23. 20×3+(40-7 x 5)=? | 28. 100-12 × 7+20—4=

10%. When the multiplier consists of two or more orders of units.

first partial product. 4 tens times 8 units are 32 tens, or 3 hundreds and 2 tens. Write the 2 tens in the tens' place, in the second partial product, and reserve the 3 hundreds to add to the product of hundreds.

4 tens times 7 tens are 28 hundreds, and 28 hundreds plus 3 hundreds reserved, are 31 hundreds, or 3 thousands and 1 hundred. Write the 1 hundred in the hundreds' place in the second partial product, and reserve the 3 thousands to add to the product of thousands.

4 tens times 6 hundreds are 24 thousands, and 24 thousands plus 3 thousands reserved, are 27 thousands, or 2 tens of thousands and 7 thousands, which write in the second partial product. The sum of the partial products is the entire product 31188.

*The operations of multiplication and division, indicated by signs, must be performed before those of addition and subtraction, unless otherwise indicated by a parenthesis or vinculum.

RULE.-I. Write the multiplier under the multiplicand 30 that units of the same order stand in the same column. When the multiplier consists of one figure.

II. Begin at the right and multiply the units of each order of the multiplicand by the multiplier. Write in the product the units of each result, and reserve the tens to add to the next result.

When the multiplier consists of more than one figure.

III. Multiply the multiplicand by the units of each order of the multiplier successively, beginning at the right, and write the right-hand figure of each partial product under the order of the multiplier used.

The sum of the partial products is the required product. PROOF.-Review the work carefully, or multiply the multiplier by the multiplicand; if the results are the same, the work is probably correct.

When there are ciphers in the multiplier, multiply by the sig. nificant figures only, since the product of any number by 0 is 0.

8. Multiply 6432 by 75; by 67; by 136.

9. Multiply 23072 by 128; by 243; by 307. 10. Multiply $420.06 by 204; by 666; by 408. What is the value

11. Of 67 hogsheads of sugar, at $37.75 a hogshead?

12. Of 2347 acres of land, at $136 an acre?

13. Of 64 horses, at $219.75 each ?

14. What will be the cost of building a line of telegraph 274 miles long, at $967 a mile?

15. If 1049 pounds of seed cotton be raised from an acre of land, how many pounds will 386 acres produce? 16. If a cotton mill manufactures 628 yards of cloth a day, how many yards can it make in 297 days? What is the product

23. Of 370607 by 4071 ?

24. Of 600326 by 2645? 25. Of 730096 by 5006 ?

17. Of 2572 bushels by 94? 18. Of $403.06 by 127? 19. Of 86072 pounds by 208? 20. Of 316 times $487.46? 21. Of 507 times 30975 days? 22. Of 325 times 6408 cents? 29. Find the cost of 386 railway coaches, at $7034.75 each.

26. Of 2407068 by 3406 ? 27. Of 408091 by 2407? 28. Of 73069 by 46035 ?

30. What cost 802 tubs of butter, at $27.08 each?

31. 236 x 63 x 28= ?

32. 439 x 0 x142= ?

33. 1927 x 613 × 802= ?

34. 4605 × 2034 × 576=?

35. How many yards of shirting in 49 bales, each bale containing 26 pieces, and each piece 57 yards?

36. What is the cost of 128 barrels of beef, each containing 216 pounds, worth 13 cents a pound?

37. Three schooners, ship 239 cords of wood each, and a fourth ships 248 cords. What is the value of the whole at $4.25 a cord?

38. If it require 108 tons of iron rail for 1 mile of rack, how many tons will be required for 476 miles, and what will be its value at $145 a ton?

39. A crop of cotton was put up in 472 bales, the average weight of which was 588 pounds. What was the weight of the whole crop, and its value at 18 cents a pound?

108. To multiply by the factors of a number.

The Factors of a number are the numbers which multiplied together will produce it. Thus, 6 and 7 are factors of 42; 2, 4, and 5 are factors of 40.

The pupil should carefully distinguish between the factors and e parts of a number. The factors are multiplied, but the parts are added, to produce a number. A factor is always a part, but a part is not always a factor.

Thus, 2 and 9, 3 and 6, 2, 3, and 3, are factors of 18; but the parts of 18 are 9 and 9, 10 and 8, 6 and 12, 7 and 11, etc.

109. PRINCIPLE.-The product of any number of factors will be the same in whatever order they are multiplied.

It will be observed that the multiplicand, multiplied by the given multiplier, or by any set of factors into which it can be separated, produces the same result.

RULE.-I. Separate the multiplier into two or more factors.

II. Multiply the multiplicand by one of the factors, the resulting product by another factor, and so continue until all the factors have been used.

The last product will be the product required.

10. What will 56 acres of land cost, at $164.50 an acre? 11. At 28 cents a pound, what will be the cost of 24 sacks of coffee, each containing 64 pounds?

12. What is the value of 107 pieces of cloth, each piece containing 42 yards, at $4.28 a yard?

110. When either the multiplicand or multiplier, or both, have ciphers on the right.

1. Multiply 286 by 100.

OPERATION.

286

100 28600

ANALYSIS.-Since removing a figure one place to the left, increases its value ten times (44), annexing a cipher to a number multiplies it by 10; annexing two ciphers multiplies it by 100, etc. Hence 286 × 100=28600, the product required.

2. Multiply 3240 by 600.

OPERATION.

3240

600 1944000

ANALYSIS. 3240=324 × 10, and 600=6×100. First multiply together the two factors 324 and 6, and then multiply their product 1944, by 10 × 100, or by 1000, by annexing three ciphers, which gives 1944000, the required product.

What is the product

3. Of 372 by 10? By 100? By 1000? By 10000 ? 4. Of 860 by 50? By 400? By 1500? By 3000?

RULE. To the product of the significant figures, annex as many ciphers as there are ciphers on the right of either or of both of the factors.