MISCELLANEOUS

339. Change 20 mi. 70 rd. 5 ft. to feet; change 106,760 ft. to integers of higher denominations.

340. Express

fraction of a rod.

rd. in yards and feet; express 3 yd. 2 ft. as a

341. Reduce .3975 of a mi. to integers of lower denominations; reduce 127 rd. 3 ft. 3.6 in. to the decimal of a mile.

342. Reduce 72 lb. avoirdupois to lb. troy; reduce 87.5 lb. apothecaries' to lb. avoirdupois.

343. Reduce 157.5 gal. to bu. approximately; reduce 16.8 bu. to gal. approximately.

344. Reduce 157.5 gal. to bu., exact to 1 decimal place; reduce 16.9 bu. to gal. exactly.

345. Reduce 38° 50′ 30′′ of arc to time; reduce 2 hr. 35 min. 22 sec. of time to arc.

346. Find the sum of 18 gal. 3 qt., and 6 gal. 3 qt. 1 pt.; find the difference between 25 gal. 2 qt. 1 pt., and 18 gal. 3 qt.

347. To Mar. 3, add 182 da.; from Sept. 1, subtract 182 da. 348. To Mar. 3, add 6 mo. 2 da.; from Sept. 5, subtract 6 mo. 2 da.

349. Find the time from the discovery of America, Oct. 21, 1492, to the Declaration of Independence, July 4, 1776.

350. Find the exact number of days from the fourth of July to Christmas.

351. How many cu cm are there in 91? 1 in 9000 cu cm? 352. Give the weight in Kg of 5 cu m of water; give the number of cu m in 5000 Kg of water.

353. How much does a grocer gain by buying 3 bu. of chestnuts at $3 a bu. dry measure, and selling for 5¢ a half pint liquid measure?

354. How much does an apothecary gain by buying 50 lb. of medicine at 20 a lb. avoirdupois weight, and retailing it at 10% an ounce apothecaries' weight?

LITERAL QUANTITIES

NOTATION AND NUMERATION

A number may be considered under conditions that are directly opposed.

The same quantity, base, may be used several times as a factor. The expression is abbreviated by writing. the base and, over it, a number, exponent, denoting how many times the base is used as a factor.

1. What does + 4 mean? -4?

aaaaa4; 2×2×2×2=24.

In the former, a is the base and 4 the exponent; in the latter, 2 is the base and 4 the exponent.

Since has an arbitrary signification, we may assume it to mean any condition which has an opposite. Thus: +4 may mean 4 to the right, 4 up, 4 to the north,

....

Since ' means the opposite of '+,' if + 4 means 4 to the right, 4 means 4 to the left; . . To find the meaning of a ‘— quire the meaning of the '+' sign in the same position.

sign, we must in

2. Write a, expressing both coefficient and exponent.

1

Ans. 1a+1. When either coefficient or exponent is omitted, '+1' is always understood.

3. If+6 mi.' means 6 mi. east, what does ' 8 mi.' mean?

4. Analyze, explain the meaning of, and read, 3 að.

3 a5 is a term; 3, the coefficient; a, the base; 5, the exponent. It means a5 + a5 + a5, or that a is taken 5 times as a factor, and that the result is taken 3 times as an addend. It is read, 3, a to the fifth power.

5. Write in simplest form that 2 is used 6 times as an addend; that a is used 6 times as an addend.

6. Write in simplest form that 2 is used 6 times as a factor; that a is used 6 times as a factor.

7. Write in simplest form that a is used 6 times as a factor, and that the result is used 5 times as an addend.

8. Analyze, explain the meaning of, and read, 4 x3.

9. What is the meaning of +5? of -5? What is the meaning of ' — $ 6,' if' + $6' means 'worth $6'?

10. After losing 34, a boy had 4 left. How much had he at first? Analyze.

11. After losing af, a boy had be left. How much had he at first? Analyze.

Ans. (a+b). 12. After losing a certain sum, a boy had 4¢ left. If he had 7 at first, how much did he lose? Analyze.

13. After losing a certain sum, a boy had be left. If he had af at first, how much did he lose? Analyze.

14. At 2 each, how much will 5 apples cost?

Ans. (a — b)¢.

x apples?

Ans. bx cents.

15. At be each, how much will x apples cost? 16. If 4 apples cost 8, how much will 1 apple cost? If x apples cost 8, how much will 1 apple cost?

17. If x apples cost be, how much will 1 apple cost?

b

Ans. -.

х

18. At 4 each, how many apples can be bought for 8¢? for x? Analyze.

19. At x each, how many apples can be bought for a¢?

20. If a boy had a marbles, and lost y of them, how many had he left? Ans. (xy) marbles. 21. If a dog runs b ft. in 1 minute, how far will he run in C min.? in 8 min.? in x min. ?

22. When eggs sell at x a dozen, what is the selling price of each egg?

Ans. ¢. 12

23. In Ex. 22, what is the selling price of 3 eggs ? 24. If eggs cost af each, how much will 1 egg cost if the price is increased 1¢? Ans. (a+1).

Ans. 2 x cents.

25. If 3 eggs sell for 64, what is the selling price of 4 eggs? of 5 eggs? of x eggs? 26. How many horses at x dollars each, must a man sell to pay for b cows at c dollars each? bc Ans. horses.

27. The product of the sum and difference of 4 and 2 is 42 - 2o. Make a similar statement with letters instead of numbers.

ADDITION

I. To add when the signs are alike, write the sum and use the common sign; to add when the signs are unlike, write the difference and use the sign of the greater.

left.

In these examples, let us assume that + means to the right, and

까

to the

To + 3 add + 2. + 3 means 3 to the right, one, two, three; + 2 means 2 to the right, one, two; counting, we have 5 to the right, or + 5.

To 3 add + 2. - 3 means 3 to the left, one, two, three; + 2 means 2 to the right, one, two; counting, we have 1 to the left, or 1.

In like manner, the other results may be proved to be - 5, and +1.

Examining these results, we see that there are two cases; where the signs are alike, and where the signs are unlike; and that the results are + 5, − 5, +1, and — 1.

II. To add quantities having a common factor, add the factors not common and retain the common factor.

6 a 2 a means 6 x a + 2 x a, or that a is taken 6 times as an addend, and then 2 times more as an addend; or 6+2, or 8, times as an addend, or 8 a.

Hence, the rule.

NOTE. The pupil should compare this with the same principle on p. 72.