Let me explain profit/loss concepts with an example:
**Scenario:**
- Cost Price (CP) = ₹500
- Markup = 20%
- Selling Price (SP) = ?
**Calculations:**
1. Markup Amount = CP × (Markup%) = 500 × 0.20 = ₹100
2. Selling Price = CP + Markup = 500 + 100 = ₹600
**If sold at 10% discount:**
- Discount Amount = SP × (Discount%) = 600 × 0.10 = ₹60
- Final Selling Price = 600 - 60 = ₹540
**Profit/Loss:**
- Final SP (₹540) - CP (₹500) = ₹40 profit
**Key Formulas:**
- Profit = SP > CP
- Loss = SP < CP
- Profit% = (Profit/CP) × 100
- Loss% = (Loss/CP) × 100
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**Number System: LCM, HCF, Divisibility**
**Example:**
Find the LCM and HCF of 24 and 36.
**Solution:**
1. **Prime Factorization:**
- 24 = 2³ × 3¹
- 36 = 2² × 3²
2. **HCF (Highest Common Factor):**
Take the *lowest* power of each prime present in both numbers.
HCF = 2² × 3¹ = 4 × 3 = **12**
3. **LCM (Least Common Multiple):**
Take the *highest* power of each prime from both numbers.
LCM = 2³ × 3² = 8 × 9 = **72**
**Divisibility Check:**
- A number is divisible by 3 if the sum of its digits is divisible by 3.
(e.g., 36: 3 + 6 = 9 → divisible by 3)
- A number is divisible by 4 if its last two digits form a number divisible by 4.
(e.g., 24: last two digits are 24 → divisible by 4)
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**Fractions & Percentages Crash Course**
*Conversions:*
- Fraction → Percent: Multiply by 100 (e.g., 3/4 = 0.75 → 75%)
- Percent → Fraction: Divide by 100 & simplify (e.g., 40% = 40/100 = 2/5)
*Applications:*
1. **Discounts:** 25% off $80 = 0.25 × $80 = **$20 saved** → Pay $60
2. **Interest:** 5% APR on $1,000 = 0.05 × $1,000 = **$50/year**
3. **Ratios:** A class has 3/5 girls → 60% girls
*Real-World Twist:*
A pizza is 8/12 eaten (→ 66.67%). You eat half the leftovers—what % remains?
*(Answer: 16.67%—try the math!)*
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**Simple vs. Compound Interest Problems**
**Simple Interest Formula:**
\[ I = P \times r \times t \]
Where:
- \( I \) = Interest earned
- \( P \) = Principal amount (initial investment)
- \( r \) = Annual interest rate (decimal form, e.g., 5% = 0.05)
- \( t \) = Time in years
**Example:**
You invest \$1,000 at a 4% annual simple interest rate for 3 years.
\[ I = 1000 \times 0.04 \times 3 = \$120 \]
Total value after 3 years = \$1,000 + \$120 = **\$1,120**
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**Compound Interest Formula:**
\[ A = P \left(1 + \frac{r}{n}\right)^{n \times t} \]
Where:
- \( A \) = Future value of investment
- \( n \) = Number of compounding periods per year
- *(Other variables same as above)*
**Example:**
Same \$1,000 at 4% annual interest, compounded quarterly (\( n = 4 \)) for 3 years.
\[ A = 1000 \left(1 + \frac{0.04}{4}\right)^{4 \times 3} \]
\[ A = 1000 \left(1.01\right)^{12} \approx \$1,126.83 \]
**Key Difference:**
- *Simple interest* earns the same amount yearly.
- *Compound interest* grows exponentially as interest earns interest.
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The Railway Recruitment Board (RRB) NTPC exam often includes questions about time/work efficiency and collective work problems. Here's how to approach them:
**Key Concepts:**
1. If Worker A completes a job in X days, their daily work = 1/X
2. Combined work rates add up: (1/X + 1/Y) for Workers A+B
3. Total time = 1/(combined rate)
**Example Problem (RRB NTPC Style):**
"A can build a wall in 10 days. B can build it in 15 days. If they work together for 4 days, then A leaves, how many more days will B need to finish?"
**Solution:**
1. A's rate = 1/10 per day
B's rate = 1/15 per day
Combined rate = (1/10 + 1/15) = 5/30 = 1/6 per day
2. Work done in 4 days together: 4 × (1/6) = 4/6 = 2/3 of wall
Remaining work: 1 - 2/3 = 1/3
3. B alone finishes remaining: (1/3)/(1/15) = 5 days
**Answer:** B needs 5 more days.
**Common Variations:**
- Adding/removing workers mid-task
- Fractional work days
- Efficiency ratios (e.g., "A is 50% faster than B")
Practice by setting up equations from rates and watching for changes in team composition.
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Here are some word/pair relationship analogies similar to the given example:
1. Chef : Kitchen :: Painter : Studio
2. Pilot : Cockpit :: Captain : Bridge
3. Actor : Stage :: Athlete : Field
4. Librarian : Library :: Bartender : Bar
5. Farmer : Farm :: Fisherman : Boat
6. Judge : Courtroom :: Scientist : Laboratory
7. Soldier : Barracks :: Student : Dormitory
8. Mechanic : Garage :: Baker : Bakery
9. Sailor : Ship :: Astronaut : Spaceship
10. Barber : Salon ::
Each pair follows the pattern "[Profession] : [Primary Workplace]" similar to "doctor:hospital::teacher:school".
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**Example Syllogism:**
1. All mammals are warm-blooded. (All A are B)
2. Some warm-blooded animals are dolphins. (Some B are C)
3. Therefore, some dolphins are mammals. (Therefore, some C are A)
**Note:** This is valid because if all A are B, and some B are C, then at least some C must inherently be A (dolphins, in this case, are indeed mammals).
**Invalid Example:**
1. All birds can fly. (All A are B)
2. Penguins are birds. (All C are A)
3. Therefore, penguins can fly. (Therefore, all C are B) → *False, as penguins cannot fly despite being birds.*
Syllogisms rely on *sound* premises (true statements) and *valid* structure to yield correct conclusions.
