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286. The terms of a ratio are together called a couplet. Two couplets whose ratios are equal are called a proportion.
The two couplets of a proportion are often written thus: 6:18= 10:30, and should be read, the ratio of 6 to 18 equals the ratio of 10 to 30.
Couplets are sometimes written thus: 20:4 :: 50 : 10, and read, 20 is to 4 as 50 is to 10.*
287. TO FIND A MISSING TERM IN A PROPORTION.
36 : 12 :: x : 25.
The ratio of the first couplet is 3; that is, the antecedent is 3 times the consequent. Siuce the ratios of the couplets are equal, the ratio of the second couplet must be 3, and its antecedent must be 3 times its consequent. Three times 25 75, the missing term.
Find the missing term.
1. 90 : 45 :: x : 180
4. 20 : 60
= x : 225 5. 30 : 50
= x : 175 6. 90 : 20
= x : 140
* The ratio sign (:) may be regarded as the sign of division ( ;-) with the hori. zontal line omitted, and the proportion sign (::) the sign of equality (=) with an erasure through its center, thus: (= =).
Since the ratio of the first couplet is 3, the ratio of the second couplet must be 3, and x must equal 1 third of 48. i third of 48 is 16.
Find the missing term.
1. 84 : 21
4. 20:60 :: 120 : x
45 : x
: x =
EXAMPLE III. 36:.
45 : 15. The ratio of each couplet is 3; so each consequent must be 1 third its antecedent, and x, 1 third of 36, or 12.
The ratio of each couplet is 4; so each antecedent must be 4 times its consequent, and x, 4 times 12, or 48.
Find the missing term.
1. x:16 ::51:17
4. x: 96 = 23 : 92
* Since the ratio is 24 (1) the consequent must be of the antecedent.
288. PRACTICAL PROBLEMS.
1. If 75 yd. of cloth cost $115.25, how much will 15 yd. cost at the same rate ?
2. If 24 acres of land cost $76.20, how much will 15 acres cost at the same rate ?
3. If 7 tons of coal can be bought for $26, how many tons can be bought for $39 ?
7 tons : x tons : : $26 : $39.
4. If 36 lb. coffee can be bought for $7, how many pounds can be bought for $171 ?
5. If sugar sells at the rate of 18 lb. for $1, how much should 63 lb. of sugar cost ?
18 lb. : 63 lb. = $1 : x dollars.
6. If a post 6 ft. high casts a shadow 4 feet long, how high is that telegraph pole which at the same time and place casts a shadow 20 feet long ?
7. If a post 5 feet high casts a shadow 8 feet long, how high is that steeple which casts a shadow 152 feet long?
8. If a train moves 50 miles in 1 hr. 20 min., at the same rate how far would it move in 2 hours ?
9. If a boy riding a bicycle at a uniform rate goes 12 miles in 1 hr. 15 min., how far does he travel in 25 minutes ?
To The TEACHER –After the pupil has solved the above problems by making use of the fact of the equality of the ratios, he should solve them by an analysis somewhat as follows: Prob. 1. Since 75 yd. cost $115.25, 1 yd. costs ts of $115.25; but 15 yd. cost 15 times as much as 1 yd., so 15 yd. cost 15 times to of $115.25.
289. MAGNITUDES Which ARE PROPORTIONAL TO the
SQUARES OF OTHER MAGNITUDES.
Observe that the ratio of the areas of the above squares is f (or ). But the area of each circle is .78 + of its circumscribed square; so the ratio of the areas of the circles is for 9). See Book II, p. 256.
1. The diameters of two circles are as 3 to 4. Compare their areas.
2. The area of a 6-inch circle is how many times as great as the area of a 3-inch circle ?
3. If a 4-inch circle of brass plate weighs 3 ounces, how much will a 6-inch circle weigh, the thickness being the same in each case ?
4. If a piece of rolled dough 1 foot in diameter is enough for 17 cookies, how many cookies can be made from a piece 2 feet in diameter, the thickness of the dough and the size of the cookies being the same in each case ?
5. If a piece of wire of an inch in diameter will sustain a weight of 1000 lbs., how many pounds will a wire of an inch in diameter sustain ?
Proportion. 290. MAGNITUDES WHICH ARE PROPORTIONAL TO THE
CUBES OF OTHER MAGNITUDES.
The solid contents of two cubes are to each other as the cubes of their lengths.
The solid contents of two spheres are to each other as the cubes of their diameters.
Observe that the ratio of the solid contents of the above cubes is
(or ??). But the solid content of each sphere is .52+ of its circumscribed cube ; so the ratio of the solid contents of the spheres is (or 37). See Book II, page 266.
PROBLEMS. 1. The diameters of two spheres are as 3 to 4. Compare their solid contents.
2. The solid content of a 6-inch sphere is how many times as great as the solid content of a 3-inch sphere?
3. If a 4-inch sphere of brass weighs 10 lbs., how many pounds will a 6-inch sphere of brass weigh?
4. If a sphere of dough 1 foot in diameter is enough for 20 loaves of bread, how many loaves can be made from a sphere of dough 2 feet in diameter?
5. If the half of a solid 8-inch globe weighs 4 lbs., how much will the half of a solid 5-inch globe weigh, the material being of the same quality ?