10. In a certain game A can give B 1 point in 5, B can give C 1 point in 8, and C can give D 3 points in 10; how many can A give D in 100? 215. The following examples are worth notice. Ex. 1. A starts at 10 o'clock to walk along a road at the rate of 4 miles an hour; B starts on a tricycle at 45 minutes past 10 and rides after A at the rate of 9 miles an hour. When will B overtake A? When B starts, A has already traveled 4 of 4 miles; that is, 3 miles. B gains on A at the rate of (9 − 4 ) 5 miles an hour. B will overtake A when he has gained 3 miles, which he will do in (35) hours = 36 minutes. Ex. 2. At what time between 4 and 5 o'clock will the hands of the clock be together? At 4 o'clock the minute-hand is 20 minute-spaces behind the hour-hand. In one hour the minute-hand passes over 60 minutespaces, and the hour-hand passes over 5 minute-spaces. Thus, the minute-hand gains 55 minute-spaces in an hour. Now, when the two hands are together, the minute-hand must have gained on the hour-hand 20 minute-spaces, and the time required for this of an hour = 21 minutes. Thus, the time required is 21 minutes past 4. Ex. 3. A train traveling at the rate of 45 miles an hour is observed to completely pass a certain telegraph post in 5 seconds; it also completely passed in 4 seconds a second train which was traveling along a parallel line of rails in the opposite direction at the rate of 30 miles an hour. How long were the trains? The time the first train takes to completely pass a post is the time it takes to travel a distance equal to the length of the train; and, since the train goes at the rate of 45 miles an hour, it goes in 5 seconds a distance = 350 of 45 miles = 110 yards. Thus, the first train is 110 yards long. Again, in the time the trains take to completely pass one another the distance traveled by the two trains together must be the sum of the lengths of the trains; and in 4 seconds the trains will together travel of 45 miles + 3 of 30 miles = 146 yd. 2 ft. Hence the length of the second train = 146 yd. 2 ft. — 110 yd. = 36 yd. 2 ft. Ex. 4. Seven fowls are worth 6 ducks, 7 ducks are worth 2 geese, 10 geese are worth 7 turkeys, and a turkey is worth 17s. 6d. ; how much is a fowl worth? 1. One boy runs at the rate of 100 yards in 15 seconds, and has a start of 40 yards in front of another boy who runs at the rate of 100 yards in 12 seconds; when will the first boy be overtaken ? 2. One cyclist rides at the rate of 15 miles an hour and starts half-an-hour after another who rides along the same road at the rate of 12 miles an hour; when will the first rider be overtaken ? 3. At what time between 5 and 6 o'clock will the hands of a clock be together? 4. At what time between 2 and 3 o'clock will the hands of a clock be at right angles? 5. A train traveling at the rate of 45 miles an hour is observed to completely pass a certain point in 9 seconds; find the length of the train. 6. A man on the platform of a station observed that a train passed him in 10 seconds, and passed completely through the station, which is 308 yards long, in 24 seconds; how long was the train, and how fast was it going? 7. A passenger train, moving at the rate of 45 miles an hour, overtook a mineral train twice as long as itself and which was going along a parallel line of rails in the same direction at the rate of 23 miles an hour; and the passenger train completely passed the mineral train in 22 seconds. How long was each train? 8. A person lights two candles, 12 and 10 inches long respectively, at 6 P.M. The former diminishes 5 inches in length in 4 hours, and the latter 1 inch in 2 hours. If kept alight, at what time will the former be two inches shorter than the latter? 9. If 3 pears are worth as much as 4 apples, 5 apples as much as 3 plums, 8 plums as much as 3 peaches, and if pears cost 36 ct. a dozen, what is the price of a peach? 10. Twelve fowls are worth as much as 11 ducks, 5 ducks are worth as much as 4 pheasants, 10 pheasants as much as 3 turkeys, and 7 turkeys as much as 10 geese; also a fowl and a pheasant are together worth 6s. 6d. Find the cost of a goose and a turkey together. CHAPTER X. PERCENTAGES. PERCENTAGE A RATIO. 216. In many cases the ratio of one number to another, or of one quantity to another of the same kind, is expressed by the number of hundredths the first is of the second, and this is called the per cent the first is of the second. 25 = 100 For example, 2 apples of 8 apples, or 25 per cent of 8 apples. This means that the ratio of 2 apples : apples is .25. The first term is sometimes called the percentage, the second is called the base, and the quotient is, as formerly, the ratio. Per cent is expressed by the sign %, or by writing the numerator as a decimal; thus, 15% = 25%, or .25; and we write '5 is 25% of 20,' or '5 is .25 of 20.' 217. The following examples will show how to express any given quantity as a per cent of any other given quantity of the same kind. Ex. 1. Five dollars is what % of $40 ? $5 = 5% of $40 = of $40 = 12 Ex. 2. In a town whose population was 243200 there were 15504 children born in a year. Find the per cent the number of births was of the population. The ratio of births to population is 15504: 243200; 218. The following examples will show how to find a given per cent of a given quantity. Ex. 1. Find 121% of $18. 121% of $18.12 of $18 = $2.25. Ex. 2. In a town whose population was 243200 the birth rate in a year was 63% of the population; how many children were born in the year. 63% of 243200.063 times 243200 = 15504. Ex. 3. Of what is 9 ct. 221%? 100 Since 9 ct. is 22%, 100%, or the whole, must be of 9 ct. =40 of 9 ct. = 40 ct. 221 219. Frequently in finding percentage it is best to multiply by the common fraction which is equivalent to the per cent expressed decimally; thus, = 61% of 96 of 96; 12% of 432 = } of 432 ; 16% of $36.85 = ¦ of $36.85 = $6.14}. [Art. 134.] EXAMPLES XCIII. Oral Exercises. What fractions are denoted by the following per cents? What per cents are equivalent to the following fractions? Find the per cent the first is of the second in 25. $10, $25; 45 ct., $2.70; $1.121⁄2, $3. |