SRCC GBO 2022

Let the number subtracted be x. Hence, the terms are 8-x, 10-x, and 14-x.

10-x = $$\ \frac{\ 2\left(8-x\right)\left(14-x\right)}{8-x+14-x}$$

On cross multiplying, we get x as -2. 

The population after 3 years will be: 10,000 $$\times\ 1.1\times\ 1.15\times\ 0.95$$, which is equal to 12,018 (approximately).

The above expression can be simplified as $$\log\ \ \frac{\ 27}{125}\times\ \ \frac{\ 21}{x}\times\ \ \frac{\ 625}{81}$$ = 0

On solving the same, we get $$\log\ \ \frac{\ 35}{x}=\ 0$$, for which x shall be 35, making $$\log1=\ 0$$

Let the total number of people be X. 

1) When X is divided by 15, we get a remainder of 10. Hence, X = 15a + 10

2) When X is divided by 20, we get a remainder of 15. Hence, X = 20b + 15

3) When X is divided by 24, we get a remainder of 19. Hence, X = 24c + 19

4) When X is divided by 28, we get a remainder of 23. Hence, X = 28d + 23

This is a classic case when the difference between the divisor and remainder is the same, which is 5 in this case. In such instances, the number is expressed as (LCM of the divisors)$$\times$$K - (difference between the divisors and remainders).

Hence, the number of people shall be 840k - 5.

The minimum number is when k=1, hence the minimum number shall be 835.

Let the sides of the polygons be 2x and 3x. 

Hence, the ratio of the sum of interior angles for the two polygons is: $$\ \frac{\ \left(2x-2\right)\times\ 180}{\left(3x-2\right)\times\ 180}$$ = $$\ \frac{\ 5}{8}$$

On solving this, x=6

This increases the number of sides in the first polygon to 12 and 18 for the second one.

To find the measure of each interior angle of the second polygon, we get $$\ \frac{\ \left(18-2\right)\times\ 180}{18}$$, which is 160 degrees.

Revenue is the product of spending and consumption. Hence, this is a case of successive percentage change.

The multiplying factor of reduction of 20% is 0.8, while that of increase by 15% is 1.15.

Assuming the initial revenue be X, we get the new revenue as X$$\times\ 1.15\times\ 0.80$$, which 0.92X.

This means there is a reduction of 8% in the revenue.

The radius of the bigger circle is 5 cm while of the smaller circle is r cm.

Since the two sides of the wall are tangential to the circles, we get a square formed by the radius of the bigger circle and the two sides wall, each with the length of 5 cm. 

Now, the diagonal of this square (s1) shall be $$5\sqrt{\ 2}$$.

Focusing on the smaller circle, on which the wall is tangential again. We get a square here as well, the diagonal of which is  $$r\sqrt{\ 2}$$.

Hence, the diagonal of s1: $$5\sqrt{\ 2}$$ = r +  $$r\sqrt{\ 2}$$ + 5

On solving this, we get r = $$15 - 10sqrt{\ 2}$$

Let us assume the number be 'x'.

Also, let us assume that the divisor is 'n'

Hence, x = n(a) + 20, where 'a' is the quotient ............ (1)

Also, 3x = n(b) + 7, where 'b' is the quotient ................(2)

Dividing (2) by (1), we get n (b-3a) = 53

Now, 53 is a prime number, and it can be broken into 53$$\times$$1.

Further, it is clear that the divisor cannot be 1, hence n = 53

In this scenario, the ant has three options: 

1) Go to C directly from A, for which it has 3 paths. 

2) Use the path A-B-C, for which it has 5$$\times\ $$4, which is 20 paths. 

3) Use the path A-E-D-C, for which it has 3$$\times\ $$5$$\times\ $$6, or 90 paths.

Adding the above numbers, we get 113 as the total number of ways in which the ant can reach C from A.

Let the number be 5x. 

As per the question, 2x = x+5, or x = 5. 

Hence, 5x = 25