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5. 30070 men went into battle; 4564 were slain, and 1300 were taken prisoners; how many were left? Ans. 24,206 men. 6. Take 229 oxen from 2006 oxen. Ans. 1777 oxen. Ans. 78,500.
7. Subtract 25 hundred from 81 thousand.
11. What number added to 9213628 will give 23475310? 12. What number subtracted from 7654321 will leave 369 ? 13. 86293210 minus 329876 equals how many?
14. 987621085 - 329875232= how many?
15. Find the sum of the last five answers. Ans. 771,984,860. 16. 3600807002 — 72824 — what?
17. 3478921+368754 — 2878796 what?
18. From 7654321 - 1234567 take 53899.
19. From 4673214+2792 take 98264.
20. 98432231-32636841 808994 what? 21. 8087670 — 7549094 — 89699 — what?
22. Find the sum of the last six answers. Ans. 77,642,007. 23. What is the difference between 19360742 and 9643278? 24. How many times can I take 7642 gallons from 21002 gallons, and what will remain?
25. If the minuend is 36 quadrillion and the subtrahend 95 million 86, what is the remainder. Ans. 35,999,999,904,999,914.
26. If the minuend be 69 trillion and the difference 85 billion, what is the subtrahend?
27. Philadelphia was founded in 1682. In what year was New York city settled, it having been settled 68 years before?
28. Victoria ascended the throne of England in 1837. How many years has she reigned?
29. Napoleon commenced his brilliant career in 1795. How many years before his final defeat in 1815?
30. The Israelites left Egypt in 1491 B. C., and 40 years after entered the land of Canaan. In what year did that event happen?
31. In the year 1851, London had 2362000 inhabitants; Pekin was estimated to have 1500000. How many more inhabitants had London than Pekin?
32. The equatorial diameter of the earth is 41843330 feet, and the polar diameter 41704788 feet; required the difference. 33. The population of St. Louis in 1850 was 77860, and in 1860, 160773; required the increase in 10 years.
34. James Nye has in his possession $172; he owes $28 to A, $36 to B, and $19 to C. After paying his debts, what will remain ?
35. I have saved from my income $362, and have $2180 in government bonds; how much more must I save that I may purchase a house worth $3500?
44. GENERAL REVIEW, No. 1.
1. Two persons, who are 200 miles apart, travel towards each other, one 46 miles, the other 51 miles a day; how far apart will they be at the end of one day?
2. If the above persons travel away from each other, how far apart will they be at the end of one day?
3. A man gave to his eldest son $3575, to his youngest son $4680, and to his daughter $2495 less than to the youngest son; his whole property was worth $20000; what sum remained?
4. A ship, which was valued at $15590, was sold at a loss of $4975; what did she bring?
5. If the subtrahend be 369 quadrillion, and the remainder 99 quadrillion 13 billion, what is the minuend?
6. The difference between two numbers is 95478. larger number is 148769; what is the smaller?
7. How many times can 18640 be subtracted from 46806, and what will remain?
8. Which of the two numbers 15672 or 10560 is nearer to 13465, and how much? ·
9. From what number must 846 be taken twice to leave 15684?
10. To what number must 962 be added three times to make 8472?
11. Which is nearer to 348628, 63248 +93264, or 6000£3 - 59321 ?
For Dictation Exercises, see Key.
45. Multiplication is the process of finding a number equal in value to one number taken as many times as there are units in another number. The number which is multiplied is called the Multiplicand, the number by which we multiply is called the Multiplier, and the result obtained is called the Product.
The multiplicand and multiplier are often called factors of the product, from the Latin facio, I make, because, being multiplied together, they make up the product. The product is also said to be the multiple of the factors. Thus, 7 times 6 = 42. Here, 7 is the multiplier, 6 the multiplicand, and 42 the product; or 7 and 6 are the factors of 42, which is their multiple.
