Simplifying Ratios

The ratio of the length of the shorter leg to the length of the longer leg for this right triangle is 4 to 6. This ratio can be written in three ways.

• 4 to 6

• 4 6 4 sixths

• 4 : 6

Example

Algebra Simplify each ratio.

1. 4 to 6

   4  to  6 = 4 6     = 2 ⋅ 2 2 ⋅ 3 Find and remove the common factor .   = 2 3   table with 3 rows and 4 columns , row1 column 1 , 4 , to , 6 , column 2 equals , column 3 4 sixths , column 4 , row2 column 1 , , column 2 equals , column 3 fraction 2 dot 2 , over 2 dot 3 end fraction , column 4 cap findandremovethecommonfactor . . , row3 column 1 , , column 2 equals , column 3 2 thirds , column 4 , end table

2. 3ab : 27ab

   3 a b : 27 a b = 3 a b 27 a b   = 3 a b ⋅ 1 3 a b ⋅ 9   = 1 9 table with 3 rows and 3 columns , row1 column 1 , 3 eh b colon 27 eh b , column 2 equals , column 3 fraction 3 eh b , over 27 eh b end fraction , row2 column 1 , , column 2 equals , column 3 fraction 3 eh b dot 1 , over 3 eh b dot 9 end fraction , row3 column 1 , , column 2 equals , column 3 1 ninth , end table

3. 4 a + 4 b a + b fraction 4 bold italic eh plus 4 bold italic b , over bold italic eh plus bold italic b end fraction

   4 a + 4 b a + b = 4 ( a + b ) a + b Factor the numerator . The denominator cannot be factored . Remove the common factor  ( a + b ) .   = 4   table with 2 rows and 4 columns , row1 column 1 , fraction 4 eh plus 4 b , over eh plus b end fraction , column 2 equals , column 3 fraction 4 open eh plus b close , over eh plus b end fraction , column 4 table with 2 rows and 1 column , row1 column 1 , cap factorthenumerator . .cap thedenominatorcannotbe , row2 column 1 , factored . .cap removethecommonfactor . open eh plus b close . , end table , row2 column 1 , , column 2 equals , column 3 4 , column 4 , end table

Exercises

Algebra Simplify each ratio.

1. 25 to 15
2. 6 : 9
3. 36 54 36 over 54
4. 0.8 to 2.4
5. 7 14 x fraction 7 , over 14 x end fraction
6. 12 c 14 c fraction 12 c , over 14 c end fraction
7. 22 x 2 22 , x squared  to 35x
8. 0.5ab : 8ab
9. 4 x y 0.25 x fraction 4 x y , over 0.25 , x end fraction
10. 1 1 2 x 1 , and 1 half , x  to 5x
11. x 2 + x 2 x fraction x squared , plus x , over 2 x end fraction
12. 1 4 r 2 1 fourth , r squared  to 6r
13. 0.72t : 7.2 t 2 7.2 , t squared
14. ( 2 x − 6 open 2 x minus 6 ) : ( 6 x − 4 open 6 x minus 4 )
15. 12xy : 8x
16. ( 9 x − 9 y open 9 x minus 9 y ) to ( x − y ) open x minus y close
17. π r r 2 + π r fraction pi r , over r squared , plus pi r end fraction
18. 8 a b 32 x y fraction 8 eh b , over 32 x y end fraction

Express each ratio in simplest form.

19. shorter leg : longer leg
20. hypotenuse to shorter leg
21. shorter leg hypotenuse fraction shorterleg , over hypotenuse end fraction
22. longer leg hypotenuse fraction longerleg , over hypotenuse end fraction
23. longer leg to shorter leg
24. hypotenuse : longer leg

    

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