2-6 Ratios, Rates, and Conversions

Quick Review

A ratio between numbers measured in different units is called a rate. A conversion factor is a ratio of two equivalent measures in different units such as 1  h 60   min , fraction 1 h , over 60 min end fraction . comma  and is always equal to 1. To convert from one unit to another, multiply the original unit by a conversion factor that has the original units in the denominator and the desired units in the numerator.

Example

A painting is 17.5 in. wide. What is its width in centimeters?

Recall that 1 in. = 2.54 cm.

17.5 in. · 2.54  cm 1 in. = 44.45  cm 17.5 , cross out enclosing , in. , end crossout , middle dot . fraction 2.54 , cm , over 1 , cross out enclosing , in. , end crossout end fraction . equals , 44.45 , cm

The painting is 44.45 cm wide.

Exercises

Convert the given amount to the given unit.

37. 6 1 2 6 , and 1 half  ft; in.
38. 4 lb 7 oz; oz
39. 135 s; min
40. 2.25 mi; yd
41. Production A bread slicer runs 20 h per day for 30 days and slices 144,000 loaves of bread. How many loaves per hour are sliced?
42. Pets A gerbil eats about 1 4 1 fourth  oz of food per day. About how many pounds of food can a gerbil eat in a year?
43. Sports If a baseball travels at 90 mi/h, how many seconds does it take to travel 60 ft?

2-7 and 2-8 Solving Proportions and Using Similar Figures

Quick Review

The cross products of a proportion are equal.

If a b = c d , eh over b , equals , c over d , comma  where b ≠ 0 b not equal to 0  and d ≠ 0 , d not equal to 0 comma  then ad = bc.

If two figures are similar, then corresponding angles are congruent and corresponding side lengths are in proportion. You can use proportions to find missing side lengths in similar figures and for indirect measurement.

Example

A tree casts a shadow 10 m long. At the same time, a signpost next to the tree casts a shadow 4 m long. The signpost is 2.5 m tall. How tall is the tree?

x 10 = 2.5 4 Write a proportion . 4 x = 10 ( 2.5 ) Cross Products Property 4 x = 25 Simplify . x = 6.25 Divide each side by  4 . table with 4 rows and 3 columns , row1 column 1 , x over 10 , column 2 equals , 2.5 over 4 , column 3 cap writeaproportion . . , row2 column 1 , 4 x , column 2 equals 10 . open 2.5 close , column 3 cap crosscap productscap property , row3 column 1 , 4 x , column 2 equals 25 , column 3 cap simplify , . , row4 column 1 , x , column 2 equals , 6.25 , column 3 cap divideeachsideby . 4 . , end table

Exercises

Solve each proportion.

44. 3 7 = 9 x 3 sevenths , equals , 9 over x
45. − 8 10 = y 5 negative 8 over 10 , equals , y over 5
46. 6 15 = a 4 6 fifteenths , equals , eh over 4
47. 3 − 7 = − 9 t 3 over negative 7 , equals , negative 9 over t
48. b + 3 7 = b − 3 6 fraction b plus 3 , over 7 end fraction . equals . fraction b minus 3 , over 6 end fraction
49. 5 2 c − 3 = 3 7 c + 4 fraction 5 , over 2 c minus 3 end fraction . equals . fraction 3 , over 7 c plus 4 end fraction
50. Models An airplane has a wingspan of 25 ft and a length of 20 ft. You are designing a model of the airplane with a wingspan of 15 in. What will the length of your model be?
51. Projections You project a drawing 7 in. wide and 4 1 2 4 , and 1 half  in. tall onto a wall. The projected image is 27 in. tall. How wide is the projected image?

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