Section 10.1: Problems Involving One Unknown Quantity

 

32.    Five times the difference of a number less 12 is equal to seven times the number increased by 6.  Find the number.

 

Let n = the number

 

Word equation: 

5(number – 12) = 7(number) + 6

 

          Equation:     5(n – 12) = 7n + 6

                           5n – 60 = 7n + 6

                           -66 = 2n

                           -33 = n

          

     Solution:  The number is –33.

 

36.    A typist can now type 10 words more than twice her beginning rate.  If her current rate is 90 words per minute, what was her beginning rate?

 

Let r = her beginning rate

 

Word equation 2(beginning rate) + 10 = 90

 

Equation: 2r + 10 = 90

         2r = 80

         r = 40

 

Her beginning rate was 40 words per minute.

42.    A gardener has a circular flower bed in the center of his rectangular vegetable garden.  The radius of the flower bed is 3 meters and the outside dimensions of the garden are 12 meters by 15 meters.  What percent of the garden is occupied by the flower garden?  Round to the nearest percent.

 

The area of the flower bed is

πr2 = 3.14(3)2 = 28.26 m2

     The area of the garden is

                L W = 12(15) = 180 m2

 

     Let p = percent of garden that is the flower bed

 

     180p = 28.26             p = 0.157

 

     The flower bed takes up about 16% of the garden.

 

44.    There are n nickels and (n + 7) dimes in a collection of new coins worth $10.  How many nickels are there in the collection?

 

Let n = # nickels

Let n + 7 = # dimes

 

0.05n + 0.10(n + 7) = 10

0.05n + 0.10n + 0.70 = 10

0.15n = 9.3

     n = 62

There are 62 nickels. (and 69 dimes)

46.    A worker at a fast-food franchise earns $4.52 per hour for the first 40 hours he works.  If he works overtime, he earns $6.78 per hour.  One week he earned $235.04.  How many hours did he work?

 

Let h = number of overtime hours he worked

 

4.52(40) + 6.78h = 235.04

180.8 + 6.78h = 235.04

6.78h = 54.24

h = 8

     He worked 48 hours. (40 hours + 8 hours OT)

 

56.    A customer at a restaurant was not pleased with the service she received for a luncheon with a special friend.  She left $60 for a bill of $52.67.  What percent (to the nearest tenth of a percent) did the customer leave for a tip?

 

She left $60 – $52.67 = $7.33 for the tip.

 

Let t = percent tip she left

 

52.67t = 7.33

     t = 0.139

 

She left a 13.9% tip.