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Geometry.

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274. THE AREA OF A RHOMBOID.*

1. One side of a rhomboid may be regarded as its base. The perpendicular distance from the base to the opposite side, is its altitude.

f

In the figure here given cf is the base and ab the altitude.

2. Convince yourself by measurements and by papercutting that from every rhomboid there may be cut a triangle, (abc), which when placed upon the opposite side, (def), converts the rhomboid into a rectangle (adeb).

Observe that the base of the rectangle is equal to the base of the rhomboid, and the altitude of the rectangle equal to the altitude of the rhomboid.

3. A rhomboid is equivalent to a rectangle having the same base and altitude. Hence, to find the area of a rhomboid, find the area of a rectangle whose base and altitude are the same as the base and altitude of the rhomboid. Or, as the rule is given in the older books,-"Multiply the base by the altitude."

PROBLEM.—If the above figure represents a piece of land, and is drawn on a scale of } inch to the rod, how many acres of land ?

* The statements upon this page apply to the rhombus as well as to the rhomboid.

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1. Mr. Watson purchased 15 shares of C., B. & Q. R. R. stock at 12 % discount. (a) How much did he pay for the stock? (b) When a 3% dividend is declared and paid, how much does he receive ? *

2. James Cooper bought 12 shares of stock in the Sugar Grove Creamery at 8 % below par, and a few days after sold the stock at 5 % above par. How much more did he receive for the stock than he gave for it?

3. A certain city borrowed a large sum of money and issued therefor 10-year 5 % bonds with the interest payable semi-annually. (a) How many coupons were attached to each bond? (b) On a $1000 bond, each coupon should call for how much money?

4. Sometimes such bonds as those described in problem 3, are offered for sale to the highest bidder, in “blocks” of $10000, $20000, or $50000. If a $20000 “block” is “bid off” at 21% premium, how much should the city receive for the “block"?

5. What must be the nominal value of 5% bonds that will yield to their owner an annual income of $750 ?

5.x
Let x = the nominal value; then = $750.

100 6. What must be the nominal value of 4 % bonds that will yield to their owner an annual income of $720 ?

7. A owns $6000 of 5 % bonds ; B owns $8000 of 41 % bonds. How much greater is the annual income from B’s bonds than from A's?

8. Find the area of a piece of land in the form of a rhomboid, whose base is 32 rods and whose altitude is 15 rods.

9. Find the area of a piece of land in the form of a rectangle, whose base is 32 rods and whose altitude is 15 rods.

* The par value of each share of stock menticned on this page is $100.

RATIO,

276. Ratio is relation by quotient. The two numbers (magnitudes) of which the ratio is to be found are called the terms of the ratio. The first term is called the antecedent and the second term the consequent. The ratio is the quotient of the antecedent divided by the consequent.

The usual sign of ratio is the colon. It indicates that the ratio of the two numbers between which it stands is to be found, the number preceding the colon being the antecedent and the number following it, the consequent. The expression, 12:4=3, is read, the ratio of 12 to 4 is 3.

EXERCISE.

Read and complete the following: 1. 12:4

4:12 = 2. 18:9

9:18 3. 15 : 5

5:15 4. 40:10

10:40 =

12:2
18:6
15:10
40:20

Note.-It will be observed that the sign of ratio is the sign of division ( + ) with the line omitted.

277. Every integral number is a ratio. The number 4 is the ratio of a magnitude 4 (inches, ounces, bushels,) to the measuring unit 1 (inch, ounce, bushel). The number 7 is the ratio of 7 yards to 1 yard; of 7 dollars to 1 dollar, or of 7 seconds to 1 second, etc.

NOTE.–The ratio aspect of numbers is not the aspect most frequently uppermost in consciousness: neither ought it to be. But the pupil should now see that number is ratio; that while it implies aggregation and often stands in consciousness for magnitude, its essence is relation-ratio.

Ratio.

278. Every fractional number is a ratio. The fraction is the ratio of the magnitude 3 to the magnitude 4.

So 12, (3), is the ratio of 12 to 4. Observe that in every case the terms of a ratio may be written as the terms of a fraction; the antecedent becoming the numerator and the consequent the denominator of the fraction. The fraction itself is the ratio.

EXERCISE I.

Make the terms of the ratio the terms of a fraction; then reduce the fraction to its siniplest form.

1. The ratio of 20 to 6 is ? 10 = 31.
2. The ratio of 6 to 20 is 2 = d.
3. The ratio of 7 to 5 is

of 5 to 7,
4. The ratio of 12 to 1 is -; of 1 to 12,

EXERCISE II.

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1. is the ratio of 5 to 7; of 10 to 14; of 15 to 21, etc. 2. i is the ratio of - to —; of - to

of

– to etc. 3. , is the ratio of to of to of - to —, etc. 4. 8 is the ratio of 8 to 1; of to —; of

to etc.

EXERCISE III.

Make the necessary reduction and find the ratio: *

1. Of 2 feet to 8 inches.
2. Of 3 yards to 6 inches.
3. Of 6 rods to 3 yards.
4. Of 2 rods 5 yards, to 1 yard 1 foot.
5. Of 3 miles 40 rods to 1 mile 80 rods.

* The comparison of two magnitudes involves their measurement by the same standard. To compare feet with inches, the inches may be changed to feet or the feet to inches, or both may be changed to yards.

Ratio.

279. Not only is number itself ratio, but a large part of the work in arithmetic is merely the changing of the form of the expression of ratios.

EXERCISE IV.

(Reducing fractions to their lowest terms.) 1. Express the ratio of 30 to 40 in its simplest form. 2. Express the ratio of 560 to 720 in its simplest form. 3. Express the ratio of 425 to 875 in its simplest form.

4. Express the ratio of 5 min. to 2 hours in its simplest form.

5. Express the ratio of 1 lb. 4 oz. to 5 lb. 8 oz. in its simplest form.

EXERCISE V.

(Reducing improper fractions to integers.) 1. Express the ratio of 400 to 50 in its simplest form. 2. Express the ratio of 375 to 25 in its simplest form. 3. Express the ratio of 256 to 16 in its simplest form.

4. Express the ratio of 3 hours 20 minutes to 50 minutes in its simplest form.

5. Express the ratio of 8 lb. 12 oz. to 1 lb. 4 oz. in its simplest form.

EXERCISE VI.

(Reducing complex fractions to simple fractions.) 1. Express the ratio of 4 to 1 in its simplest form. 2. Express the ratio of to in its simplest form. 3. Express the ratio of 23 to 8} in its simplest form.

4. Express the ratio of i of an inch to 1 foot in its simplest form.

NOTE.-Observe that the denominator in fractions corresponds to the consequent in ratio.

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