Unit 12: Ratios and Proportions in Business


Performance Objectives

After completing Unit 12, you will be able to:



RATIOS
A ratio is a comparison of two amounts and is often written in fraction form.

Example: $1.00 profit for every $5.00 in sales.

This can be written as 1/5 and it is read "one out of five" or "one to five." Ratios in fraction form should be reduced to lowest terms.

Example: $4.00 cost for $10.00 in sales.

 
 4 
10
= 2
5
recall that this is the same as 4/10 = 2/5
1.  Express in ratio form: 2 computer operators for every 15 office employees. __________

2. How would you express $3 cost for every $6 in sales? ___________

EXPRESSING RATIOS IN THE SAME UNITS   Whenever possible, express a ratio in the same units. If the two amounts refer to time, then be sure both amounts are expressed in equivalent units.

Example 1:  15 minutes for each hour would be written as 15/60, because an hour is 60 minutes.

Example 2: 50 days out of a year would be written as 50/365.

3.  Express as a ratio: 10-minute rest break for every 2 hours. ________

4. Express as a ratio: 2 days' vacation for every month (30 days). __________

EXPRESSING RATIOS USING WHOLE NUMBERS
For convenience, ratios are usually stated using whole numbers. Example: $2.50 profit from every $10 of sales can be expressed as:

 
$2.50
$10
= 2.50 x 100
10 x 100
=
 250 
1000
= 1
4
.
(Multiply the numerator and denominator by 100 to get rid of the decimal point.) $1.00 in profit for every $4.00 in sales.

5. Express the following as a ratio using whole numbers: 1 1/3 minutes out of each hour. Finish the problem and reduce.

6. Express as a ratio: 50 cents out of every $10. ______________

How would the answer be read? ____________

ALL RATIOS HAVE A DENOMINATOR
All ratios can be expressed as a fraction.

Example: ''Five hours for each hour of setup time'' is written as 5/1.

7. How would you express "Twelve hours for the completion of one assembled electronic calculator"? __________

8. Express as a ratio: $15 for 3 crates of fruit. ______________

9. Express as a ratio: 2 feet on a blueprint plan are represented by 1/2 inch. Be sure that the units are the same, in this case inches, and that the numerator and denominator are whole numbers. ______________

10. How would you read the answer to Problem 9? _________________________

PROPORTIONS
A proportion is formed by two equal ratios. There are four terms in a proportion since each ratio has two terms. Thus if you know three of the terms, you can easily find the fourth term.


 
11. Is
  A 
250
= 2
5
a proportion with one unknown? ____________
12. What term is applied to
100
300
=
  R 
680
? ____________
13. What is the unknown in the proportion 
 6 
40
=
P
? ____________

14. Analyze the following statement and write a proportion: If 5 persons can pick 68 boxes of strawberries in an hour, how many boxes can 8 persons pick? Before you write the proportion, decide on some letter, such as N, for the unknown number of boxes. Then write the proportion.

 
5 persons
68 boxes
= 8 persons
N boxes
, or
 5 
68
=
8
N
(a)  The number of persons is on the (top/bottom)  _________ of each of the equal ratios.

(b)  The number of boxes is on the (top/bottom)  ________ of each of the  equal ratios.

15. Analyze and write a proportion for the following statement, letting P represent pounds: If $1.25 will buy 4 pounds of fruit, how many pounds of fruit will $3 buy? _______________

SOLVING PROPORTIONS BY CROSS MULTIPLICATION
Proportions are solved for the unknown term by cross multiplication. After cross multiplication, the resulting equation is then solved for the unknown. In any proportion, the product of the first and fourth terms equals the product of the second and third terms, as shown in the example below.

16. Cross multiply the proportion below, and observe that the resulting left side and the right side are equal. To show the cross, draw one line from 3 to 20 and another line from 5 to 12.

 
3
5
=
12
20
(a)  Cross multiply. _____________

(b)  Compute. _____________

(c)  Are the sides equal to one another? ____________

17. Cross multiply the following proportion and solve.
 
3
5
=
36
N
(a)  Cross multiply. __________

(b)  Compute. ______________

(c)  Solve for N. ______________
 
(d) Proof: Does 3
5
= 36
60
? Reduce 36
60
to find out. __________

18. You can prove your answer to Problem 17 by putting the value of the unknown back in the proportion and cross multiplying to see whether the sides are equal to one another. Try it below. ________________

19. Solve for the unknown in the following proportions by cross multiplying. Then prove.

(a) G
5
= 600
12
_________
(b) 17
D
=
 85 
100
_________
(c)  83 
249
=
  S 
150
_________
(d) 150
600
= 620
C
_________
20. Solve for the unknown. Round to two decimal places.
 
(a)
_D_
15
= 3
8
_________
(b) 120
25
= 72
T
_________
(c) 620
350
=
_S_
50
_________
PROBLEM SOLVING USING PROPORTIONS
The analysis of a business problem that can be solved by a proportion follows the thought process shown in the example below. Note: The answers to many proportion problems involving the work of people are often only estimates, because many factors, such as training, motivation, and work conditions, are not taken into account. Managers and supervisors, however, can use the estimates with other available information in the decision-making process.

