Q.A sum of ₹7,440 is the amount received after 3 years on a certain principal lent at 8% per annum simple interest. Find the principal.

Step 1 — condition for doubling

Doubling means the interest earned equals the principal $P$. With $SI=P\cdot r\cdot t$:

Step 2 — condition for becoming four times

Four times the sum means total $=4P$, so interest $=4P-P=3P$:

Step 3 — solve for $t$

Under simple interest the multiple grows linearly, so 4× takes $3\times$ the doubling time.

Step 1 — set up

First part $₹12,000$ at 10%, second part $y$ at 20%. The combined interest equals 14% of the total invested:

Step 2 — expand

Step 3 — solve for $y$

Step 4 — total invested

The question asks for the total amount, so add both parts:

Step 1 — Write the simple-interest formula

Step 2 — Substitute the given data

Rate $R=p\%$, time $T=p$ years, and the interest is $SI=\text{₹}\,p$:

Step 3 — Solve for the principal $P$

Multiply both sides by $100$ and divide by $p^{2}$:

Why the other options are wrong: (A) ₹$p$ ignores the formula entirely; (B) ₹$100p$ forgets to divide by $p^{2}$; (C) ₹$\dfrac{100}{p^{2}}$ cancels the $p$ in the interest twice instead of once.

Step 1 — Amount formula for simple interest

Step 2 — Compute the growth factor

$R=7\%$, $T=5\tfrac12=5.5$ years:

Step 3 — Solve for the principal

Check: $48000 \times 1.385 = 66480$ ✓.

Why the other options are wrong: (B) 50,000, (C) 45,000 and (D) 51,000 do not reproduce ₹ 66,480 when multiplied by $1.385$.

Step 1 — Rate that doubles the money (simple interest)

Doubling means interest earned $=$ principal, so $P = P\cdot\tfrac{x\cdot 5}{100}$.

Step 2 — Rate that triples the money

Tripling means interest earned $= 2P$, so $2P = P\cdot\tfrac{y\cdot 8}{100}$.

Step 3 — Difference of the rates

Why the other options are wrong: (D) 4% would arise from wrongly using $100/8$ for the tripling case; (B) 8% and (C) 3% do not match either computed rate.

Step 1 — Find the interest earned

Step 2 — Apply the simple-interest formula for time

Step 3 — Substitute

Why the other options are wrong: (A) 4.50 Yrs simply echoes the rate; (C) 6.50 Yrs and (D) 8.10 Yrs come from arithmetic slips — only ₹7,500 of interest at ₹900/year gives ≈8.34 years.

Step 1 — Interest on the first loan (full year)

Step 2 — Interest on the second loan (4 months, double rate)

Lent after 8 months → time = 4 months = $\tfrac{4}{12}=\tfrac13$ year, rate = $2r$:

Step 3 — Add and solve

Why the other options are wrong: (A) 3.6% and (D) 4.12% come from mistiming the second loan (using 6 months or the full year instead of 4 months).

Step 1 — Convert the time to years

$4$ years $4$ months $= 4 + \tfrac{4}{12} = \tfrac{13}{3}$ years.

Step 2 — Apply the simple-interest formula

Step 3 — Solve for the principal

$\dfrac{13P}{100} = 2{,}470 \Rightarrow P = \dfrac{2{,}470\times100}{13} = ₹19{,}000.$

Step 1 — Write the monthly simple interest

Here $r = 18\%$, so monthly interest $= P \times \dfrac{18}{100} \times \dfrac{1}{12} = P \times 0.015$.

Step 2 — Set equal to ₹1,200 and solve

So the required sum is ₹80,000.

Why the other options are wrong: ₹50,000 gives ₹750/month, ₹60,000 gives ₹900/month, ₹66,000 gives ₹990/month — none reach ₹1,200.

Step 1 — Compute the simple interest

Step 2 — Total amount to be cleared

Step 3 — Value of the cow

Step 1 — Write the simple interest

With rate R% for R years:

Step 2 — Apply the one-fourth condition and solve