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Given:

1^st Proportional(a) = 2; 6

2^nd Proportional(b) = 5; 8

3^rd Proportional(c) = 6; 9

Formula Used:

4^th Proportional(d) = (b x c)/a

Calculation:

According to the formula,

(5 x 6)/2 = 15

The fourth proportion is 15

Again,

(8 x 9)/6 = 12

The fourth proportion is 12

Ratio = 15 : 12

⇒ 5 : 4

∴ The ratio is 5 : 4

Given:

The given cubic sum is as follows,

 13 + 23 + ......... 103

Concept:

The sum of the cube of the first n natural number is ${[\frac{n(n+1)}{2}]}^2$

Solution:

As per the given cubic sum n = 10

Substituting the value of n in ${[\frac{n(n+1)}{2}]}^2$ we will get,

${[\frac{n(n+1)}{2}]}^2={[\frac{10\times 11}{2}]}^2$

$55^2=3025$

Hence, option 1 is correct.

Shortcut Trick

Recognize the Pythagorean triplet: 5, 12, 13.

Given hypotenuse = 13 cm and one side = 5 cm, the other side (height) must be 12 cm.

Area = (1 ÷ 2) × base × height = (1 ÷ 2) × 5 × 12 = 30 cm².

∴ The correct answer is 30 cm².

Alternate Method

Given: Hypotenuse (h) = 13 cm, Base (b) = 5 cm.

Formula Used: Pythagoras Theorem: h² = b² + p² | Area = (1 ÷ 2) × b × p

⇒ p² = 13² − 5²

⇒ p² = 169 − 25 = 144

⇒ p = √144 = 12 cm

⇒ Area = (1 ÷ 2) × 5 × 12

⇒ Area = 5 × 6 = 30 cm²

∴ The correct answer is 30 cm².

Additional Information

Pythagorean Triplets

Common sets of integers that satisfy a² + b² = c² include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Area of Triangle (Heron's Formula)

For any triangle with sides a, b, and c, Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c) ÷ 2.

Circumradius of Right Triangle

The radius of the circumcircle of a right-angled triangle is always half of the hypotenuse (R = h ÷ 2).

Shortcut Trick

For 1 year compounded half-yearly, the number of periods (n) = 2 and rate (r) = 25 ÷ 2 = 12.5% = 1/8.

The difference between CI and SI for 2 periods is given by: Difference = P × (r/100)².

⇒ 4375 = P × (1/8)²

⇒ 4375 = P × (1/64)

⇒ P = 4375 × 64 = 280000.

∴ The correct answer is ₹ 280000.

Alternate Method

Given: Rate (R) = 25% per annum, Time (T) = 1 year, Difference (CI − SI) = ₹ 4375.

Formula Used: SI = PRT/100 | CI = P[(1 + r/100)ⁿ − 1]

⇒ Simple Interest (SI) = (P × 25 × 1) ÷ 100 = 0.25P

⇒ For Compound Interest (half-yearly), Rate (r) = 25 ÷ 2 = 12.5% and n = 2.

⇒ CI = P[(1 + 12.5/100)² − 1] = P[(1 + 1/8)² − 1]

⇒ CI = P[(9/8)² − 1] = P[81/64 − 1] = 17P/64

⇒ Difference = CI − SI = 17P/64 − 0.25P

⇒ 4375 = (17P − 16P) ÷ 64 = P/64

⇒ P = 4375 × 64 = 280000

∴ The correct answer is ₹ 280000.

Additional Information

Compound Interest Half-Yearly

When interest is compounded half-yearly, the annual rate is divided by 2 (R/2) and the time period is multiplied by 2 (2T).

SI vs CI Difference Formula

For 2 years (or 2 compounding periods), the difference is P(r/100)². For 3 periods, it is P(r/100)² × (3 + r/100).

Effective Rate of Interest

The effective rate for two successive periods of r% is calculated as (r + r + r²/100)%.

Shortcut Trick

Initial Ratio of Rida and Riya = 3 : 5

New Ratio after Rs. 5,000 increment = 5 : 7

Both ratios increased by 2 parts (5 − 3 = 2 and 7 − 5 = 2).

2 parts = Rs. 5,000 ⇒ 1 part = Rs. 2,500

Riya's present salary = 5 parts × 2,500 = Rs. 12,500

∴ The correct answer is Rs. 12,500.

Alternate Method

Given:

Initial ratio of salaries of Rida and Riya = 3 : 5

Increment in the salary of each = Rs. 5,000

New ratio of salaries = 5 : 7

Rida = 3 parts

Riya = 5 parts

• 2 parts

• 2 parts

Rida = 5 parts

Riya = 7 parts

2 parts = Rs. 5,000

Calculations:

⇒ Let the present salaries of Rida and Riya be 3x and 5x, respectively.

⇒ After adding Rs. 5,000 to each, the new salaries become (3x + 5000) and (5x + 5000).

