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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.

$9

$5

$4

If I have $9 and spend $5, I retain the difference between $9 and $5, which is $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.

752 is read "7 minus 5 equals 2." The form, 752, is called what?

PRINCIPLES.

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

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21

PRINCIPLE.-Only like orders can be subtracted.

We therefore write the 3 units under the 4 units and the 3 tens under We then say "4 units 3 units 1 unit; 5 tens - 3 tens Hence the remainder is 21."

the 5 tens.

2 tens.

Proof.

PRINCIPLE.-The subtrahend + the remainder the minuend.

=

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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 the difference is 142." Show proof.

- 4 tens; 4 hundreds

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3 hundreds = 1 hundred.

Hence

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NOTE.-Let the pupil first indicate the solution of each problem by

using the minus sign,

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

Process.

6878

4345 2533

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1. Since the army went into battle with 6878 men and came out with only 4345, the number missing was 6878 minus 4345.

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

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5 units 3 units, 7 tens 4 tens

5 hundreds,

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3 tens, 8 hundreds thousands 4 thousands

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number of men missing was 2533."

2 thousands.

3 hundreds

Hence the

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2. A grain dealer, having 7890 bushels of wheat, sold 6370

bushels. How many bushels had he remaining?

1

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.

594

368

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The difficulty is that 8 units cannot be taken from 4 units. But one of the 9 tens 10 units; 10 units + 4 units 14 units. Hence we write :

3 hundreds + 6 tens + 8 units.

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