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Solving Simple Proportions (page 4 of 5)

Sections: Ratios, Proportions, Checking proportionality, Solving proportions

Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying, and solving the resulting equation. You'll probably start out by just solving proportions, like this:

  • Find the unknown value in the proportion:  2 : x = 3 : 9.

      • 2 : x = 3 : 9

    First, I convert the colon-notation ratios to fractions:

      .2/x  =  3/9

    Then I solve:

      .2/x  =  3/9
      18 = 3x
      6 = x

  • Find the unknown value in the proportion:  (2x + 1) : 2 = (x + 2) : 5

      • (2x + 1) : 2 = (x + 2) : 5

    First, I convert the colon-notation ratios to fractions:

      (2x + 1)/2   =  (x + 2)/5

    Then I solve:

      .(2x + 1)/2   =  (x + 2)/5
      5(2x + 1) = 2(x + 2)
      10x + 5 = 2x + 4
      8x = –1
      x = –1/8

Once you've solved a few proportions, you'll move into word problems, where you'll have to invent the proportion, extracting it from the word problem, before solving it.

  • If twelve inches correspond to 30.48 centimeters, how many centimeters are there in thirty inches?

      • .(inches)/(centimeters) : 12/30.48  =   30/c
      .12/30.48  =  30/c
      12c = (30)(30.48)
      12c = 914.4
      c = 76.2

    Thirty inches corresponds to 76.2 cm.

  • A metal bar ten feet long weighs 128 pounds. What is the weight of a similar bar that is two feet four inches long?

    First, I need to convert the "two feet four inches" into feet: four inches is four-twelfths, or one-third, of a foot, so:

      2 feet + 4 inches = 2 feet +   1/3 feet =  7/3 feet

    Then set up the proportion and solve:

      .(length)/(weight) :  10/128  =   ( 7/3 )/w
      .10/128  =   ( 7/3 )/w
      10w = 128( 7/3 )
      10w =  896/3
      w =  896/30

    Since this is a word problem, you may want to round to a practical number. The answer is:
    The bar will weigh 896/30 pounds, or about 29.87 pounds.

  • The tax on a property with an assessed value of $7000 is $1100. What is the assessed value of a property if the tax is $1400?

      • .(value)/(tax) :  7000/1100  =  v/1400
      .7000/1100  =  v/1400
      9,800,000 = 1100v
      8909.09090909... = v

    Since this is a dollars-and-cents problem, I need to round the final answer to two decimal places:

    The assessed value is $8909.09. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

  • One piece of pipe 21 meters long is to be cut into two pieces, with the lengths of the pieces being in a 2 : 5 ratio. What are the lengths of the pieces?

    I'll label the length of the short piece as "x". Then the long piece must have a length of 21 – x.

      (short piece) : (long piece):  2 : 5 = x : (21 – x)
      .2/5   =  x/(21 – x)
      2(21 – x) = 5x
      42 – 2x = 5x
      42 = 7x
      6 = x

    Then the length of the long piece is given by:

      21 – x = 21 – 6 = 15

    The pieces are 6 meters long and 15 meters long.

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Cite this article as:

Stapel, Elizabeth. "Solving Simple Proportions." Purplemath. Available from
    http://www.purplemath.com/modules/ratio4.htm. Accessed
 

 

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