Example: Man Barnes produced 20 copies of a form letter on a photocopying machine in 11 seconds. How many seconds will she need to produce 420 copies of the same letter?

This problem can be analyzed and solved in seven steps, as shown below.

21. Using the seven steps described above, analyze and solve the following problem. Jerry Kahak duplicated 1,200 copies of a notice on a duplicating machine in 32 minutes. He wants to know how long it will take to duplicate 4,000 copies of a similar notice. Compute the answer to the nearest whole minute. Check with the answers only after trying to complete the seven steps.
 
Thought Process
Result
(1) State what happened the first time.
_____________________
(2) Set up a new ratio.
_____________________
(3) State the new condition or plan.
_____________________
(4) Set up this statement as ratio.
_____________________
(5) Use the resulting two ratios to set up a proportion.
_____________________
(6) Solve by cross multiplication. (Round to a whole number.)
_____________________
(7) Check whether the answer makes sense. (see below)
_____________________

(a)  Is a run of 4,000 copies roughly 3 or more times the size of a run of 1,200 copies?  _______

(b)  Are 107 minutes 3 or more times 32 minutes? ____________

(c)  Does the answer 107 minutes appear to make sense? ________

22. Rochi Corporation spent $1,000,000 on product research last year and their sales were $56,000,000. The corporation believes that additional research will proportionately increase sales. If the Rochi management spends $1,500,000 on research, what might it expect its sales to be this year? Use the seven steps you just learned to analyze and solve this proportion problem.
 
Thought Process
Result
(1) State what happened the first time.
_____________________
(2) Set up a new ratio.
_____________________
(3) State the new condition or plan.
_____________________
(4) Set up this statement as ratio.
_____________________
(5) Use the two ratios to set up a proportion.
_____________________
(6) Solve by cross multiplication. (Round to a whole number.)
_____________________
(7) Check whether the answer makes sense. (see below)
_____________________
(a)  Is the amount to be spent on research about one-half greater or smaller? _______

(b)  Are the projected sales about one-half greater or smaller? _________

(c) Does the answer $84,000,000 make sense? ____________

23. Last month 4 office workers turned out 1,200 statements in 3 days. If there are 1,720 statements to be turned out in 3 days, how many extra office workers will be needed?
(a)  Set up a proportion and solve.  _________________

(b)  Round your answer to the nearest whole number of persons. _______

(c)  Is the work load about one-half greater? ________

(d)  Is the number of people one-half greater?  ____________

(e)  Does your answer seem reasonable? _____________

24. The Koicheks have been saving $25 out of their combined monthly income of $2,100. If they continue to save at the same rate, how much should they save out of their new combined monthly income of $2,400? (Compute the new amount saved to the nearest whole cent.)
(a)  Set up the proportion, solve, and check if the answer makes sense.  ___________

(b)  What is the answer? ___________

(c)  Estimate the percent increase in combined monthly income--is it about 5%, 10%, or 15%? __________

(d)  What is the increase in the planned savings, 5%, 10%, or 15%?  ____________

(e)  Does the amount of planned savings seem to be correct? ___________

SELECTION OF FACTS TO SOLVE A PROBLEM
When many facts are given, be certain to select only the necessary facts. In the following problem, although you are told that 10,000 copies of the report were printed, this number is not needed to solve the problem. It must be assumed the 10,000 copies is the job for both situations. The fact that 10,000 copies is being run is not needed in the solution.

25. Last year the printing of 10,000 copies of a 120-page annual report required 24 hours of printing time. How much time will be required for a 90-page report?

(a)  Is there any unnecessary information? _____________

(b)  What is it? _______________

(c)  Set up the proportion, solve, and prove. _________________

(d)  Does the answer 18 hours seem reasonable? ____________

26. At Stereo House, 650 out of 1,650 radios sold this year were AM/FM. Also, 350 cassette players were sold.
(a)  Assuming radio buyers behavior does not change, how many AM/FM radios are expected to be sold next year if total radio sales equal the projection of 3,000 radios? _____________

(b)  Is 650 about one-third of 1,650? __________

(c)  Is 1,182 about one-third of 3,000? __________

(d)  Does your answer make sense? ___________

WORKER-DAYS
A worker-day is the amount of work done by one worker in one day.

Example: How many worker-days were used by 3 workers in 5 days? 3 x 5 = 15 worker-days.

27.

(a) How many worker-days are there when 1 typist works 3 days? ___________

(b) How many worker-days are there when 3 typists work 8 days? _________

(c) How many worker-days are there when 6 office workers work 21 days? _________

28. An office efficiency committee observed that 5 typists can type 800 letters in 4 days. If they employed an additional 3 temporary typists for 1 day, how many letters could they expect from all 8 typists that day? Assume that the temporary typists are as efficient as the in-house typists and that there is sufficient space and equipment for them. First analyze the problem, using the same seven steps. Study the first four steps and then finish out the remaining three steps.
 
