Rational Equations Using Proportions

Key Questions

  • Answer:

    Solving proportions is like solving fractions:

    a:b -> c:d can be rewritten as a/b = c/d and now you can solve for any of the variables.

    See an example below:

    Explanation:

    3 to 4 is like what to 16?

    This can be rewritten as a proportion:

    3:4 -> c:16

    Which can be rewritten as:

    3/4 = c/16

    Which can be solved as:

    color(red)(16) xx 3/4 = color(red)(16) xx c/16

    cancel(color(red)(16))color(red)(4) xx 3/color(red)(cancel(color(black)(4))) = cancel(color(red)(16)) xx c/color(red)(cancel(color(black)(16)))

    12 = c

    c = 12

    3 to 4 is like 12 to 16?

  • We multiply the numerator of each (or one) side by the denominator of the other side.

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    For example, if I have want to solve for x for the following equation:

    x/5=3/4

    I can use cross-multiplication, and the equation becomes:

    x*4=3*5

    4x=15

    x=15/4=3.75

  • A proportion is a statement that two ratios are equal to each other.
    For example 3/6=5/10 (We sometimes read this "3 is to 6 as 5 is to 10".)

    There are 4 'numbers' (really number places) involved. If one or more of those 'numbers' is a polynomial, then the proportion becomes a rational equation.

    For example: (x-2)/2=7/(x+3) ("x-2 is to 2 as 7 is to x+3").

    Typically, once they show up, we want to solve them. (Find the values of x that make them true.)

    In the example we would "cross multiply" or multiply both sides by the common denominator (either description applies) to get:
    (x-2)(x+3)=2*7. Which is true exactly when
    x^2+x-6=14 Which in turn, is equivalent to
    x^2+x-20=0 (Subtract 14 on both sides of the equation.)
    Solve by factoring (x+5)(x-4)=0
    so we need x+5=0 or x-4=0 the first requires
    x=-5 and the second x=4.

    Notice that we can check our answer:
    (-5-2)/2=-7/2 and 7/(-5+3)=7/-2=-7/2. So the ratios on both sides are equal and the statement is true.

Questions