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Given

Time (T1) = 7 years

Time (T2) = ?

Formula used

S.I = PRT/100

Where SI = simple interest

P = principal

R = interest rate (in percentage)

T = time duration (in years)

Solution

Let the principal = a

Amount = 2a

S.I = 2a - a = a

a = (a × r × 7)/100

r = 100/7%

Now since the amount becomes 8 times,

S.I = 8a - a = 7a

7a = (a × 100/7 × T₂)/100

T₂ = 49 years

The correct option is 2.

Formula used:

(a +b)² = a² + b² + 2ab

Calculation:

$\sqrt{4-\sqrt{7}+\sqrt{19+2 \sqrt{84}}}$

⇒ $\sqrt{4-\sqrt{7}+\sqrt{12+7+2 \sqrt{84}}}$

⇒ $\sqrt{4-\sqrt{7}+\sqrt{(\sqrt12)^2+(\sqrt7)^2+2 \sqrt{12} \sqrt{7}}}$

⇒ $\sqrt{4-\sqrt{7}+\sqrt{(\sqrt12+\sqrt7)^2}}$

⇒ $\sqrt{4-\sqrt{7}+\sqrt12+\sqrt7}$

⇒ $\sqrt{4+\sqrt12}$

⇒ $\sqrt{1+3+\sqrt12}$

⇒ $\sqrt{1+3+2\sqrt3}$

⇒ $\sqrt{(1)^2+(\sqrt3)^2+2\times1\times\sqrt3}$

⇒ $\sqrt{(1+\sqrt3)^2}$ = $1+\sqrt{3}$

∴ The correct answer is option (1).

Shortcut Trick

Sum of ratio parts = 2 + 3 + 5 = 10 units.

Each variable appears twice in the sum, so 2 × (a + b + c) = 10 units.

Since a + b + c = 20, then 2 × 20 = 40 = 10 units ⇒ 1 unit = 4.

Therefore, a + b = 2 units = 2 × 4 = 8.

Value of c = (a + b + c) − (a + b) = 20 − 8 = 12.

∴ The correct answer is 12.

Alternate Method

Given: (a + b) : (b + c) : (c + a) = 2 : 3 : 5 and a + b + c = 20.

⇒ Let a + b = 2k, b + c = 3k, and c + a = 5k.

⇒ Adding the ratios: (a + b) + (b + c) + (c + a) = 2k + 3k + 5k.

⇒ 2(a + b + c) = 10k.

⇒ Given a + b + c = 20, substitute: 2(20) = 10k.

⇒ 40 = 10k ⇒ k = 4.

⇒ Now, a + b = 2k = 2 × 4 = 8.

⇒ We know c = (a + b + c) − (a + b).

⇒ c = 20 − 8 = 12.

∴ The correct answer is 12.

Additional Information

Constant of Proportionality

When terms are given in a ratio (x : y = m : n), they can be expressed as x = mk and y = nk, where k is the common multiplier.

Cyclic Ratio Sums

For expressions involving pairs like (a+b), (b+c), and (c+a), the sum of all pairs is always equal to 2(a + b + c).

Finding Individual Variables

To find a single variable like c from the total sum S = a+b+c, subtract the sum of the other two variables: c = S − (a+b).

Given:

The ratio of the radius and height of the cylinder = 3: 2

Radius = 21 cm

Formula used:

The curved surface area of the cylinder = 2πrh

Here,

r = radius

h = height

Calculation:

Let the radius of the base and height of the cylinder be 3x and 2x

So,

3x → 21

2x → 21/3 × 2 = 14

Curved surface area = 2π × 21 × 14

 

⇒ 2 × (22/7) × 21 × 14

⇒ 2 × 22 × 21 × 2

⇒ 1848

∴ The curved surface area (in cm2) of the cylinder is 1848.

Shortcut Trick

Marked Price (MP) = Rs. 10000

Price after 1^st discount (15%) = 10000 × (85 ÷ 100) = Rs. 8500

Final Selling Price (SP) = Rs. 7650

Second discount amount = 8500 − 7650 = Rs. 850

x% = (850 ÷ 8500) × 100 = 10%

∴ The correct answer is 10 percent.