⇒ According to the question, (3x + 5000) ÷ (5x + 5000) = 5 ÷ 7

⇒ By cross-multiplying, we get:

⇒ 7 × (3x + 5000) = 5 × (5x + 5000)

⇒ 21x + 35000 = 25x + 25000

⇒ 25x − 21x = 35000 − 25000

⇒ 4x = 10000

⇒ x = 2500

⇒ Riya's present salary = 5x = 5 × 2500 = 12500

∴ The correct answer is Rs. 12,500.

Additional Information

Constant Difference in Ratios

When a constant value is added to or subtracted from both terms of a ratio, the difference between the terms stays constant. If the initial part difference is unequal, multiply by cross differences to balance them.

Cross-Multiplication Method

For two numbers a and b, if x is added to both to form a new ratio c : d, then (a + x) ÷ (b + x) = c ÷ d. This is solved by directly equating d(a + x) = c(b + x).

Age Problems Relation

This specific ratio change shortcut is heavily used in Age problems, where the gap in ages between two people remains strictly constant regardless of the time elapsed.

Given:

The ratio of the speed of a boat to that of the current water is 35 : 8.

The boat goes along with the current in 5 hours 10 minutes.

Formula used:

Speed of boat in still water (B) / Speed of current (C) = 35 / 8

Time = Distance / Speed

Calculation:

Let speed of current be 8x, then speed of boat in still water = 35x

Speed downstream = 35x + 8x = 43x

Speed upstream = 35x - 8x = 27x

Time taken to go along with the current = 5 hours 10 minutes = 5 + 10/60 = 31/6 hours

⇒ Distance = Speed × Time

⇒ Distance = 43x × 31/6

Time taken to come back = Distance / Speed upstream

⇒ Time = (43x × 31/6) / 27x

⇒ Time = (43 × 31) / (6 × 27)

⇒ Time = 1333 / 162

⇒ Time = 8.2284 hours

⇒ Time = 8 hours 13 minutes 48 seconds

∴ The correct answer is option (4).

Formula used:

Perimeter of Square = 4a where, a is the side of Square.

Area of Square = a² where, a is the side of Square.

Perimeter of Rectangle = 2(L + B) where, L and B are the

Length and Breadth of Rectangle 

Area of Rectangle = L × B 

Calculations:

Perimeter of Square = 4a = 120 cm 

a = 30 cm

Area of Square = a2 = 900cm²

 The length of a rectangle is two times the side of the square. = 60cm

Perimeter of Rectangle = 2(L + B)  = 2(60 + B) = 156 cm 

60 + B = 156/2 = 78cm 

B = 78 - 60 = 18 

Area of Rectangle = L × B = 60 × 18 = 1080cm²

The difference between the areas is:

A_rectangle - A_square = 1080 - 900 = 180 cm²

Hence, the difference between the areas of the rectangle and the square is 180 cm².

Given:

A sold a mobile phone to B at a gain of 25%

B sold the mobile to C at a loss of 10%

Total money paid by C = Rs. 5625

Calculation:

Let, the A paid for the mobile phone = x

So, x × (125/100) × (90/100) = 5625

⇒ x = 5625 × (100/125) × (100/90)

⇒ x = 5000

∴ A paid Rs. 5000 (in Rs.) for the phone

Given:

Which of the following number is divisible by 11?

Options: 1) 6589, 2) 9164, 3) 8962, 4) 4857

Formula used:

A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11.

Calculation:

1. 6589

Odd positions: 6 + 8 = 14

Even positions: 5 + 9 = 14

⇒ 14 - 14 = 0

∴ 6589 is divisible by 11

2. 9164

Odd positions: 9 + 6 = 15

Even positions: 1 + 4 = 5

⇒ 15 - 5 = 10

∴ 9164 is not divisible by 11

3. 8962

Odd positions: 8 + 6 = 14

Even positions: 9 + 2 = 11

⇒ 14 - 11 = 3

∴ 8962 is not divisible by 11

4. 4857

Odd positions: 4 + 5 = 9

Even positions: 8 + 7 = 15

⇒ 9 - 15 = -6

∴ 4857 is not divisible by 11

∴ The correct answer is 6589.

Given:

Speed of Mohit from his home to office = 10 m/s

Speed of Mohit from his office to home = x km/hr

Average speed of Mohit for the whole journey = 12 m/s

Concept:

Unit conversion from Km/hr to m/s, then multiply from 5/18.

Unit conversion from m/s to Km/hr, then multiply from 18/5.

Formula used:

Distance = Speed × Time

Average speed = 2 × v₁ × v₂/(v₁ + v₂)

Where v₁ and v₂ are the speeds of journey for equal distance

Calculation:

Average speed of Mohit = 12 × 18/5 Km/hr

Speed of Mohit from his home to office = 10 × 18/5 = 36 Km/hr

⇒ Average speed = 2 × 36 × x/(36 + x)

⇒ 12 × 18/5 = 2 × 36 × x/(36 + x)

⇒ 6 × 18 = 180 × x/(36 + x)

⇒ 6 × (36 + x) = 10x

⇒ 216 = 4x

⇒ x = 54 km/hr

∴ Speed of Mohit from his office to home is 54 km/hr.