Rational Equations Using Proportions

Key Questions

  • How do you solve proportions?

    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?

    smendyka · 1 · Jun 3 2018
  • What is cross multiplication?

    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#

    Tanish J. · 2 · Oct 26 2014
  • What is Rational Equations Using Proportions?

    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.

    Jim H · 1 · Mar 6 2015

Questions

Rational Equations and Functions