Ratio
Ratio is a proportional relationship between two numbers or quantities. Ratio describes how two different things relate numerically to one another. For ex-ample, the statement, “the ratio of trucks to tank cars is two to one” means that the number of trucks is twice as great as the number of tank cars. So, if the ratio statement is about a company that owns tank cars and trucks, then the company may own 50 tank cars and 100 trucks.
The ratio of one quantity to another is obtained by dividing the first quantity by the second. Thus, ratio can also be defined as the relative size of two quantities expressed as the quotient of one divided by the other. For example, the ratio of
$6 to $2 is three to one because $6⁄$2 equals 3⁄1.
A colon placed between two quantities expresses the ratio of one quantity to the other. For example, the ratio of $6 to $2 can be written as $6:$2 and spoken as “six dollars is to two dollars.” This ratio may also be written as the fraction $6⁄$2.
Example Problem: What is the ratio of 12 to 4?
Solution: Applying the definition for ratio, divide 12 by 4, which is written as 12⁄4
= 3. In effect, dividing 12 by 4 reduces the ratio to 3 to 1. That is, the ratio of 12 to 4 is the same as the ratio of 3 to 1.
Proportion
Proportion is a relation of equality between two ratios. The ratio of 2 to 4 is the same as the ratio of 6 to 12. That is, 2 to 4 is proportional to 6 to 12 or 2 is to 4 as 6 is to 12 (2:4 = 6:12). Proportions are mostly used when one ratio is known and only part of another ratio is known. Since ratios in a proportion are equal in value, the missing part of a ratio can be found. As with ratios, proportions can be expressed in two ways: a colon inserted between the two numbers or as a fraction:
2:4 = 6:12
or 12⁄4 = 6⁄12.
Written in either form, this proportion is read as “two is to four as six is to twelve.”
The first and fourth terms of a proportion (in this case, 2 and 12) are the extremes, and the second and third terms (4 and 6) are the means. A key con-sideration is that in any proportion, the product of the extremes is equal to the product of the means. For example, in the proportion 2:4 = 6:12,
2 × 12 (the extremes) = 24 4 × 6 (the means) = 24
24 = 24.
Direct Proportion
Proportions can be either direct or indirect (also called inverse). In direct propor-tion, each term increases or diminishes as the term on which it depends increases
Ratio and Proportion 63
64 NUMBER RELATIONS
or diminishes. For example, figure 3.2 shows that a small cost is to a large cost as a small number of objects (bolts in this case) is to a large number of objects.
Put another way, fewer bolts cost less than more bolts and more bolts cost more than fewer bolts. This statement is a direct proportion because cost increases as the number of bolts increases and vice versa.
To solve direct proportion problems, three terms must be known. If three terms are known, then the fourth term can be determined.
Example Problem: If 12 hacksaws cost $155.88, how much do 100 hacksaws cost?
Solution: The three known quantities in this problem are a small number of hacksaws, a small cost, and a large number of hacksaws. Because the number to express the cost of the large number of hacksaws is unknown, call the unknown quantity x and set up the proportion: 12:100 = $155.88:x. Note that this propor-tion reads 12 is to 100 as $155.88 is to x.
Mathematically, the proportion can be written as:
12 155.88 ––– = ––––––.
100 x
To solve the proportion, merely cross-multiply the ends and the means.
(Cross-multiplying means to multiply the numerator on one side of the fraction by the denominator on the other side. In this problem, the result of the cross-multiplication is:
12x = 15,588.
As you will learn in chapter 5, “Principles of Algebra,” to solve 12x = 15,588, divide 15,588 by 12, which equals 1,299. Thus, x = $1,299 the cost of 100 hacksaws.
Inverse Proportion
In inverse, or indirect, proportion, each term increases as the term on which it depends decreases; or, each term decreases as the term on which it depends increases. For example, figure 3.3 shows that a slow speed is to a fast speed as
$2 : $4 = 10 BOLTS : 20 BOLTS
SMALL LARGE SMALL LARGE
COST is to COST as No. is to No.
