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Given

A certain sum of money becomes three times of itself in 20 years at simple interest. 

Concept used

Principal + interest = Amount

Calculation

Let the principal be P 

Amount = 3P 

Interest = 3P - P

Interest = 2P

Rate = $\frac{{{\mathop{\rm int}} erest\ \times 100}}{{principal \times time}}$

Rate = $\frac{{2p \times 100}}{{p \times 20}}$

Rate = 10%

Now let after t years the amount becomes double

Si = 2p - p = p

Time = $\frac{{{\mathop{\rm int}} erest\ \times 100}}{{principal \times rate}}$

Time = $\frac{{p \times 100}}{{p \times 10}}$

Time = 10 years.

Given:

Tap P filled = 6 hours

Tap Q filled = 8 hours

Tap R empty = 12 hours

Concept used:

Total capacity = LCM

The efficiency of inlet tap always positive and efficiency of outlet tap is always negative

Formula used:

Efficiency = Total capacity/Time

Calculation

Total capacity = LCM of 6, 8, 12 = 24

 

Total efficiency of P, Q, and R = 4 + 3 – 2 = 5

Total time taken by P, Q, and R to fill the tank = (24/5) hours

∴ Total time taken by P, Q, and R to fill the tank is (24/5) hours

Given:

The length of the train = 1350 m

Speed of men = 5 km/h

Time to cross the men = 135 seconds

Calculation:

Let the speed of the train be x km/h

Relative speed = (x - 5) km/h

⇒ (x - 5) = ( 1350 × 18)/(135 × 5 )

⇒ X - 5 = 36

⇒ X = 36 + 5 = 41 km/h

 Hence, 41 Km/h is the speed of train

Given:

Marked price of the sweatshirt = Rs. 2999

Discount percentage = 40%

Formula:

Amount to be paid after discount = Marked price - (Discount percentage × Marked price)/100

Solution:

Substituting the given values into the formula:

2999 - (40/100) × 2999

2999 - (0.4) × 2999

2999 - 1199.6

Rs. 1799.40

Therefore, Harshi paid Rs. 1799.40 after applying the 40% discount on the marked price.

Let my age 11 years ago be x years

Age of my father 11 years ago = 4x years

My present age be x + 11 years

My Father’s present age be 4x + 11

According to the question,

x + 11 + 5 + 4x + 11 + 5 = 87

5x = 55

x = 11 years

Present age of my father = 4 × 11 + 11 = 55 years

Shortcut Trick

Rate for half-yearly = 30% ÷ 2 = 15%.

Ratio of Amount to Principal = 1216.70 ÷ 800 = 12167 ÷ 8000.

Since 15% = 3/20, the multiplier is 23/20.

Observe that (23/20)³ = 12167/8000.

So, 3 half-years = 1.5 years.

∴ The correct answer is 1.5 years.

Alternate Method

Given: Principal (P) = ₹800, Amount (A) = ₹1216.70, Rate (R) = 30% p.a.

Formula Used: A = P(1 + r/100)ⁿ, where r is the rate per period and n is the number of periods.

⇒ For half-yearly compounding, Rate (r) = 30% ÷ 2 = 15% per half-year.

⇒ 1216.70 = 800(1 + 15/100)ⁿ

⇒ 1216.70 ÷ 800 = (115/100)ⁿ

⇒ 12167 ÷ 8000 = (23/20)ⁿ

⇒ (23/20)³ = (23/20)ⁿ

⇒ n = 3 half-years.

⇒ Time = 3 ÷ 2 = 1.5 years.

∴ The correct answer is 1.5 years.

Additional Information

Compounding Frequency

When interest is compounded half-yearly, the annual rate is divided by 2 and the time (in years) is multiplied by 2.

Effective Rate of Interest

The effective annual rate for 30% compounded half-yearly is [ (1 + 0.15)² − 1 ] × 100 = 32.25%.

Rule of 72

A quick way to estimate the time to double an investment: Time ≈ 72 ÷ Interest Rate.

