When

The sign X, is called the sign of multiplication. placed between two numbers, it denotes that they are to be multiplied together. Thus, 12 x 3, denotes that 12 is to be multiplied by 3.

The parenthesis is used to indicate that the sum or difference of two or more numbers is to be regarded as a single number. Thus, (2 + 3 + 5) × 6,

shows, that the sum of 2, 3, and 5, is to be multiplied by 6. And (5 − 3) × 6,

denotes that the difference between 5 and 3, is to be multiplied by 6.

The sign, is called the sign of division. When placed between two numbers, it denotes that the one on the left is to be divided by the one on the right. Thus, 45, denotes that 4 is to be divided by 5.

Properties of the 9's.

34. In any number, written with a single significant figure, as, 4, 40, 400, 4000, &c., the excess over exact the number of units in the significant figure. number may be written thus,

9's is equal to For, any such

Each of the numbers 9, 99, 999, &c., contains an exact number of 9's; hence, when multiplied by 4, the several products will contain an exact number of 9's: therefore,

33. What is the sign of Equality? What is the sign of Addition? What of Subtraction? What of Multiplication? For what is the parenthesis used? What is the sign of Division?

34. What will be the excess over exact 9's in any number expressed by a single significant figure? How may the excess over exact 9's be found in any number whatever?

The excess over exact 9's, in each number, is 4 ; and the same may be shown for each of the other significant figures.

If we write any other number, as

6253,

we may read it, 6 thousands, 2 hundreds, 5 tens, and 3. Now, the excess of 9's in the 6 thousands, is 6; in 2 hundreds, it is 2; in 5 tens, it is 5; and in 3, it is 3: hence, in them all, it is 16, which is one 9, and 7 over: therefore, 7 is the excess over exact 9's in the number 6253. In like manner,

The excess over exact 9's, in any number whatever, is found by adding together the significant figures, and rejecting the exact 9's from the sum.

NOTE. It is best to reject or drop the 9, as soon as it occurs: thus, we say, 3 and 5 are 8 and 2 are 10; then, dropping the 9, we say, 1 to 6 is 7, which is the excess; and the same for all similar operations.

1. What is the excess of 9's in 48701? In 67498?

2. What is the excess of 9's in 9472021? In 2704962 ? 3. What is the excess of 9's in 87049612? In 4987051?

REDUCTION.

35. REDUCTION is the operation of changing a number from one unit to another, without altering its value.

36. REDUCTION DESCENDING is the operation of changing a number from a greater unit to a less.

37. REDUCTION ASCENDING is the operation of changing a number from a less unit to a greater.

38. If we have 4 yards, in which the unit is 1 yard, and wish to change to feet, the units of the scale will be 3, since 3 feet make 1 yard; therefore, the number of feet will be

4 x 3 12 feet.

35. What is Reduction?-36. What is Reduction Descending? — 37. What is Reduction Ascending?

If it were required to reduce 12 feet to inches, the units of the scale would be 12, since 12 inches make 1 foot: hence,

4 yards = 4 × 3 = 12 feet 12 x 12 = 144 inches. If, on the contrary, we wish to change 144 inches to feet, and then to yards, we would first divide by 12, the units of the scale in passing from inches to feet; and then by 3, the units of the scale in passing from feet to yards. Hence,

1st. To reduce a number from a higher unit to a lower:

Multiply the units of the highest denomination by the number of units in the scale, and then add to the product the units of the next lower denomination. Proceed in the same manner through all the denominations till the number is brought to the required denomination.

2d. To reduce a number from a lower unit to a higher:

Divide the given number by the number of units in the scale, and set down the remainder, if there be one. Divide the quotient thus obtained, and each succeeding quotient in the same manner, till the number is reduced to the required denomination: the last quotient, with the several remainders annexed, will be the answer.

Examples.

1. Reduce £3 14s. 4d. to pence. We first multiply the £3 by 20, which gives 60 shillings. We then add 14, making 74 shillings: we next multiply by 12, and the product is 888 pence : to this we add 4d. and we have 892 pence, which are of the same value as £3 14s. 4d.

If, on the contrary, we wish to change 892 pence to pounds, shillings, and pence, we should first divide by 12: the quotient is 74 shillings, and 4d. over. We next divide by 20, and the quotient is £3, and 14s. over: hence, the result is £3 14s. 4d.,

which is equal to 892 pence.

The reductions, in all the denominate numbers, are made in the same manner.

6. In $426, how many cents? How many mills?

7. In 36 eagles 8 dollars and 6 dimes, how many cents?

8. In 8750 mills, how many dollars and cents?

9. In 43 eagles 3 dollars and 5 mills, how many mills? 10. In £37 9s. 8d., how many pence ?

11. In 1569 farthings, how many pounds, shillings, pence, and farthings?

12. In 7 T. 14 cwt. 1 qr. 20 lb. Avoirdupois, how many pounds?

13. In 15445 lb. Avoirdupois, how many tons, cwts., qrs., and lbs. ?

14. How many grains of silver in 4 lb. 6 oz. 12 dwt. and 7 gr.?

15. How many pounds, ounces, pennyweights, and grains of gold in 704121 grains?

16. In 5 13 13 10 2 gr. Apothecaries' weight, how many grains?

17. In 174947 grains, how many pounds, ounces, drams, scruples, and grains?

18. In 6 yards 2 feet 9 inches, how many inches?

19. In 5 miles, how many rods, yards, feet, and inches? 20. In 2730 inches, how many yards, feet, and inches? 21. In 56 square feet, how many square yards?

22. In 355 perches, or square rods, how many acres, roods, and perches?

23. In 456 square chains, how many acres?

24. In 3 A. 2 R. 8 P., how many perches?

25. In 14 tons of round timber, how many cubic inches?

26. In 31 cords of wood, how many cubic feet?

27. In 56320 cubic feet, how many cords? 28. In 157 yards of cloth, how many nails? 29. In 192 ells Flemish, how many yards?

30. In 97 yd. 3 qr., how many ells English?

31. In 4 hhd. wine measure, how many quarts?

32. In 7560 pints, wine measure, how many hogsheads? 33. In 7 hogsheads of ale, how many pints?

34. In 74304 half-pints of ale, how many barrels? 35. In 31 bushels, dry measure, how many pints?

36. In 2110 pints, dry measure, how many bushels? 37. In 2 solar years of 365 d. 5 h. 48 m. 48 sec., each, many seconds?

how

38. How many months, weeks, and days in 254 days, reckoning the month at 30 days?