GED Mathematical Reasoning: Proportions

 

A proportion is a statement that two ratios or rates are equal. Showing on the slide is an example of a proportion:

\frac{12}{9} = \frac{4}{3}

Visually, notice that a proportion is composed of two fractions and an equal sign.

It is important to note, though, that a proportion may or may not be TRUE.  The example presented here IS true since 12 over 9 and 4 over 3 are equivalent fractions, meaning:  ‘12 over 9’   is, in fact, equal to  ‘4 over 3.’

Another way to analyze a proportion is to consider the cross products. The cross products of a TRUE proportion are equal. To find each cross product, multiply the two numbers appearing diagonally from one another.

In this example: 12 times 3 is equal to 36. And 9 times 4 is also equal to 36.

\frac{12}{9} = \frac{4}{3}

12 \times 3 = 9 \times 4

 36 = 36

To solve a proportion for the missing number follow these steps:

STEP 1: Let the missing number be represented by a letter, like “x” or “n.”

STEP 2: Write the cross products separated by an equal sign. One of the cross products will contain a letter. Leave that one alone, but go ahead and do the multiplication for the cross product containing only numbers. We’ll call that number the “complete cross product.”

STEP 3: To find the missing number, divide the ‘complete cross product’ by the number that’s with the letter.

 

Example 1

What is the value of x in the following proportion?

\frac{x}{32} = \frac{3}{4}

To solve a proportion for the missing number, follow these steps:

Step 1: Let the missing number be represented by a letter, like “x” or “n”. In our example, this is already done.

Step 2: Write the cross products separated by an equal sign. One of the cross products will contain a letter. Leave that one alone, but go ahead and do the multiplication for the cross product containing only numbers. We’ll call that number the “complete cross product.”

\frac{x}{32} = \frac{3}{4}

 x \times 4  = 32 \times 3

 x \times 4 = 96

Step 3: To find the missing number, divide the ‘complete cross product’ by the number that’s with the letter

To finish our example and find the missing number “x,” we’ll divide 96 by the number that’s with x, which is a 4. 96 divided by 4 equals 24.

x = \frac{96}{4} = 24

Therefore, the missing number is 24, and the complete proportion can be written as:

\frac{24}{32} = \frac{3}{4}

And we can check our work by verifying that the cross products are equal: 24 times 4 equals 96. 32 times 3 is also equal to 96.

 

Example 2

A roadmap scale shows that a length of 3 centimeters on the map is equal to a distance of 120 miles. If two cities are approximately 5 centimeters apart on the map, how far apart are they in miles?

 

To answer this question, we need to first set up a proportion based on the information given. For this example, we’ll begin by writing a fraction that represents the ratio of centimeters to miles. We are given that 3 centimeters is equal to 120 miles, which we can write as 3 centimeters over 120 miles.

\frac{3\, centimeters}{120\, miles}

To continue, we’ll insert an equal sign.

\frac{3\, centimeters}{120\, miles} =

To write the second fraction in the proportion we must be consistent — place centimeters in the numerator and miles in the denominator, just like the first fraction. And for the value that we don’t know and are trying to find, use a letter like “x” or “n.”

So for the second fraction, we’ll write “5 centimeters” over “n miles:”

\frac{3\, centimeters}{120\, miles} = \frac{5\, centimeters}{n\, miles}

\frac{3}{120} = \frac{5}{n}

And we’ll do so by writing the cross products separated by an equal sign: 3 times n equals 120 times 5.

3 \times n = 120 \times 5

We’ll leave the left side alone and multiply the right side.  120 times 5 is 600 and we’ll place that on the right side as our “complete cross product”.

3 \times n = 600

To find the missing number “n,” we’ll divide 600 by the number that’s with n, which is a 3. 600 divided by 3 equals 200.

 n = \frac{600}{3} = 200

Therefore, the number of miles represented by 5 centimeters on the map is 200 miles. The complete proportion can be written as: 3 centimeters over 120 miles equals 5 centimeters over 200 miles

\frac{3\, centimeters}{120\, miles} = \frac{5\, centimeters}{200\, miles}

And we can check our work by verifying that the cross products are equal:

3 times 200 equals 600. 120 times 5 is also equal to 600.

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