Take note Key Concept: Properties of Proportions

a, b, c, and d do not equal zero.

Property

1. a b = c d eh over b , equals , c over d  is equivalent to b a = d c . b over eh , equals , d over c , .
2. a b = c d eh over b , equals , c over d  is equivalent to a c = b d . eh over c , equals , b over d , .
3. a b = c d eh over b , equals , c over d  is equivalent to a + b b = c + d d . fraction eh plus b , over b end fraction . equals . fraction c plus d , over d end fraction . .

How to Apply It

Write the reciprocal of each ratio.

becomes 3 2 = 6 4 . 3 halves , equals , 6 fourths , .

Switch the means.

becomes 2 4 = 3 6 . 2 fourths , equals , 3 sixths , .

In each ratio, add the denominator to the numerator.

2 3 = 4 6 2 thirds , equals , 4 sixths  becomes 2 + 3 3 = 4 + 6 6 . fraction 2 plus 3 , over 3 end fraction . equals . fraction 4 plus 6 , over 6 end fraction . .

Problem 5 Writing Equivalent Proportions

In the diagram, x 6 = y 7 . x over 6 , equals , y over 7 , .  What ratio completes the equivalent proportion x y = □ □ ? x over y , equals , fraction white square , over white square end fraction , question mark  Justify your answer.

• Method 1

  x 6 = y 7   x y = 6 7 Property of Proportions  ( 2 ) table with 2 rows and 2 columns , row1 column 1 , x over 6 , equals , y over 7 , column 2 , row2 column 1 , x over y , equals , 6 sevenths , column 2 cap propertyofcap proportions . open 2 close , end table

  Plan

  How do you decide which property of proportions applies?

  Look at how the positions of the known parts of the incomplete proportion relate to their positions in the original proportion.

• Method 2

  x 6 = y 7   7 x = 6 y Cross Products Property 7 x 7 y = 6 y 7 y To solve for  x y ,  divide each side by  7 y . x y = 6 7 Simplify . table with 4 rows and 3 columns , row1 column 1 , x over 6 , column 2 equals , y over 7 , column 3 , row2 column 1 , 7 x , column 2 equals 6 y , column 3 cap crosscap productscap property , row3 column 1 , fraction 7 x , over 7 y end fraction , column 2 equals . fraction 6 y , over 7 y end fraction , column 3 cap tosolvefor . x over y , comma . divideeachsideby . 7 y . , row4 column 1 , x over y , column 2 equals , 6 sevenths , column 3 cap simplify , . , end table

The ratio that completes the proportion is 6 7 . 6 sevenths , .

✓ Got It?

5. For parts (a) and (b), use the proportion x 6 = y 7 . x over 6 , equals , y over 7 , .  What ratio completes the equivalent proportion? Justify your answer.
   1. 6 x = □ □ 6 over x , equals , fraction white square , over white square end fraction
   2. □ □ = y + 7 7 fraction white square , over white square end fraction , equals . fraction y plus 7 , over 7 end fraction
   3. Reasoning Explain why 6 x − 6 = 7 y − 7 fraction 6 , over x minus 6 end fraction . equals . fraction 7 , over y minus 7 end fraction  is an equivalent proportion x 6 = y 7 . x over 6 , equals , y over 7 , .