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Given:

Certain operations are performed in two numbers such as addition and substraction and the ratio of those are given.

Calculation:

Let the numbers after addition of 2 be 'x - 2' and '2x - 2'

According to the question,

⇒ $\dfrac{x -5}{2x -5}$ = $\dfrac{2}{5}$

⇒ 5 × (x - 5) = 2 × (2x - 5)

⇒ 5x - 25 = 4x - 10

⇒ x = 15

So, the numbers are 15 - 2 = 13, 30 - 2 = 28 

∴ The answer is 28.

Given:

The average age of 3 brothers = 15

Formula used:

Average = sum of all observations/number of observations

Calculation used:

Let the age of the three brothers be x years, (x + 3) years, and (x + 6) years.

According to questions:

⇒ [x + (x + 3) + (x + 6)] / 3 = 15

⇒ x + (x + 3) + (x + 6) = 45

⇒ 3x = 36

⇒ x = 12

So, ages of the brothers are 12 years, 15 years and 18 years

∴ The age of youngest brother is 12 years.

Shortcut Trick

Product of two numbers = LCM × HCF

Second Number = (450 × 30) ÷ 90 = 150

Difference = 150 − 90 = 60

∴ The correct answer is 60.

Alternate Method

Given:

LCM of the two numbers = 450

HCF of the two numbers = 30

First number = 90

Formula Used:

First Number × Second Number = LCM × HCF

Calculations:

⇒ Let the second number be x.

⇒ 90 × x = 450 × 30

⇒ 90 × x = 13500

⇒ x = 13500 ÷ 90

⇒ x = 150

⇒ Difference = Second number − First number

⇒ Difference = 150 − 90 = 60

∴ The correct answer is 60.

Additional Information

LCM and HCF Formula

For any two given numbers a and b, the product of the numbers is always equal to the product of their LCM and HCF: a × b = LCM(a, b) × HCF(a, b).

Co-prime Numbers

Two numbers are said to be co-prime if their HCF is 1. In such cases, their LCM is simply the product of the two numbers.

LCM and HCF of Fractions

LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

HCF of fractions = HCF of Numerators ÷ LCM of Denominators.

Given:

Amount in two years = Rs. 1000

Amount in 3 years = Rs. 1250

Formula used:

Amount = P(1 + R/100)^T

Compound Interest = P((1 + R/100)^T – 1)

Calculation:

Amount = P(1 + R/100)^t 

1000 = P(1 + R/100)²      ....(I)

1250 =  P(1 + R/100)³     ....(II)

Dividing equation (II) from (I), 

(1 + R/100) = 1250/1000

⇒ R/100 = 5/4 – 1

⇒ R/100 = 1/4

⇒ R = 25% 

∴ The annual rate of interest is 25%.

Let the number of 2 - rupee, 50 - paise and 25 - paise coins be a, b and c

According to the problem statement

⇒ 2a + b/2 + c/4 = 51.25

Given that they are in ratio 3 : 5 : 7, so 2 - rupee, 50 - paise and 25 - paise are 3x, 5x and 7x respectively, put this in the above equation

⇒ 2 × 3x + 5x/2 + 7x/4 = 51.25, Multiply both sides by 4

⇒ 24x + 10x + 7x = 205

⇒ 41x = 205

⇒ x = 5

⇒ 50 - paise coins are 5x = 5 × 5 = 25

∴ There were 25 coins of 50 - paise

Given:

Base Area = 154 sq cm

CSA = 550 sq cm

Formula used:

CSA(cone) = π x r x l

r = radius of the base

l = slant height of the cone

Base area = $\frac{22}{7}$ x r x r 

Volume(Cone) = $\frac{1}{3}\times \frac{ 22}{7}$ x r x r x h

Calculation:

Base area = $\frac{22}{7}$ x r x r = 154

r = 7 cm

CSA(cone) = $\frac{22}{7}$ x r x l = 550

l = 25 cm

h² = 25² - 7² 

h = 24 cm

Volume(Cone) = $\frac{1}{3}\times \frac{ 22}{7}$ x r x r x h = $\frac{1}{3}\times \frac{ 22}{7}$ x 7 x 7 x 24

Volume = 1232 cm³ 

Hence, the correct answer is option 1).

Calculation:

Let Raj initially have Rs. 100x

Raj initially had Rs. 100x. After spending 10% of his money, he was left with 90x.

Out of this remaining amount, he spent 70%, which left him with 30% of the original 90x, which was equal to Rs. 2646.

30% of 90x = 2646

⇒ 27x = 2646

⇒ x = 98

So , 100x = 100 × 98 = 9800

∴ Option 4 is the correct answer.

Alternate Method

Let the sum of money be x

According to the question,

x × 9/10 × 3/10 = 2646

⇒ 27x/100 = 2646

⇒ x = 9800

Given:

An inlet pipe takes 8 hours to fill a tank. An outlet pipe takes 12 hours to empty.

Calculation:

 ⇒ Time is taken to fill in 1 hour = 1/8

 ⇒ Time taken to empty in 1 hour =  -1/12

 ⇒ If both pipes are opened simultaneously tank be filled in 1 hour = 1/8 – 1/12 = 1/24 hours

∴ The total time taken is 24 hours.

Given:

${7^{x - 2}} \div ;{7^3} = ;\sqrt[5]{{{7^{10}}}}$

Concept used:

a^x ÷ a^y = a^{x – y}

$\sqrt[y]{{{a^x}}} = {a^{\frac{x}{y}}}$

Calculations:

7^{[(x – 2) – 3]} = 7^(10/5)

∴ x – 2 – 3 = 10/5

⇒ x – 5 = 2

⇒ x = 7

∴ The value of x is 7

Shortcut Trick

The Volume of a cylinder is proportional to the square of its radius (V ∝ r²) when height (h) is constant.

Since the radius is halved, the ratio of radii is 1 ÷ 2.

Therefore, the ratio of their volumes = (1)² ÷ (2)² = 1 ÷ 4.

∴ The correct answer is 1  ∶ 4.

Alternate Method

Given:

Original Radius = r

Reduced Radius = r ÷ 2

Height = h (remains same)

Formula Used:

Volume of Right Circular Cylinder = πr²h

⇒ Volume of Original Cylinder (V₁) = π × r² × h

⇒ Volume of Reduced Cylinder (V₂) = π × (r ÷ 2)² × h

⇒ V₂ = π × (r² ÷ 4) × h

⇒ Ratio (V₂ ÷ V₁) = [π × (r² ÷ 4) × h] ÷ [π × r² × h]

⇒ Ratio = (1 ÷ 4) ÷ 1 = 1 ÷ 4

∴ The correct answer is 1  ∶ 4.

Additional Information

Volume Proportionality

For a cylinder, the Volume varies directly with the Height and with the Square of the Radius. If only height is changed by factor k, volume changes by k. If only radius is changed by factor k, volume changes by k².

Curved Surface Area (CSA)

The Curved Surface Area of a cylinder is 2πrh. If the radius is halved and height remains same, the CSA is also halved (Ratio 1 ÷ 2).

Total Surface Area (TSA)

The Total Surface Area is given by 2πr(r + h), which includes the area of the two circular bases (2 × πr²) and the curved surface area.