Given:

The ratio of the areas of two squares = 16 ∶ 1

Formula used:

Area of square = (Side of the square)²

The perimeter of square = 4 × (Side of the square)

Calculations:

Let the side of the two squares be a and b respectively

The ratio of the areas of two squares

⇒ a² ∶ b²  =16 ∶ 1

⇒ a ∶ b = 4 ∶ 1

The ratio of the perimeter of two squares

⇒ 4a ∶ 4b = (4 × 4) ∶ (4 × 1)

⇒ 16 ∶ 4

⇒ 4 ∶ 1

∴ The ratio of the perimeter of two squares is 4 ∶ 1

Given:

The price of two varieties of a commodity is Rs.7/kg and Rs.12/kg

Final price of the mixture = Rs.10/kg

Concept used:

Alligation method

Calculation:

We can use the given data in the allegation method as -

∴ The two varieties should be mixed in the ratio 2 ∶ 3

Additional Information

Given:

80 liters of milk solution has 60% milk

Calculation:

Let's denote the amount of pure milk to be added as x liters.

Initially, there are 80 liters of milk solution,

which contains 60% milk. So, there are (80 × 0.60),

⇒ 48 liters of milk in the 80 liters of milk solution.

After adding x liters of pure milk, the total volume of the milk solution will be (80 + x) liters.

We want this new solution to have 80% milk in it.

So, the equation will be:

(48 + x) / (80 + x) = 0.80

Now, let's solve for x:

⇒ 48 + x = 0.80 × (80 + x)

⇒ 48 + x = 64 + 0.80x

⇒ x - 0.80x = 64 - 48

⇒ 0.20x = 16

⇒ x = 16 / 0.20

⇒ x = 80

∴ Need to add 80 liters of pure milk to the milk solution to make it 80% milk.

Given:

 x = 1, y = 2 

Calculations:

x² + y² 

= (1)²  + (2)²

= 1 + 4

= 5

∴ The value of x² + y² is 5.

Given:

Consumption of sugar for 24 days is 6 kg.

Calculations:

If Consumption of sugar for 24 days is 6 kg

∴ The consumption of sugar for 18 days is 4.5 kg.

Calculation:

Prime factors of 552 = 2 × 2 × 2 × 3 × 23 = 8 × 69

Prime factor of 759 = 3 × 11 × 23 = 11 × 69

Prime factor of 828 = 2 × 2 × 2 × 3 × 3 × 23 = 24 × 69

The highest common factor of 552, 759, and 828 is 69.

∴ The value of x lies between 65 and 70.

Given:

The ratio of two numbers = 9 : 13

HCF = 6

Concept used:

If a and b are two numbers, then a × b = HCF and LCM

HCF of two numbers is highest common factor of both of them

Calculation:

Let the numbers be 9x and 13x respectively

Then, HCF = x

As we have given, x = 6

First number = 9x = (9 × 6) = 54

Second number = 13x = (13 × 6) = 78

Now,

⇒ a × b = HCF × LCM

⇒  54 × 78 = 6 × LCM

⇒ LCM = (9 × 78)

⇒ LCM = 702

∴ The required LCM is 702

Calculation:

Divisibility by 5:

A number is divisible by 5 if its last digit is 0 or 5.

The last digit of 1254216 is 6, which is not 0 or 5.

Therefore, 1254216 is not divisible by 5.

Divisibility by 11:

A number is divisible by 11 if the difference between the sum of its digits in odd positions

And the sum of its digits in even positions is either 0 or a multiple of 11.

In 1254216:

Sum of odd-positioned digits: 1 + 5 + 2 = 8

Sum of even-positioned digits: 2 + 4 + 6 = 12

Difference: 12 - 8 = 4

Since 4 is neither 0 nor a multiple of 11, 1254216 is not divisible by 11.

Divisibility by 16:

We've established in the previous response that 1254216 is not divisible by 16.

The last four digits (4216) are not divisible by 16 (4216 ÷ 16 = 264, and 264 × 16 = 4224, not 4216).

Divisibility by 8:

A number is divisible by 8 if its last three digits are divisible by 8.

The last three digits of 1254216 are 216. To check if 216 is divisible by 8, divide 216 by 8: 216 ÷ 8 = 27.

Since 27 is a whole number (integer), 1254216 is divisible by 8.

∴The number 1254216 is divisible only by 8.

Given:

Initial salary = Rs. 8100

Increased salary = Rs. 9000

Formula used:

Percentage increased = (Initial salary – Increased salary)/Initial salary × 100

Calculation:

Initial salary = Rs. 8100

Increased salary = Rs. 9000

Percentage increased = (Initial salary – Increased salary)/Initial salary × 100

⇒ Required percent = (9000 – 8100)/8100 × 100

⇒ Required percent = 900/8100 × 100

⇒ Required percent = 1/9 × 100

⇒ Required percent = 100/9%

N^th term = (2n - 1)² = 4n² + 1 - 4n

For total number of term,

2n - 1 = 15

⇒ n = 8

We know that,

= 4 (1² + 2² + ... + 8²) + (1 + 1 + ..8 times) - 4(1 + 2 + 3 + .. + 8)

⇒ 4[(8 × 9 × 17)/6] + 8 - 4[(8 × 9)/2] = 816 + 8 - 144 = 680