The excess over exact 9's, in each number, is ; 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 r 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 Dumber 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 × 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 highest denomination by the number of units in the scale at that place, 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 quotien is 71 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 Ibs.?

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, how many seconds?

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

ADDITION.

39. ADDITION is the operation of finding the sum of two or more numbers.

THE SUM of two or more numbers, is a number containing as many units as all the numbers taken together.

Operations of Addition.

The operations of Addition depend on four principles, viz.:

1. A single number expresses a collection of like units.

2. Like units alone can be added together; that is, units must be added to units, tens to tens, dollars to dollars, &c.

3. Every number expressed by two or more figures, is the sum of Its various units.

4. The sum of several numbers is equal to the sum of all their parts

1. What is the sum of 769 and 487?

ANALYSIS.-Write the numbers, so that the like units

may fall in the same column, thus:

Sum of the units

Sum of the tens

Sum of the hundreds

Entire sum

The example may be done in another way, thus: Set down the numbers as before: then say, 7 and 9 are 18: set down 6 in the units' place, and the 1 ten under the 8 in the column of tens. Then say, 1 to 8 are 9, and 6 are 15. Set down the 5 in the column of tens, and the 1 hundred in the column of hundreds. We then add the hundreds, and find their sum to be 12: hence, the entire sum of 1256.

OPERATION.

769

487

16

14

11

1256

OPERATION.

769

487

1 1

1256

NOTE.-1. Observe, that units of the same value are always written in the same column.

2. When the sum in any column equals or exceeds the units of the scale 10, it produces one or more units of a higher order, which belong to the next column at the left. In that case, write down the excess, and add the higher units to the next column. This is called carrying to the next column. The number to be carried, should not, in practice be written under the column at the left, but added mentally.