Thought Process
Solution
(1) State what happened the first time. 20 worker-days were required to produce 800 letters. 5 typists (workers) x 4 days = 20 worker-days.
(2) Set up a new ratio.
 20 
800
(3) State the new condition or plan. How many letters will be produced in 8 worker days? (5 original workers + 3 temporary workers for 1 day = 8 worker-days.)
(4) Set up another ratio, letting L present the number of letters to be produced in the new condition.
8
L
(5) Use the two ratios to set up a proportion.
_________(a)__________
(6) Solve by cross multiplication. 
_________(b)__________
(7) Check whether the answer makes sense.
_________(c)__________

29. Using the procedure described in Problem 28, solve the following problem.
Under normal conditions, 16 factory workers can produce 5,120 units in 10 days. During a seasonal rush of orders, if the work force is increased to 24 and operates for 35 days, how many units (U) will be produced? (Make a ratio and solve.) ______________________________________

30. Assume that 3 salespersons were able to make 62 successful sales in 5 days. For a special sales promotion 6 additional salespersons are employed for 11 days. The sales manager believes the new salespersons will perform as well as the regular sales staff, and other factors are equal. How many successful sales can the sales manager expect for the 11 days? Compute to the nearest whole number. (Make a ratio and solve.) ______________________________________

CROSS MULTIPLICATION FOR OTHER PROBLEMS
Other equations may also be solved by cross multiplication. Study the problem below.

31. If you have a problem like 425 = 17A/200, you can use cross multiplication to solve it. Write 425 as 425/1 .The problem becomes 425/1 = 17A/200.

32. Solve the following equations using cross multiplication.

 
(a) 600 = 15D
7
D = _________
(b) 7,500
20H
= 5
H = _________
(c) 12A
7
= 24
A = _________
(d) 35
13
= 16.4
5M
M = _________
SOLVING INVERSE PROPORTION PROBLEMS
A proportion in which one ratio increases while the related ratio decreases is called an inverse proportion. For example, the faster a car is driven, the lower the gas mileage; the more persons are put on a particular job, the less the time required to complete it. If the analysis of a problem indicates that it is an inverse proportion, invert one of the two ratios and solve by cross multiplication. Note that inverse proportion problems require that each ratio contain similar information. For example, if you are concerned with the speed of a car and its gas mileage, put the speeds in one ratio and the mileages in the other ratio.

Example: If 5 employees can turn out a job in three 8-hour days, or 24 hours, how many hours will it take 12 employees to do the same job? (Assume the additional employees are equally efficient and other factors are equal.)

Analysis: Clearly 12 employees will take less time to do the job than 5 employees; therefore you must use an inverse proportion. Set up the two ratios in the following manner.

 
Step 1: Use employees for the first ratio. Remember 
that each ratio must contain similar facts, in this case 
the number of employees.
 5 employees 
12 employees
=
 5 
12
Step 2: Use hours for the second ratio, letting N 
equal the hours taken by the 12 employees.
3 x 8 hours
N hours
24
N
Step 3: Invert one ratio and set up a proportion.
 5 
12
=
N
24
Step 4: Solve. 12N = 5 x 24;
N = 10 hours.
 
Step 5: Check to see if your answer 
makes sense.
If it takes 5 employees 24 hours, 
then it will take 12 employees 
10 hours. Yes, the answer is 
reasonable.
33. Last month, it took 6 office employees 8 days to get out a 100,000-piece mailing. How many days will it take if S additional employees are put on the job? Remember, the 100,000 pieces is the job and therefore does not have to be taken into account. Use the steps above to analyze and solve this problem.
Step 1: Use employees for the first ratio. ____________

Step 2: Use days for the second ratio. ____________

Step 3: Invert one ratio and set up a proportion. ___________

Step 4: Solve. __________

Step 5: Check to see if your answer is reasonable. ____________

34. 22 employees completed a large mailing project in 12 days. How many days will it take if 14 more employees are added to get the same project done next month? Show your work below.
 
 
 

35. Last month, 17 office employees of East National Corporation completed the preparation and printing of an advertising circular in 5 1/2 days. This month the president wants a similar one out in 2 days. How many additional employees must be transferred from another office?

(a)  First find out how many employees are needed altogether; then find out how many additional employees are needed. _______________

(b)  Check: Does it take more or fewer employees to get the work out in less time? _____________

(c)  Does this make sense? ___________

36. On the average, it takes 3 secretaries 5 days to handle the weekly correspondence in an office. If another secretary is added to the office staff, how many days and hours will it take all 4 to handle the correspondence? Assume that a day has 7 working hours. Convert the answer in hours to days and hours by dividing by 7 hours, which is a normal working day. Show your work. Remember to set up the first ratio with the number of employees and the second ratio with numbers of hours. Invert one ratio and establish the proportion. ______________________________________



You have finished Unit 12. Please select one of the choices below.