The Numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, then the denominator becomes eight times the numerator, then find the fraction.

Let A, B, C and D pay ?x, ?y, ?z and ?a
According to the question,
x = 1/2(y + z + a)
2x = (y + z + a) ..........(i)
y = 1/3(x + z + a)
3y = (x + z + a) ...........(ii)
z = 1/4(x + y + a)
4z = (x + y + a) ..........(iii)
Also, x + y + z + a = 60 ...............(iv)
Now, put the value of x + y + a = 4z from (iii) in (iv)
Then, 4z + z = 60 ? 5z = 60
? z = 12
Similarly, on putting the value of x + z + a = 3y from (ii) in (iv), we get
3y + y = 60 = ? 4y = 60
? y = 15
Again, on putting the value of (y + z + a) = 2x from (i) in (iv), we get
2x + x = 60 ? 3x = 60
&ther4; x = 20
Now, x + y + z + a = 60
On putting the value of x, y and z, we get
12 + 15 + 20 + a = 60
? a = 60 - 47 = ? 13

Let the original fraction be p/q.
Numerator is increased by 200%.
? Numerator = p + 200% of p
= p + 200p/100
= p + 2p
= 3p
and denominator of the fraction is increased by 150%.
Denominator = q + 150q/100
= (100q + 150q)/100
= 250q/100
= 5q/2
Then, according to the question,
3p/5q/2 = 9/35
3p x 2/5q = 9/35
6p/5q = 9/35
? p/q = 9 x 5/35 x 6
p/q = 3/14

Let the number of pages be p.
Number of pages read on first day = 3p/8
Remaining pages = p - 3p/8 = (8p - 3p)/8 = 5p/8
Number of pages read on second day = 4/5 x 5p/8 = p/2
Now , remaining pages = p - [ 3p/8 + p/2]
= p - [(3p + 4p)/8] = [p- 7p/ 8]
= [(8p - 7p)/ 8] = p/8
According to the question
p/8 = 40
? p = 40 x 8
? p = 320

Let total length of pencil be x cm.
Then, black part = x/8
Remaining part pencil after black part = x - x/8 = 7x/8
then White part = 1/2 (7x/8)
= 7x/16
Now Remaining part of pencil is blue = total length of pencil - (length of black part + length of white part)
length of blue part of pencil = x - (x/8 + 7x/16)
length of blue part of pencil = x - (2x + 7x)/16
length of blue part of pencil = x - 9x/16
length of blue part of pencil = (16 x - 9x)/16
? Length of blue part = 7x/16;

According to question
Length of blue part = 3¹/₂
7x/16 = 3¹/₂
? 7x/16 = 7/2
? x/16 = 1/2
? x = 16/2
? x = 8 cm

Shantanu's pocket allowance = ? 3832.50
Total days in 2011 (general year) = 365 days
Allowance per day = 3832.5/365 = ? 10.5

? = 783/9 x 0.75 = 116

Let us assume length of pole is L.
Length of the pole in Mud is = L x 4/7 = 4L/7
Length of the Pole which is pulled out = L x 1/3 = L/3
Length of the pole is still in mud = Length of the pole in Mud - Length of the Pole which is pulled out
Length of the pole is still in mud = 4L/7 - L/3
According to question,
Length of the pole is still in mud = 250
4L/7 - L/3 = 250
(4L x 3 - L x 7)/7 x 3 = 250
12L - 7L = 250 x 21
5L = 250 x 21
L = 250 x 21/5
L = 50 x 21
L = 1050

x³ + 1 + 2x =6x + 1/x
x³ + 1 + 2x = (6x² + 1)/x
(x³ + 1 + 2x)x = 6x² + 1
x⁴ - 4x² - 6x² -1 =0
x⁴ - 4x² + x - 1 =0
Degree of polynomial is highest exponent degree term i.e.,4.

x = 1 + ?2
? x⁴ - 4x³ + 4x² = x²(x² - 4x + 4)
= x²(x - 2)²
= (1 + ?2)²(1 + ?2 - 2)²
=(?2 + 1)² (?2 - 1)²
=[(?2)² - (1)²]²
=(2 - 1)² =1

x - y = 0.9 ...(i)
and 11(x + y)^-1=2
? 11/ (x + y) = 2
? 2(x + y) =11
? x + y = 11/2 ...(ii)
On solving Eqs.(i) and (ii),we get
x = 3.2
and y = 2.3