The numerator of a fraction is two less than its denominator. When both numerator and denominator are increased by 3, the fraction is incre
ased by 3/20. Find the original fraction. Pls include solution and workings thx :)
first denominator -- x
first numerator ---- x-2
new denominator -- x+3
new numerator ---- x+1
(x-2)/x + 3/20 = (x+1)/(x+3)
times 20x(x+3) , the LCD
20(x+3)(x+1) + 3x(x+3) = 20x(x+1)
expanding and simplifying gave me
x^2 + 3x - 40 = 0
(x-5)(x+8) = 0
x = 5 or x=-8
if x=5, the original fraction was 3/5
if x=-8 the original fraction was -10/-8 or 5/4
check for 3/5 , new fraction would be 6/8 or 3/4
3/5 + 3/20 = 12/20 + 3/20 = 15/20 = 3/4
but for 5/4, new fraction would be 8/7
5/4 + 3/20 = 28/20 = 7/5 β 8/7
BUT, if we take the unsimplified fraction -10/-8 , new fraction would be -7/-5 = 7/5
So the original fraction would be 3/5 for sure, but
also the unsimplified fraction -10/-8
There are 2 solutions and I've done till the first one .. is it alright ??? The last thing was .. if x=-8 the original fraction was -10/-8 or 5/4 !!!
Is it completed till here ???
Pls help β€
Too much hard solution
i do not understand
:-: this is making my head much more worse....
I can't solve this question just and just bcz its fraction in this too huh plz help me how to solve it plz π₯Ίπ₯Ίππ
To solve this problem, let's first assume the fraction is represented as (x-2)/x, where x represents the denominator.
According to the problem, when both the numerator and denominator are increased by 3, the new fraction is increased by 3/20. So, the new fraction is ((x-2)+3)/(x+3), which can be simplified as (x+1)/(x+3).
Now we can set up an equation based on the information given:
((x+1)/(x+3)) - ((x-2)/x) = 3/20
To simplify, we can get rid of the fractions by multiplying every term by the common denominator, which in this case is x(x+3):
(x+1)(x) - (x-2)(x+3) = (3/20) * x(x+3)
Simplifying further:
x^2 + x - (x^2 - x - 6) = (3/20)(x^2 + 3x)
x^2 + x - x^2 + x + 6 = (3/20)x^2 + (9/20)x
Combining like terms:
2x + 6 = (3/20)x^2 + (9/20)x
Multiplying both sides by 20 to eliminate the fractions:
40x + 120 = 3x^2 + 9x
Rearranging the equation:
3x^2 + 9x - 40x - 120 = 0
3x^2 - 31x - 120 = 0
Now, we can factorize or use the quadratic formula to solve for x. However, this equation does not factor nicely, so we will use the quadratic formula:
x = [-b Β± β(b^2 - 4ac)] / (2a)
Using a = 3, b = -31, and c = -120:
x = [31 Β± β((-31)^2 - 4(3)(-120))] / (2*3)
Simplifying further:
x = [31 Β± β(961 + 1440)] / 6
x = [31 Β± β(2401)] / 6
x = [31 Β± 49] / 6
This yields two possible values for x: x = 6 and x = -3. However, since we are referring to the denominator of a fraction, it must be a positive number. Thus, x = 6.
Therefore, the original fraction is (x-2)/x = (6-2)/6 = 4/6, which can be simplified to 2/3.
Numerator = x New Numerator = x+3
Denominator = x+2 New Denominator = x+5
x/(x+2) +3/20 = (x+3)/(x+5)
(20x+3x+6)/(20x+40) = (x+3)/(x+5)
Cross Multiply
(x+5)(23x+6) = (x+3)(20x+40)
23x^2+6x+115x+30 = 20x^2+40x+60x+120
23x^2 +121x+30 = 20x^2 +100x+120
Take one of the equations to the other side to simplify:
23x^2 +121x+30 -(20x^2 +100x+120) =0
3x^2 -21x-90 =0
Divide by 3
x^2 -7x-30=0
x^2 +10x-3x-30=0
x(x+10) -3(x+10) =0
(x+10)(x-3) =0
x= -10 or x =3
x =3 (because its positive)
so
x+2 =3+2 =5
The Original fraction was : x/x+2 3/5 <---- Answer
Check:
3/5+3/20 = 3+3/3+5
15/20 = 6/8
3/4 =3/4
Correct!
3/5
Let denominator and numerator be x
The numerator is 2 less than its denominator so
X-2/x
It will increased by 3 and will become 3/20 so