The sign of multiplication is a small, oblique cross, X, read, times, or, multiplied by. Thus, 7 X 6 may be read either. 7 times 6, or 7 multiplied by 6. In the former case 7 is the multiplier and 6 the multiplicand, while in the latter 6 is the multiplier and 7 the multiplicand. The product is the same, whichever is the multiplier.
NOTE. In the process of multiplication, the multiplier must be an abstract number. We cannot multiply pencils by pencils, or pencils by apples, but either may be multiplied by an abstract number, and give a product of the same denomination as the concrete factor. (Art. 4.) 46. ILLUSTRATIVE EXAMPLE, I.
Multiply 2364 by 7.
2 tens and 8 units.
Seven times 4 units We write the 8 in the units' place, and reserve the 2 tens for the tens' place. 7 times 6 tens 42 tens, which, with the 2 reserved tens, 44 tens 4 hun
dreds and 4 tens; we write the 4 tens in the tens' place, and reserve the 4 hundreds for the hundreds' place. 7 times 3 hundreds 21
Multiple, 16548 Product.
hundreds, which, with the 4 reserved hundreds, 25 hundreds.2
NOTE. This result might be obtained by finding the sum of the num
ber 2364 taken seven times; that is, by adding 2364 to itself six times. Hence, Multiplication may be regarded as a short way of performing Addition.
1. 267 by 2; by 3; by 4; and add the products. Ans. 2,403.
3. 3401 by 8; by 9;
4. 90021 by 10; by 11; 5. 66285 by 12; by 8; 6. 4364 X 8
7. 7762 X 9
8. 5391 X 4 = what?
9. 3409 X 5 what?
14. Add the last eight products, and multiply by 7.
Multiply 3648 by 294.
15. 123456 X 6 = ?
9832 X 7
8349 X 6
48. ILLUSTRATIVE EXAMPLE, II.
22078 X 11
19869 X 12
987654321 X 7 =?
Here we are to multiply, not only by units, but by tens and hundreds. We write the numbers units under units, tens under tens, &c., and multiply first by the units, as before, and then by the tens. It is evident that the product of any number multiplied by tens will be ten times as great as if multiplied by the same number of units; multiplied by hundreds, one hundred times as great as if multiplied by units; multiplied by
thousands, one thousand times as great, etc. Hence, when a number is multiplied by tens, hundreds, or thousands, the products thus obtained are written one, two, or three places farther to the left than when multiplied by units; or, in other words, we multiply by the other terms as we multiply by the units, placing the first figure of each product under the term by which we multiply. The sum of these partial products is the entire product. Hence the
RULE FOR MULTIPLICATION.
Write the multiplier under the multiplicand. Beginning at the right, multiply each term of the multiplicand by each term of the multiplier, successively, placing the right hand figure of each partial product under the term by which you multiply, carrying as in addition. Add all the partial products, and the result will be the entire product.
49. PROOF I. Take the multiplicand for the multiplier, and the multiplier for the multiplicand. If the result thus obtained be like the first result, the work is probably correct.
50. PROOF II. By casting out the 9's. This method is much the easier, though not always sure.
NOTE. - To cast out the 9's from any number, commence at the left, and add the digits towards the right. When their sum equals 9 or more, reject 9 and add the remainder to the next digit, and so on. The last remainder is called the excess of 9's.
TO PROVE MULTIPLICATION BY CASTING OUT THE 9's.
Cast out the 9's from each of the factors. Then multiply the remainders, should there be any, cast out the 9's from the product, and note the last remainder. Cast out the 9's from the answer, and if the remainder equals the one obtained above, the work may be presumed to be right; thus,
4, 1st remainder. 2681 26+8 16: 9+7. 7+1= 8, 2d remainder.
3+2= 5, last remainder.
97009304 Ans. 7+3=10=9+1. 1+4= 5, which equalling the remainder above, the work is right.
NOTE. For demonstration of rule, see Appendix.