After completing Unit 12, you will be able to:
• Simplify business ratios.
• Set up common business problems in proportion form.
• Solve proportion problems for the missing terms by cross multiplying.
• Solve inverse proportion problems.
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.
1. Express in ratio form: 2 computer operators for every 15 office employees. __________
4
10= 2
5recall that this is the same as 4/10 = 2/5
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: 10minute 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:
(Multiply the numerator and denominator by 100 to get rid of the decimal point.) $1.00 in profit for every $4.00 in sales.
$2.50 $10 = 2.50 x 100 10 x 100 = 250 1000= 1
4.
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 

=  2
5 
a proportion with one unknown? ____________ 
12.  What term is applied to 

= 

? ____________ 
13.  What is the unknown in the proportion 

= 

? ____________ 
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.
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? _______________(a) The number of persons is on the (top/bottom) _________ of each of the equal ratios.
5 persons 68 boxes = 8 persons N boxes , or 5 68= 8 N(b) The number of boxes is on the (top/bottom) ________ of each of the equal ratios.
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. _____________17. Cross multiply the following proportion and solve.(b) Compute. _____________
(c) Are the sides equal to one another? ____________
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. ________________(a) Cross multiply. __________
3
5= 36
N(b) Compute. ______________
(c) Solve for N. ______________
(d) Proof: Does 3
5= 36
60? Reduce 36
60to find out. __________
19. Solve for the unknown in the following proportions by cross multiplying. Then prove.
20. Solve for the unknown. Round to two decimal places.
(a) G 5 = 600 12 _________
(b) 17 D= 85
100_________
(c) 83 249 = S
150_________
(d) 150 600 = 620 C _________
PROBLEM SOLVING USING PROPORTIONS
(a) _D_
15= 3 8 _________
(b) 120 25 = 72 T _________
(c) 620 350 = _S_
50_________
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.


(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? _______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.(b) Are 107 minutes 3 or more times 32 minutes? ____________
(c) Does the answer 107 minutes appear to make sense? ________


(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 onehalf greater or smaller? _______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?(b) Are the projected sales about onehalf greater or smaller? _________
(c) Does the answer $84,000,000 make sense? ____________
(a) Set up a proportion and solve. _________________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.)(b) Round your answer to the nearest whole number of persons. _______
(c) Is the work load about onehalf greater? ________
(d) Is the number of people onehalf greater? ____________
(e) Does your answer seem reasonable? _____________
(a) Set up the proportion, solve, and check if the answer makes sense. ___________SELECTION OF FACTS TO SOLVE A PROBLEM(b) What is the answer? ___________
(c) Estimate the percent increase in combined monthly incomeis 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? ___________
25. Last year the printing of 10,000 copies of a 120page annual report required 24 hours of printing time. How much time will be required for a 90page report?
(a) Is there any unnecessary information? _____________26. At Stereo House, 650 out of 1,650 radios sold this year were AM/FM. Also, 350 cassette players were sold.(b) What is it? _______________
(c) Set up the proportion, solve, and prove. _________________
(d) Does the answer 18 hours seem reasonable? ____________
(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? _____________WORKERDAYS(b) Is 650 about onethird of 1,650? __________
(c) Is 1,182 about onethird of 3,000? __________
(d) Does your answer make sense? ___________
Example: How many workerdays were used by 3 workers in 5 days? 3 x 5 = 15 workerdays.
27.
(a) How many workerdays are there when 1 typist works 3 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 inhouse 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.(b) How many workerdays are there when 3 typists work 8 days? _________
(c) How many workerdays are there when 6 office workers work 21 days? _________


(1) State what happened the first time.  20 workerdays were required to produce 800 letters. 5 typists (workers) x 4 days = 20 workerdays. 
(2) Set up a new ratio. 
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 workerdays.) 
(4) Set up another ratio, letting L present the number of letters to be produced in the new condition. 
L 
(5) Use the two ratios to set up a proportion. 

(6) Solve by cross multiplication. 

(7) Check whether the answer makes sense. 

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.
SOLVING INVERSE PROPORTION PROBLEMS
(a) 600 = 15D 7 D = _________
(b) 7,500 20H = 5 H = _________
(c) 12A 7 = 24 A = _________
(d) 35
13= 16.4 5M M = _________
Example: If 5 employees can turn out a job in three 8hour 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.
33. Last month, it took 6 office employees 8 days to get out a 100,000piece 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. Remember
that each ratio must contain similar facts, in this case
the number of employees.5 employees 12 employees= 5 12Step 2: Use hours for the second ratio, letting N
equal the hours taken by the 12 employees.3 x 8 hours
N hours24 N Step 3: Invert one ratio and set up a proportion. 5
12= N 24Step 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.
Step 1: Use employees for the first ratio. ____________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.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. ____________
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. _______________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. ______________________________________(b) Check: Does it take more or fewer employees to get the work out in less time? _____________
(c) Does this make sense? ___________
CHECKPOINT for Unit 12