The sum of three numbers is 300. If the ratio between first and second be 5:9 and that between second and third be 9:11, then second number is

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.

Let us assume father age is F and his son age is S.
According to question,
The sum of the ages of father and his son is 4 times the age of the son,
F + S = 4S
F = 3S.............. (1)
Average age of father and son is 28.
(F + S)/2 = 28
F + S = 56
Put the value of F from equation from (1),
3S + S = 56
4S = 56
S = 14 years
Age of Son = 14 years.

Method 1
Let us assume the number be 3p , 3(p+1) and 3(p+2)
According to question,
3p + 3(p+1) + 3(p+2) = 72
? 3p + 3p + 3 +3p + 6 = 72
? 9p +9 = 72
? 9p = 72 - 9
? 9p = 63
? p = 63/9 = 7
? Largest number = 3(p + 2)
Put the value of p in above equation.
? Largest number = 3 x ( 7 + 2 )
? Largest number = 3 x 9
? Largest number = 27

Method 2
Let us assume the numbers P , P + 3 ,P+ 6
According to Question,
sum of three numbers = 72
P + P + 3 + P + 6 = 72
? 3P + 9 = 72
? 3P = 72 - 9
? 3P = 72 - 9
? P = 63/3
? P = 21
So largest Number = P + 6 = 21 + 6 = 27

Method 1
Let us assume the ratio factor is x.
Therefor the present ages of Vikas and Vishal be 15x years and 8x years.
After 10 years
vikas's age = 15x + 10 and Vishal' age = 8x + 10
According to question,
(15x+10)/(8x+10) = 5/3
? 3(15x + 10) = 5(8x + 10)
? 45x + 30 = 40x + 50
? 5x =20
? x = 20/5
? x = 4
Therefore Present age of Vikas = 15x = 15 x 4 = 60 years
and Present age of Vishal = 8x = 8 x 4 = 32 years

Method 2
Let us assume the present age of Vikas = x years and Vishal's present year = y years
According to question,
Present age of Vikas/ Present age of Vishal = 15/8
? x/y = 15/8
? 8x = 15y
? 8x - 15y = 0
? x = 15y/8 .................................(1)
After 10 years Vikash age = x + 10 and vishal age = y + 10
Ratio of age after 10 years = 5/3
(x + 10)/(y + 10) = 5/3
? 3(x + 10) = 5(y + 10)
? 3x + 30 = 5y + 50
? 3x - 5y = 50 - 30
? 3x - 5y = 20 ................................(2)
Put the value of x from equation (1) in above equation (2).
? 45y/8 - 5y = 20
? 45y - 40y = 20 x 8
? 5y = 20 x 8
? y = 4 x 8 = 32
Put the value of Y in equation (1)
x = 15 x 32/8 = 15 x 4 = 60

Therefore Present age of Vikas = x = 60 years
and Present age of Vishal = y = 32 years

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 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

Cost of 5 pens + 8 pencils = ?31
On multiplying by 3
15 pens + 24 pencils = 3 x 31 =
?15 pens + 24 pencils = 93

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