## Properties of Proportions

The four properties that follow are not difficult to justify algebraically, but the details will not be presented here.

No. If this were a proportion,

If 8/10 = 4/5, then 5/10 = 4/8, or if 8/10 = 4/5, then 8/4 = 10/5.

Use the switching property of proportions and switch the means positions, the 5 and the

First, apply the converse of the

Next, proceed in one of the following two ways:

Recall that

Let

Apply the

The cities are 16⅔ miles apart.

Source-Cliffs Notes http://www.cliffsnotes.com/study_guide/Properties-of-Proportions.topicArticleId-18851,articleId-18810.html

*Property 1 (Means-Extremes Property, or Cross-Products Property):*If*a/b*=*c/d*, then*ad =bc*.*Conversely, if ad = bc*≠ 0, then and .**Example 1:**Find*a*if*a*/12 = 3/4.*By Property 1:***Example 2:**Is 3 : 4 = 7 : 8 a proportion?No. If this were a proportion,

*Property 1*would produce*Property 2 (Means or Extremes Switching Property):*If*a*/*b*=*c*/*d*and is a proportion, then both*d*/*b*=*c*/*a*and*a*/*c*=*b*/*d*are proportions.**Example 3:**8/10 = 4/5 is a proportion.*Property 2*says that if you were to switch the 8 and 5 or switch the 4 and 10, then the new statement is still a proportion.If 8/10 = 4/5, then 5/10 = 4/8, or if 8/10 = 4/5, then 8/4 = 10/5.

**Example 4:**If*x*/5 =*y*/4, find the ratio of*x*/*y.*Use the switching property of proportions and switch the means positions, the 5 and the

*y.**Property 3 (Upside-Down Property):*If*a*/*b*=*c*/*d*, then*b*/*a*=*d*/*c*.**Example 5:**If 9*a*= 5*b*≠ 0, find the ratio .First, apply the converse of the

*Cross Products Property*and obtain the following:Next, proceed in one of the following two ways:

- Apply
*Property 3*to 9/5 =*b*/*a:*

- Turn each side upside-down.
- 5/9 =
*a/b*, or*a/b*= 5/9

- Apply
*Property 2*to 9/*b*= 5/*a:*

- Switch the 9 and the
*a.* *a*/*b*= 5/9

- Switch the 9 and the

*Property 4 (Denominator Addition/Subtraction Property):*If*a*/*b*=*c*/*d*, then (*a*+*b*)/*b*= (*c*+*d*)/*d*or (*a*−*b*)/*b*= (*c*−*d*)/*d.***Example 6:**If 5/8 =*x*/*y*, then 13/8 = ?**Example 7:**In Figure 1 ,*AB*/*BC*= 5/8. Find*AC*/*BC*.
| |||

Recall that

*AB*+*BC*=*AC**(Segment Addition Postulate).***Example 8:**A map is scaled so that 3 cm on the map is equal to 5 actual miles. If two cities on the map are 10 cm apart, what is the actual distance the cities are apart?Let

*x*= the actual distance.Apply the

*Cross-Products Property*.The cities are 16⅔ miles apart.

Source-Cliffs Notes http://www.cliffsnotes.com/study_guide/Properties-of-Proportions.topicArticleId-18851,articleId-18810.html

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