# Math

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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 :)

• Math -

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 ≠ 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

• Math -

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

• Math -

3/5

• Math -

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!

• Math -

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!

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