Method Of Case Study Solving Proportions

Finding the missing value in a proportion is much like finding the missing value for two equal fractions. There are three main methods for determining whether two fractions (or ratios) are equivalent.

  • Vertical
  • Horizontal
  • Diagonal (often called "cross-products")

 

Method 1: Vertical

Are these two ratios equivalent?

Since the numerator and denominator are related (by multiplying or dividing by 2), we know these two ratios are equivalent.

Method 2: Horizontal

Are these two ratios equivalent?

Since the numerators are related (by multiplying or dividing by 3) to each other and the denominators are related to each other, we know these two ratios are equivalent.

Method 3: Cross-products

Are these two ratios equivalent?

Since the cross-products are equal to each other, the two ratios are equivalent.

All three methods work for every problem, but often one method is easier to use than the others. Always watch for the easiest method to use!

 


Example 1

The following two ratios are equivalent. Find the missing value.

Answer: Let's use the vertical method on this. Since 3 x 4 = 12, then 5 x 4 = 20. So the missing value is 20.


Example 2

The following two ratios are equivalent. Find the missing value.

Answer: We will use the horizontal method on this one. Since 5 x 2 = 10, then 4 x 2 = 8. So, the missing value is 8.


Example 3

The following two ratios are equivalent. Find the missing value.

Answer: In this problem it is easiest to look at the cross-products.


 

Directions:

  1. Look at the proportion on the right.
  2. Solve it. (You may use the calculator below.)
  3. Then enter your answer and press the "Enter" or "Return" key on your keyboard.
  4. Press the "New problem" button to get a new problem. (Duh.)

 

This free script provided by
JavaScript Kit

 

 


 

Self-Check


Q1: Find the missing value. $\,\,\,\frac{9}{12}=\frac{12}{m}\,\,\,$[show answer]

$m=16$


Q2: Find the missing value. $\,\,\,\frac{18}{12}=\frac{k}{4}\,\,\,$[show answer]

$k=6$


Q3: Find the missing value. $\,\,\,\frac{a}{24}=\frac{5}{20}\,\,\,$[show answer]

$a=6$


 

 

 

 

    What is a Proportion?

    There are a few different ways to define a proportion.  One way to describe a proportion is that it's an equation with two equal ratios.  In other words, a proportion is when you have two fractions with an equals sign in the middle.  Some proportions just have two fractions set equal to each other. Proportions can also have variables in one or both of the fractions.  This lesson will show you how to solve for a variable that's in a proportion.


    How do you Solve a Proportion?

    There's more than one way to solve a proportion.  One way is to cross-multiply.  There's a property you can use called the Means Extremes Property. It says that the cross products of a proportion will be equal.

    Why is this called the Means Extremes Property? The proportion can also be written with colons as a:b = c:d. The extremes are the terms that are furthest apart on the outside: the a and the d.  The means are the terms on the inside: the b and c.  This property says the product of the means is equal to the product of the extremes: ad = bc.

    If you have a variable in your proportion, you can cross-multiply and get an equation that is much easier to solve.  Open the next tab to see some examples.

    Example 1


    There's more than one way to solve this proportion. To solve it by cross-multiplying, you multiply diagonally and set the two cross-products equal to each other.  Multiply the x and the 3 together and set it equal to what you get when you multiply the 2 and the 9 together.

    A common mistake that students make when they cross-multiply is that they forget about the equals sign and write the two parts as a new fraction.  Make sure to always put the equals sign in between the two cross-products.  If you're solving an equation and your equals sign has disappeared, you have a problem!

    Example 2


    First, cross-multiply.  Be extra careful when you simplify each side.  Make sure to distribute the 5 and the 3.

    Example 3


    In this example, you end up multiplying x by itself when you cross-multiply.  x times another x is x-squared.  You can undo squaring the x by taking the square root of both sides.

    Be extra careful if you end up with an x-squared term when you cross-multiply.  This is a quadratic equation and quadratics often have 2 solutions.  In this case, we're looking for a number that equals 36 when we square it.  We can take the square root of both sides, but we need to remember that a calculator will only give the positive solution.  6 squared equals 36, but -6 times -6 also equals a positive 36. You can find similar equations in the lesson on completing the square if you have more questions on this step.

    Example 4


    ​In this problem, you have to multiply x + 3 by x + 9 when you cross-multiply.  When you multiply two binomials by each other (x + 3 and x + 9 are called binomials because they have 2 terms), you need to distribute twice - this is often called the FOIL method.  Make sure to check out the lesson on multiplying binomials if you need help with this step.

    When we cross-multiplied, we ended up with a quadratic equation (it has an x-squared term).  There are several different ways to solve quadratic equations.  You can complete the square, use the Quadratic Formula, or solve by factoring.  It's usually fastest to try and factor if you can, so we recommend always trying to factor first.  In this case, the quadratic we have on the left side can be factored.  Once you have it factored, set each factor equal to 0 and solve.
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