OBJECTIVES
Upon completion of chapter three, the student will be able to—
1. Discuss the relation of percent to the whole, and calculate a given percent of a given number.
2. Change percent to hundredths and hundredths to percent.
3. Solve for base, rate, or percentage in percent problems.
4. Read, write, and determine ratios of one quantity to another.
5. Solve problems involving direct proportion.
6. Solve problems involving inverse proportion, including pulley and gear ratio problems.
7. Find the average, or mean, of a set of statistics.
8. Find the median and mode in a body of data.
9. Use reference tables for extracting information.
10. Interpolate additional numerical data from information given in a table.
11. Name the parts of a table and construct a table from given data.
12. Extract approximate statistics from a graph, noting trends.
13. Determine the best method for depicting numerical information.
14. Plot and draw a bar graph, a line graph, and a circle graph.
INTRODUCTION
Number relationships, charts, graphs, and tables can help in making calculations and decisions. Charts, graphs, and the like can show trends, illustrate deductions, and help make decisions based on numerical facts. Common relationships are percentage, ratio, proportion, average, mean, median, and mode. Graphs, tables, and charts are often used to depict relationships.
PERCENTAGE
Percent expresses a proportion of an amount in hundredths. Percent means for or out of each hundred. The symbol for percent is % and it expresses quantity in relation to a whole. It is only used specifically and always with a number. For example:
26 percent = ²⁶⁄₁₀₀ = 26% = 0.26.
14½ percent = ¹⁴⁵⁄1,000 = 14.5% = 0.145.
To change from percent to hundredths, shift the decimal point two places to the left and drop the percent symbol. For example, to change 32% to hundredths,
60 NUMBER RELATIONS
drop the % symbol and move the decimal point two places to the left, which yields 0.32. In other words, 32% = 0.32. To change a decimal fraction to percent, move the decimal point two places to the right and add the percent symbol; for example, 0.64 becomes 64%.
Example Problem: Find 14% of 430.
Solution:
14% = 0.14 0.14
×
430 = 60.2.Thus, 14% of 430 is 60.2.
If you have a calculator with a % key, all you do is enter:
430
×
14% = 60.2.Example Problem: Find 4½% of 85.
Solution:
0.045
×
85 = 3.825, or, using the calculator, 85×
4.5% = 3.825.Base, Rate, and Percentage
Base, rate, and percentage are terms used in percentage problems. Base is the quantity of which a percentage is desired. Rate is a desired percentage of the base. And, percentage is the product of the rate times the base. For example, figure 3.1 shows that 6% of 300 is 18. In this case, 6% is the rate, 300 is the base, and 18 is the percentage. The relationship of base, rate, and percentage can be expressed as
percentage = rate × base.
Three types of percentage problems involve finding one of these elements when the other two are known. First, when the base and rate are known, percent-age is found by multiplying the base times the rate.
Example Problem: How much is 8% of $625?
Solution: In this example, 8% is the rate and $625 is the base. So, rate times base is
$625
×
0.08 = $50.00.Eight percent of $625 is $50.00.
Example Problem: If a woman earns $120 and saves 12½% of it, what percentage of her earnings did she save?
Solution:
12½% = 12.5% = 0.125 $120
×
0.125 = $15.00.The percentage saved is $15.00.
RATE × BASE = PERCENTAGE
6% of 300 is 18 Figure 3.1 Percent relations
500
Percentage 61
The second percentage problem involves solving for rate when the base and percentage are known. The formula for finding rate when base and percentage are known is
rate = percentage ÷ base.
Example Problem: What percent of $500 is $125?
Solution: Here, the percentage is $125 and the base is $500. So, the percentage divided by the base is
125 ÷ 500 = 0.25, or 25%
or 125 = 0.25, or 25%.
Example Problem: An oil tank contains 130 barrels, of which 2.6 barrels are sedi-ment and water (S&W). What percent of S&W does it contain?
Solution:
2.6
–— = 0.02, or 2% S&W.
130
The third percentage problem involves knowing the rate and percentage but not the base. The formula for finding base is
base = percentage ÷ rate.
Example Problem: An oilwell produces 95% water, and the allowable oil production for a specified period is 325 barrels. What is the total fluid production needed to produce this much oil?
Solution: Oil comprises 5% (rate) of the fluid, so 100% – 95% = 5%.
The allowable, 325 barrels, is the percentage; and the total fluid production is the base. Thus,
325
base = –— = 6,500 barrels.
0.05
Example Problem: 43.75 is 5% of what number?
Solution:
43.75 ––— = 875.
0.05
Practice Problems
Round off all answers to two places.
1. An item costs $52 plus 5% sales tax. What is its total price?
_________________________________________________________
2. How much is 8½% of 200?
_________________________________________________________
62 NUMBER RELATIONS
3. Six percent of a person’s earnings goes into a retirement fund. If this person earns $1,940 per month, what amount goes toward the person’s retirement each month?
_________________________________________________________
4. An alloy contains 83.3% tin, 5.6% copper, and 11.1% antimony. How much of each metal is contained in 500 pounds of this alloy?
a. Tin __________________________________________________
b. Copper _______________________________________________
c. Antimony _____________________________________________
5. In a shipment of pressure gauges, 15% were damaged, leaving 170 that could be used. How many gauges were in the shipment?
_________________________________________________________
6. A supply house gives a discount of ¹⁄5 off from the prices listed in its catalog.
What will be the net price of four 3-inch gate valves that list at $92 each?
_________________________________________________________
7. A carpenter orders 240 two-by-fours but sends 36 of them back because they are defective. What percent did he return?
_________________________________________________________
8. A job in a machine shop required 27 hours to complete. Of this, 14½ hours were spent on the lathe, 2¼ hours on the drill press, 6¾ hours on the shaper, and 3½ hours in welding. What percent of the time did each process require?
a. Lathe ________________________________________________
b. Drill press _____________________________________________
c. Shaper _______________________________________________
d. Welding ______________________________________________
9. A certain oil yields 39% gasoline. How many barrels of this crude does it take to produce 500 barrels of gasoline?
_________________________________________________________
10. The following problems provide practice in mental arithmetic. Try to figure out the answers in your head and then write them down.
a. Find 25% of 88. (Hint: Multiplying by 0.25 is the same as dividing by 4.) ________________________________________________
b. If a person earning $400 per week receives a 25% raise and then a 20% cut in salary, what is the person’s new salary?______________
c. If 42 gallons of oil are removed from a full 100-barrel tank, what percent of oil is left in the tank? ___________________________
d. If ⅛ of a lease has been surveyed, what percent remains to be sur-veyed? ________________________________________________
e. What percent of 110 is 55? _______________________________