrong
2: Missed the $5^0$ case when counting combinations.
4: Arithmetic error while subtracting the two factor sets.
5: Included powers that violate the condition.

• Time-Saving Tip
Instead of subtracting total sets, just fix the exponent of the prime that causes the restriction (here, power of 2 must be exactly 1) and calculate the combinations for the remaining primes directly.

• Additional Info
The three specific factors are $18 \times 1, 18 \times 5$, and $18 \times 25$, which are 18, 90, and 450.

• Key Fact
Let CP be 100. Profit is 40%, so SP = 140. Discount is 18%, meaning SP is 82% of MRP. $0.82 \times \text{MRP} = 140 \implies \text{MRP} = \frac{140}{0.82}$. The ratio of CP to MRP is directly $(100 - D) : (100 + P) = (100 - 18) : (100 + 40) = 82 : 140 = 41 : 70$. If the discount is 12%, the new SP will be 88% of the MRP. $\text{New SP} = 70 \times 0.88 = 61.6$. Profit = $61.6 - 41 = 20.6$. Profit % = $\frac{20.6}{41} \times 100 = 50.24\% \approx 50.2\%$.

• Related Formula(s)
$\frac{\text{CP}}{\text{MRP}} = \frac{100 - \text{Discount \%}}{100 + \text{Profit \%}}$

• Shortcut / Trick
Use the CP/MRP ratio shortcut: $\frac{CP}{MP} = \frac{100-18}{100+40} = \frac{82}{140} = \frac{41}{70}$. New SP is 88% of 70. Mentally: $70 \times 0.88 = 61.6$. Cost is 41. Profit is 20.6 on base 41. $20.5$ is exactly 50% of 41, so $20.6$ is slightly above 50%. Option 50.2% is the direct match.

• Why wrong options are wrong
48.5%: Calculated error by taking base as MRP instead of CP for profit.
52.4%: Additive error assuming 6% less discount directly means 6% more profit on base 140.
54.6%: Miscalculated the 88% of 70.

• Time-Saving Tip
The actual selling price in rupees (if given) is entirely useless when finding the new profit percentage. Work entirely in ratios of CP and MRP to save time.

• Additional Info
If he gave no discount (0% discount), he would sell at MRP, making his profit $\frac{70-41}{41} \times 100 = 70.7\%$.

• Key Fact
Let the speed of the thief be $4x$ and the speed of the policeman be $5x$. Since the time taken to catch the thief is the same for both (30 minutes), the ratio of distances covered by them is equal to the ratio of their speeds. Distance ratio = 4:5. The thief covered 6 km, so 4 parts = 6 km. Therefore, 1 part = $\frac{6}{4} = 1.5$ km. The policeman covered 5 parts = $5 \times 1.5 = 7.5$ km. The initial gap is the extra distance the policeman had to run, which is $7.5 - 6 = 1.5$ km.

• Related Formula(s)
$\text{Distance} = \text{Speed} \times \text{Time}$
$\text{If Time is constant, } D_1 : D_2 = S_1 : S_2$

• Shortcut / Trick
Speed ratio is 4:5. So distance ratio is 4:5. The difference is 1 part (which is the initial gap). If 4 parts = 6 km, then 1 part = 1.5 km directly. The 30 minutes given in the question is completely irrelevant data meant to waste your time.

• Why wrong options are wrong
1.0 km: Calculation error from dividing 6 by 5 or similar mixing of ratios.
1.2 km: Used 30 minutes to find speed and miscalculated the relative gap.
2.0 km: Arbitrary error.

• Time-Saving Tip
When time is constant, always equate the ratio of speeds directly to the ratio of distances. Do not calculate actual speeds unless explicitly asked.

• Additional Info
If asked for the speeds: Thief speed = $6 \text{ km} / 0.5 \text{ hr} = 12 \text{ km/h}$. Police speed = $7.5 \text{ km} / 0.5 \text{ hr} = 15 \text{ km/h}$.

• Key Fact
Let the amount invested at 4% be $x$. Then the amount at 5% is $5000 - x$. Interest from first part = $x \times 4\% = 0.04x$. Interest from second part = $(5000 - x) \times 5\% = 250 - 0.05x$. Total interest = $0.04x + 250 - 0.05x = 223$. $-0.01x = 223 - 250 = -27$. $0.01x = 27 \implies x = 2700$.

• Related Formula(s)
$\text{Simple Interest} = \frac{P \times R \times T}{100}$

• Shortcut / Trick
Use Alligation method. If all 5000 was at 4%, interest = 200. If all 5000 was at 5%, interest = 250. Actual interest = 223. Ratio of amounts = $(250 - 223) : (223 - 200) = 27 : 23$. Total parts = $27 + 23 = 50$. 50 parts = 5000 $\implies$ 1 part = 100. Amount at 4% (corresponding to 27 parts) = $27 \times 100 = 2700$.

• Why wrong options are wrong
Rs. 2300: This is the amount invested at 5%, a common error if you mix up the ratio order in alligation.
Rs. 2500: Assumes equal distribution (half and half).
Rs. 2800: Calculation error during subtraction.

• Time-Saving Tip
For questions splitting a principal into two rates given total interest, alligation on the total absolute interest (like 200 vs 250) is much faster than alligation on percentages, which creates messy fractions.

• Additional Info
If the total interest was for 2 years (e.g., 446 Rs), divide it by 2 first to get the 1-year interest before applying this method.