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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