Given- We have to find the simplified value of $\sqrt {5 + \sqrt {11 + \sqrt {19 + ;\sqrt {29 + ;\sqrt {49} } } } }$

Calculation-

Considering the given equation -

$\sqrt {5 + \sqrt {11 + \sqrt {19 + ;\sqrt {29 + ;\sqrt {49} } } } }$

$⇒ \sqrt {5 + \sqrt {11 + \sqrt {19 + ;\sqrt {29 + ;7 } } } }$

$⇒ \sqrt {5 + \sqrt {11 + \sqrt {19 + \sqrt {36 } } } }$

$⇒ \sqrt {5 + \sqrt {11 + \sqrt {19 + ;6} } }$

$⇒ \sqrt {5 + \sqrt {11 + 5} }$

$⇒ \sqrt {5 + 4}$

⇒ √9

⇒ 3

∴ The simplified value is 3.

Shortcut Trick

Average Speed for equal distance segments = (2 × S₁ × S₂) ÷ (S₁ + S₂)

Average Speed = (2 × 40 × 60) ÷ (40 + 60) = 4800 ÷ 100 = 48 kmph

Total Distance = Average Speed × Total Time = 48 × 8 = 384 km

∴ The correct answer is 384 kms.

Alternate Method

Given:

Speed 1 (S₁) = 40 km/hr

Speed 2 (S₂) = 60 km/hr

Total Time (T) = 8 hours

Formula Used:

Time = Distance ÷ Speed

Calculations:

Let the total distance be 2D. Then each half is D.

⇒ Time taken for first half (t₁) = D / 40

⇒ Time taken for second half (t₂) = D / 60

⇒ t₁ + t₂ = 8

⇒ (D / 40) + (D / 60) = 8

⇒ (3D + 2D) / 120 = 8

⇒ 5D / 120 = 8

⇒ D / 24 = 8

⇒ D = 24 × 8 = 192 km

⇒ Total Distance = 2D = 2 × 192 = 384 km

∴ The correct answer is 384 kms.

Additional Information

Average Speed (Equal Distances)

When a journey is divided into two equal distances with speeds x and y, the average speed is given by (2xy) ÷ (x + y).

Average Speed (General Formula)

Average Speed = Total Distance Covered ÷ Total Time Taken.

Relative Speed

If two objects move in opposite directions, relative speed = S₁ + S₂. In same direction, relative speed = |S₁ − S₂|.

Given:

Side of a rhombus is 20 cm.

One diagonal is 24 cm.

Formula used:

Area of a rhombus = 1/2 × product of diagonals

In Pythagoras theorem,

H²= P² + B²

Where, H = Hypotenuse

P = Perpendicular

B = Base

Calculation:

Here, we have a rhombus ABDC

We know that all sides of a rhombus are equal

⇒ AB = BD = DC = CA = 20 cm

Two diagonals are AD and BC

Now,

BM = MC = BC/2 =24/2 = 12 cm (As in rhombus diagonals bisect each other at 90 degrees)

Hence,

Consider triangle AMC, 

As it is a right-angled triangle 

⇒ AC² = AM² + MC²

⇒ ( 20)² = AM² + (12)²

⇒ AM² = 400 - 144 = 256

⇒ AM = $\sqrt{256}$ = 16 cm

Now,

AM = MD = 16 cm

AD = (AM +MD) = (16 + 16) = 32 cm

Area of a rhombus =1/2 × product of diagonals

⇒ 1/2 × ( AD × BC)

⇒ 1/2 × 32 × 24 = 384 cm²

∴ the area of the rhombus is 384 cm².

Given:

Marked Price = Rs.400 

Discount percent = 20% 

Loss = Rs.48 

Formulas used:

Selling Price = Marked Price × (100 - Discount)%

Loss = Cost Price - Selling Price 

Loss percentage = Loss/Cost Price × 100

Calculation:

Selling Price = 400 × (100 - 20)% 

⇒ 400 × 4/5 

⇒ Rs.320 

Cost Price = Selling Price +  Loss 

⇒ CP = Rs.320 + Rs.48 

⇒ Rs.368 

∴ Loss percentage = Loss/Cost Price × 100 

⇒ 48/368 × 100 

⇒ $13\frac{1}{23}\%$

∴ The required result is $13\frac{1}{23}\%$.