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(9.)

.85 463.27 39.99

1.58 6,598.86 9,005.79 95,783.04 2,469.98

956.83 14,816.00 3,947.25

(10.)
$901.09

91.85

387.24 19,877.46

19.90 101.99 3,972.87 79,841.24

18.72 3,120.50

14.12

(11.)

.3789 .7398 4.217 3.95 45.007 4.256 3.520 23.3 29.317 343.28 1899.

REVIEW.
1. Define the following terms :
1. Like numbers.

5. Sign of Addition.
2. Unlike numbers. 6. Equation.
3. Addition.

7. Indicated process.
4. Sum.

8. Process. 2. Repeat the principles of Addition. 3. Repeat the rule for Addition.

4. Invent five problems in Addition and indicate their solution.

SUBTRACTION.

INDUCTIVE STEPS. 1. How many are 6 units less 3 units ? 7 tens less 5 tens ? 8 millions less 4 millions ?

2. If you have $9 and spend $5, how many dollars do

you retain ?

Process.

Solution. 89 If I have $9 and spend $5, I retain the difference between $5

$9 and $5, which is $4. $4 Is that explanation analytical or synthetical ?

3. $5 + $4 = how many dollars ? 4. Was it analysis or synthesis that gave you the $9?

5. Does the synthesis, then, prove the correctness of the analysis?

6. Robert is 10 years of age and Richard is 8. What is the difference of their ages? Write and explain the process. Prove the correctness of the result.

7. There were 7 bunches of ripe grapes on a vine; a fox took 2 bunches. How many bunches remained ?

8. Have you been finding the difference between like numbers ?

9. Finding the difference between two numbers is called Subtracting.

10. What is the difference between 6 horses and 3 sheep ?

11. Subtraction of numbers makes what requirement as to their units ?

DEFINITIONS. 1. Subtraction is the process of finding the difference between two like numbers.

2. The greater number is called the Minuend; the less number is called the Subtrahend; the result is called the Difference or Remainder.

3. The Sign of Subtraction is a short horizontal line, called minus (less), and is always placed after the minuend and before the subtrahend.

7–5= 2 is read “7 minus 5 equals 2.” The form, 7 — 5= 2, is called what?

PRINCIPLES.

1. Only like numbers and orders can be subtracted. 2. Subtrahend + Remainder Minuend.

=

1. From 54 subtract 33.

Process.

Explanation. 54

The minuend, 54 = 5 tens + 4 units; 33 the subtrahend, 33 = 3 tens + 3 units.

2 tens + 1 unit 21. 21

PRINCIPLE.Only like orders can be subtracted. We therefore write the 3 units under the 4 units and the 3 tens under the 5 tens. We then say "4 units — 3 units 1 unit; 5 tens 3 tens

2 tens. Hence the remainder is 21."

Proof. PRINCIPLE.The subtrahend + the remainder

33 + 21 = 54.

the minuend.

2. From 469 subtract 327.

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Process.

Explanation. 469

469 4 hundreds + 6 tens + 9 units.

327 327

3 hundreds + 2 tens + 7 units.

1 hundred + 4 tens + 2 units 142. 142

PRINCIPLE.Only like orders can be subtracted. We therefore write the 4, 6, and 9 of the minuend, and under them the 3, 2, and 7 of the subtrahend, with units under units, tens under tens, and hundreds under hundreds. We now say “9 units — 7 units 2 units; 6 tens — 2 tens - 4 tens; 4 hundreds 3 hundreds 1 hundred. Hence the difference is 142."

Show proof.

EXERCISES.

Copy, subtract, explain, prove:

(1.) (2.) (3.) (4.) 824 569 997 965 413 245 743 752

(5.) 896 544

(6.) 8953 3420

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PROBLEMS. NOTE.—Let the pupil first indicate the solution of each problem by using the minus sign,

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1. An army went into battle with 6878 men, and came out with only 4345 men. How many men were missing ?

Process Indicated.
4345 men = the number missing.

6878 men

Process.

Explanation.

6878 1. Since the army went into battle with 6878 men and 4345

came out with only 4345, the number missing was 6878

minus 4345. 2533 2. Since the unit of both the numbers is one man, the

numbers are like and can be subtracted, the less from the greater; units from units, tens from tens, etc. Therefore we say “8 units 5 units 3 units, 7 tens 4 tens

3 tens, 8 hundreds 3 hundreds 5 hundreds, 6 thousands 4 thousands 2 thousands. Hence the number of men missing was 2533."

=

Proof.

=

PRINCIPLE.—Subtrahend + Remainder Minuend.

4345 + 2533 = 6878.

2. A grain dealer, having 7890 bushels of wheat, sold 6370 bushels. How many bushels had he remaining ?

Process Indicated. 7890 bushels 6370 bushels the bushels remaining. 3. Watches were invented at Nuremburg in 1477. How many years ago ?

4. If I borrow $6798, and afterwards pay $3534, how much do I still owe?

5. Under a call for volunteers, California's quota was 3237 men; Arkansas's quota, 2025 men, Find the difference?

6. The population of Spain in 1820 was about 11,000,000; at present (1899) it is 17,550,216. Find the increase.

7. The exports of the United States from the Philippine Islands last year amounted to $4,982,857; their imports, to $162,446. Find the excess of the exports over the imports.

8. The population of Havana is 198,720, of Santiago, 71,300. Find the difference.

9. The telescope was invented in 1610. How many years between that date and 1899 ?

10. Harvey discovered the circulation of the blood in 1619. How many years after the invention of the telescope ?

CHIEF DIFFICULTY OF SUBTRACTION.

1. From 594 take 368. Process.

Explanation. 594

ANALYTIC AND SYNTHETIC. 368

594 5 hundreds + 9 tens + 4 units.

368 3 hundreds + 6 tens + 8 units. 226.

The difficulty is that 8 units cannot be taken from 4

units. But one of the 9 tens 10 units; 10 units + 4 Proof.

units = 14 units. Hence we write :

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594 5 hundreds + 8 tens + 14 units

368 3 hundreds + 6 tens + 8 units ing we have 2 hundreds + 2 tens + 6 units

594

226.

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