The cost of 5 pens and 8 pencils is ?31.What would be the cost of 15 pens and 24 pencils?

Let us assume 1 pen cost be ? x and 1 pencil cost be ? y.
According to question,
The Cost of 36 pens and 42 pencils is 460.
36x + 42y = 460 ...................(1)
then the cost of 18 pens and 21 pencils = 18x + 21y.
Now divide the Equation (1) by 2. We will get,
18x + 21y = 230
So the Cost of 18 pens and 21 pencils = 18x + 21y = 230

Let the number be x.
According to question,
x - 20 = 7x/12
? x - 7x/12 = 20
? (12x - 7x)/12 = 20
? 5x/12 = 20
? x = 20 x 12/5
? x = 4 x 12
? x = 48
Sum of the digits of 48 = 4 + 8 = 12

Let us assume the ratio factor is x.
Then the three numbers be 5x , 9x and 11x respectively.
Then according to question,
5x + 9x + 11x = 300
? 25x = 300
? x = 12
So, the second number is 9x = 9 x 12 =108

Let the middle number = x , then first number = x - 2 and last number = x + 2
(x - 2) + x + (x + 2) = 176/4 - 14
x - 2 + x + x + 2 = 44 - 14
3x = 30
x = 10
So middle number = x = 10

Let the two numbers be x and y .
According to question,
? xy = 192...................... (1)
x + y = 28........................(2)
As we know that,
(x - y)² = (x + y)² - 4xy
Put the value of xy and x + y from equation (1) and (2), we will get
(x - y)² = (28)² - 4 x 192
? (x - y)² = 784 - 786
? (x - y)² = 16
? x - y = 4......................(3)
Add the equation (2) and (3), we will get
x + y + x - y = 28 + 4
2x = 32
x = 16
Put the value of x in equation (2), we will get
16 + y = 26
y = 28 - 16
y = 12

So numbers are x = 16 and y = 12.
Smaller number is 12.

2x + 3y = 29 ...(i)
and y = x + 3 ...(ii)
Putting the value of y from Eq. (ii) to Eq. (i), we get
2x + 3y = 29
2x + 3(x + 3) = 29
2x + 3x + 9 = 29
? 5x = 20
? x = 4

3x + 7y = 75 ..(i)
5x - 5y = 25
? x - y = 5 ...(ii)
On multiplying Eq. (ii) by 7 and adding to Eq. (i), we get
3x + 7y = 75
7x - 7y = 35
------------------
10x = 110
x = 11
? Putting the value of x in Eq. (ii), we get
11 - y = 5
? y = 6
? x + y = 6 + 11 = 17

we have, x + y - 7 = 0
? x + y = 7 ...(i)
and 3x + y - 13 = 0
? 3x + y = 13 ..(ii)
By subtracting Eq. (i) from Eq. (ii), we get
3x + y = 13
x + y = 7
----------------------
2x = 6 ...(ii)
? x = 3

On putting the value of x in Eq. (i), we get
3 + y = 7
? y = 4

Now, 4x² + y² + 4xy
= 4 x (3)² + (4)² + 4 x 3 x 4
= 4 x 9 + 16 + 48
= 36 + 16 + 48 = 100

Given, 3^{x + y} = 81
? 3^{x + y} = 3⁴
? x + y = 4 ...(i)
and 81^{x - y} = 3 or (3⁴)^{x - y} = 3
? x - y = 1/4
On solving the Eqs. (i) and (ii), we get
2x = 17/4 ? x = 17/8

Given equation are 6x + 3y = 7xy
? (6/y) + (3/x) = 7 .....(i)

And 3x + 9y = 11xy
? (3/y) + (9/x) = 11 .....(ii)

On multiplying Eq. (ii) by 2 and subtracting from Eq. (i), we get
(6/y) + (3/x) = 7
(6/y) + (18/x) = 22
-------------------------
- 15/x = - 15
? x = 1

On putting this value in Eq. (i), we get
(6/y) + 3 = 7
? 6/y = 4
? y = 6/4 = 3/2