Figure 3.2 Direct proportion
25 mph : 50 mph = 84 min. : 42 min.
SLOW FAST LONG SHORT
SPEED is to SPEED as TIME is to TIME
Figure 3.3 Inverse proportion
a long time is to a short time. That is, the slower the speed is, the more time it takes and the faster the speed is, the less time it takes. The ratios vary in the op-posite order—as one quantity goes up, the other goes down and vice versa. The speed of a vehicle is inversely proportional to the time needed to travel a certain distance. For example, a car moving at 25 miles per hour can cover 35 miles in 84 minutes, while at 50 miles per hour it can go the same distance in 42 minutes.
Another example of inverse proportion involves belt- or chain-driven pulleys. The pulley directly connected to a motor or other power source is the driver pulley. The other pulley is the driven pulley. As the driver pulley rotates, the belt or chain drives the other, driven pulley. The bigger the pulley is, the slower it rotates. For example, the 12-inch pulley (pulley B) in figure 3.4 rotates slower than the smaller, 8-inch pulley (pulley A). Increasing the size of pulley A decreases its speed; likewise, decreasing the size of pulley B increases its speed.
Example Problem: If the speed of pulley B in figure 3.4 is 250 revolutions per minute (rpm), and it is 12 inches in diameter, how many rpm does pulley A turn if it is 8 inches in diameter?
Solution: Since the speeds of the two pulleys are inversely proportional to their diameters, write the proportion:
Then substitute the known values and let x stand for the speed of pulley A:
x:250 = 12:8, or —– = ––
8x = 250 × 12 = 3,000 x = 3,000 ÷ 8 = 375.
The speed of pulley A is 375 rpm.
A simple rule to remember in calculating speeds and diameters of pul-leys is that the speed of the driver pulley multiplied by its diameter is equal to the speed of the driven pulley multiplied by its diameter. The unknown or missing quantity is designated by a symbol such as x, and its value is found as previously shown.
Gears are similar to pulleys (fig. 3.5). Gears, like pulleys, rotate. However, instead of belts or chains, gears have teeth, or cogs, that intermesh. As the driver gear turns, the intermeshing teeth cause the driven gear to rotate. A rule for determining speeds and diameters of gears is that the speed of the driver mul-tiplied by its number of teeth is equal to the speed of the driven gear mulmul-tiplied by its number of teeth.
Example Problem: When gear A in figure 3.5 makes 5 revolutions, how many revolutions does gear B make?
66 NUMBER RELATIONS
Solution: Using the gear ratio rule, set up the proportion:
SA × TA = SB × TB where
SA = speed of gear A
TA = number of teeth in gear A SB = speed of gear B
TB = number of teeth in gear B.
Substituting the known values and letting x equal the unknown:
5 • 45 = x • 31 31x = 225 x = 7.258.
Gear B turns 7.258 revolutions each time gear A makes 5 revolutions.
Practice Problems
1. What is the ratio of 125 to 5?
_________________________________________________________
2. Jones’ salary is $250 and Smith’s is $150. What is the ratio of Jones’ salary to Smith’s?
_________________________________________________________
3. Soft pine wood weighs 35 pounds per cubic foot; steel weighs 490 pounds per cubic foot. Find the ratio of these densities.
_________________________________________________________
4. In treating drilling mud, 60 sacks of barite are added to 100 barrels of mud to increase its weight 1 pound per gallon. How many sacks are re-quired to raise the weight of 850 barrels of mud from 9.4 to 11 pounds per gallon?
_________________________________________________________
5. If 3 inches on a blueprint represents 12 feet, how many feet does a line that is 10 inches long represent?
_________________________________________________________
6. A job must be completed in 6 days. Records show that 20 workers com-pleted a similar job in 15 days. How many workers must be put on the job to insure its completion in the given time?
_________________________________________________________
7. A car traveling at 48 miles per hour takes 50 minutes to go from lease A to lease B. At what speed must it go to make the trip in 30 minutes?
_________________________________________________________