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Process.

Explanation. Paid.

$3586 Since he paid $3586 at one time and Paid.

2000

$2000 at another time, he paid at both times

$3586 + $2000, or $5586. Total paid. 5586

Since his note, or debt, was $6000, he

still owes the difference between $6000 and Note.

$6000 $5586, or $414. Paid.

5586

2. A man finished a journey of Balance due,

$414

972 miles in 3 days; the first day he travelled 398 miles; the second day, 409 miles. How many miles did he travel the third day?

Process Indicated. 972 miles (398 miles + 409 miles) miles travelled third day. 3. A produce dealer had in bank $6032, and checked out on one day $2360, and on the next day, $2307. How much had he left in bank?

4. A cattle dealer had 982 cattle, bought 621 more, lost by disease 32, sold 416. How many remained ?

5. How much does the sum of 3694 and 5005 exceed the difference of 10,532 and 3903?

6. In 1880 there were 16,120 Indians and 75,025 Chinese in California. How many were there of both, and how many more Chinese than Indians ?

7. A. sells a house to B. for $3486 ; B. sells it to C. at a gain of $360; C. sells it to D. at a loss of $285. What does D. pay for the house?

? 8. I have a yearly income of $10,000. I pay $450 for rent, $230 for fuel, $50 for medical attendance, and $4786 for all my other expenses.

What have I saved at the end of the year?

9. A tract of land containing 2753 acres was divided among four persons, A., B., C., D. A.'s share was 679 acres,

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B.'s was 47 acres more, C. had 75 acres less than B., and D. had the remainder. What were the shares of B., C., and D.?

10. A man has an income of $1845; he spends $645 for board, $456 for clothing, and $297 for other expenses. What has he saved at the end of the year ?

11. A man deposits in bank $2374. At one time he draws out $897, at another, $543, and at a third time, $689. How much has he remaining in bank?

12. I bought 24 shares of bank stock for $2863, and paid a broker $22 for purchasing the same; afterwards sold it for $3000. What was my profit?

13. A farmer invests $18,975 as follows: in land, $11,893; in horses, $1575; in mules, $4297 ; in stock, $937; the remainder in tools. How much did he expend for tools ?

14. A man bought three houses; for the first he gave $3585; for the second, $5260; for the third, as much as for the other two. He sold them all for $15,280. Did he gain or lose, and how much ?

15. From the sum of 874 and 398 subtract their difference.

16. I have a bin that holds 936 bushels. I put into it 383 bushels, and again 457 bushels. How much more will the bin hold ?

17. Some excursionists made a journey that cost them $492.97. Railroad fares cost $203.26 ; hack hire cost $48.36; steamboat fare cost $72.46. The remainder was expended for food. What did their food cost ?

18. A merchant bought silk for $486, muslin for $286, linen for $346, and sold the whole for $1200. How much did he gain?

19. An estate worth $23,460 was bequeathed to a wife and two children. The widow received $7820; the son received $3400 less; and the daughter, the balance. Find the daughter's share.

a

REVIEW. 1. Define the following terms: 1. Minuend.

6. Analysis. 2. Subtrahend.

7. Synthesis. 3. Difference

8. Solution. 4. Remainder.

9. Parenthesis. 5. Minus.

10. Vinculum. 2. Repeat the principles of Subtraction. 3. Repeat the rule for Subtraction.

4. Invent five problems involving both Addition and Subtraction, and indicate the process of solution.

,

MULTIPLICATION.

INDUCTIVE STEPS.

1. How many are 3 + 3?
2. How many are 3 + 3 + 3?
3. How many are two times 3 ?
4. How many are three times 3 ?

5. 3+3+3=9, and 3 times 3=9. The first operation is addition ; the second is multiplication. Which is the shorter ?

6. Is Multiplication a short kind of addition ? 7. What will 4 apples cost at 2 cents apiece ? Process.

Explanation. ADDITION. MULTIPLICATION. Since 1 apple costs 2 cents, 4 apples

will cost 4 times 2 cents, which are 8 2 cents 2 cents

In the preceding process, is 2 cents

cents

2 cents concrete? Is 8 cents 2 cents 4

concrete ? Is the 4 concrete or 8 cents 8 cents abstract ?

cents.

8. What is the cost of 5 yards of ribbon at 10 cents a yard ?

Write and explain the process. In the preceding process, did you write 5 yards or simply 5? What kind of number is 5?

You took 10 cents five times; hence 5 is called the Multiplier, and the process is called Multiplication.

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DEFINITIONS. 1. Multiplication is a short process of finding the sum of two or more equal numbers; or of taking a number as many times as there are units in another number.

2. The Multiplicand is the number to be taken or repeated.

3. The Multiplier is the number which shows how many times the multiplicand is to be taken or repeated.

4. The Product is the result of the multiplication.

5. The Multiplicand and Multiplier are called Factors of the product.

6. The Sign of Multiplication is an oblique cross, X.

5 X 4=20, is read “5 times 4 are 20”; or, “5 multiplied by 4 equals 20."

7. What are the factors of 20 ?
Is 4 X 5 = 5 X 4 a correct equation ?

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a

The 20 stars as arranged equal 5 stars in a line taken 4 times, or 4 stars in a column taken 5 times.

Hence the product is the same in whatever order the factors are taken.

8. What is a parenthesis and what does it signify ? A vinculum?

134 — (9 + 6) X 3 signifies that you must take 9 + 6 three times and subtract the result from 134.

Process.
(9 + 6) X 3 = 45; 134 — 45 = 89.

PRINCIPLES.

1. The multiplier must be considered an abstract number. 2. The product and multiplicand are like numbers. 3. Either factor may be taken as the multiplier.

EXERCISES.

FOR ANALYTIC AND SYNTHETIC EXPLANATION. 1. What is the product of 347 multiplied by 3?

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1st
2d
3d

Explanation.
Process. Process. Process.
347

1st PROCESS.–We say,

duct is the result of multiplication. 3

Since multiplication is a short process 347 21

of adding equal numbers, we can find 347 12 347 the product by addition; adding, we

have 1041." 347 9

3
2D PROCESS. - We say,

6. Since 1041 1041 1041 347 must be taken 3 times, each order

of units must be taken 3 times. 3 times 7 units 21 units; 3 times 4 tens 12 tens; 3 times 3 hundreds = 9 hundreds; adding, we have 1041."

3D PROCESS.—The shortest process is generally the best in practice. We say, “3 times 7 = 21; we reserve the 2; 3 times 4 12, 12 and 2 reserved are 14; we reserve the 1; 3 times 3 9, 9 and 1 reserved Hence the product is 1041.”

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= 10.

Proof.

The first and second processes are proof of the accuracy of the third.

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