Alternate Method

Given: MP = Rs. 10000, SP = Rs. 7650, d₁ = 15%, d₂ = x%

Formula: SP = MP × (100 − d₁) ÷ 100 × (100 − d₂) ÷ 100

⇒ 7650 = 10000 × (85 ÷ 100) × (100 − x) ÷ 100

⇒ 7650 = 85 × (100 − x)

⇒ 100 − x = 7650 ÷ 85

⇒ 100 − x = 90

⇒ x = 10

∴ The correct answer is 10 percent.

Additional Information

Successive Discounts

When two discounts a% and b% are given, the single equivalent discount is (a + b − ab ÷ 100)%.

Marked Price & Selling Price

Selling Price is the final amount paid by the customer after all discounts are subtracted from the Marked Price.

Given:

Marked price of a chair is 10 percent more than its cost price

The selling price of the chair is Rs. 300

The discount is Rs. 140

Formula used:

Loss = Cost price - Selling price

Calculation:

Let cost price of the chair is X

The marked price of the chair is = 110 percent of the cost price = 1.1X

⇒ 1.1X - 140 = 300

⇒ 1.1X = 440

⇒ X = 400

The cost price is = 400

The selling price is = 300

The loss percentage = ${400\ -\ 300\over 400}\times 100\ =\ 25 \%$

∴ The required result will be 25%.

Given Expression:

$(6-3)\div[(9-6)\div{(6-4)\div\left(2+\frac{8}{13}\right)}]$

Concept used:

Calculation:

$(6-3)\div[(9-6)\div{(6-4)\div\left(2+\frac{8}{13}\right)}]$

= 3 ÷ [(3) ÷ {(2) ÷ ($\frac{26+8}{13}$) }]

= 3 ÷ [(3) ÷ {2 ÷ $\frac{34}{13}$)}]

= 3 ÷ [3 ÷ {2 × $\frac{13}{34}$}]

= 3 ÷ [3 ÷ {$\frac{13}{17}$}]

= 3 ÷ [3 × $\frac{17}{13}$]

= 3 × [$\frac{13}{3 × 17}$]

= $\frac{13}{17}$

Thus, the required value is $\frac{13}{17}$.

Hence, the correct answer is option 3).

Given data:

Two numbers: 0.75 and 3

Concept: The mean proportional (geometric mean) of two numbers a and b is √(ab).

⇒ Mean proportional = √(0.75 ×  3) = 1.5

Therefore, the mean proportional of 0.75 and 3 is 1.5.

Shortcut Trick

In Compound Interest, if a sum becomes 'n' times in 'T' years, it becomes 'n^k' times in 'k × T' years.

Here, the sum doubles (2 times) in 10 years (n = 2, T = 10).

To become four times (4 = 2²), the power k = 2.

Time required = k × T = 2 × 10 = 20 years.

∴ The correct answer is 20 years.

Alternate Method

Given:

Initial Sum = P

Amount after 10 years = 2P

Target Amount = 4P

Formula Used:

A = P(1 + r/100)^t

Calculations:

⇒ 2P = P(1 + r/100)¹⁰

⇒ 2 = (1 + r/100)¹⁰ ---- (i)

⇒ Let the required time be 't' years for the sum to become 4P.

⇒ 4P = P(1 + r/100)^t

⇒ 4 = (1 + r/100)^t

⇒ (2)² = (1 + r/100)^t

⇒ Substituting value of 2 from (i):

⇒ [(1 + r/100)¹⁰]² = (1 + r/100)^t

⇒ (1 + r/100)²⁰ = (1 + r/100)^t

⇒ By comparing powers, t = 20 years.

∴ The correct answer is 20 years.

Additional Information

Effective Rate of Interest

When interest is compounded more than once a year, the effective annual rate is (1 + r/n)ⁿ − 1, where n is the frequency of compounding.

Rule of 72

A quick way to estimate how long it takes to double money at CI: Time ≈ 72 ÷ Interest Rate.

SI vs CI for Doubling

In Simple Interest, if a sum doubles in T years, it becomes n times in (n−1) × T years. For 4 times, it would take (4−1) × 10 = 30 years.

Given:

4374562 × 64 = ? × 7777

Calculation:

4374562 × 64 = ? × 7777

Let the unknown number be x.

⇒ 4374562 × 64 = x × 7777

⇒ x = $\dfrac{4374562 \times64}{7777}$ 

⇒ x = 35999.995 ≈ 36000

∴ Option (3) is the correct answer.