Problem: A fraction is such that if the double of the numerator and triple of denominator is changed by 10% and -30% respectively then we get 33% of 16/21. Find the fraction?

Solution:

Let the fraction be represented as x/y.

According to the given condition,
2x + 3y - (10/100)(2x + 3y) - (-30/100)(2x + 3y) = (33/100) * (16/21)

Simplifying the above equation, we get
2x + 3y - (3/20)(2x + 3y) + (3/10)(2x + 3y) = (16/100)
2x + 3y + (3/20)(2x + 3y) = (16/100) + (3/10)(2x + 3y)
2x + 3y + (3/10)(2x + 3y) = (16/100) + (3/20)(2x + 3y)
10x + 15y + 9x + 13.5y = 8 + 3x + 4.5y
19x + 23.5y = 8 + 3x + 4.5y
16x + 19.5y = 8

Now, we need to find the fraction x/y which satisfies the above equation.

Method 1: Using Trial and Error
We can try different values of x and y such that they satisfy the above equation. For example,
If we take x = 1 and y = 2, then
16(1) + 19.5(2) = 51
which is not equal to 8
If we take x = 2 and y = 3, then
16(2) + 19.5(3) = 73.5
which is not equal to 8
If we take x = 4 and y = 1, then
16(4) + 19.5(1) = 75.5
which is not equal to 8
If we take x = 1 and y = 4, then
16(1) + 19.5(4) = 84
which is not equal to 8

Hence, we can see that it is difficult to find the fraction using the method of trial and error.

Method 2: Using Linear Equations
We can rewrite the equation 16x + 19.5y = 8 as
y = (-16/19.5)x + (8/19.5)

Now, we know that x/y = x/((-16/19.5)x + (8/19.5))
Simplifying the above equation, we get
x/y = (19.5x)/(8 - 16x)

Now, we can substitute different values of x such that the above equation satisfies the condition x/y. For example,
If we take x = 1, then
x/y = (19.5(1))/(8 - 16(1)) = -0.975
which is not a valid fraction
If we take x =

Is the answer 6/25??

let the fraction be x/y
twice the numerator is 2x and it is changed by 10% so numerator be 2x*1.1=2.2x
thrice the denominator is 3y and it is changed by -30% so denominator be 3y*(1-0.3)=2.1y

2.2x/2.1y=( 33/100)*(16/21)
22x/21y= (33*16)/(100*21)
x/y=6/25