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2. What is the first principle of subtraction? 3. On what principle does the proof depend? 4. From 703 take 549.

Process.

6913

703

549

154

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The difficulty is that we cannot take 9 units from 3 units, nor 4 tens from 0 tens. But one of the 7 hundreds :

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10 tens;

13 units.

Hence we

Subtracting we

have 1 hundred + 5 tens + 4 units = 154.

5. Give the principles of subtraction and prove the work. 6. From 367.280 take 298.356.

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7. Where must the point always be placed in the remainder?

RULE FOR SUBTRACTION.

1. See that the numbers to be subtracted are like numbers.

2. Write the subtrahend under the minuend, units under etc.

units,

3. Beginning at the right, subtract each lower figure from the one above it.

4. When necessary, increase the upper figure by 10 and diminish by 1 the next upper figure on the left.

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$35.56 $40.19
$40.19 $74.36 $82.99
$32.49 $38.02 $58.19 $58.03 $19.78

$53.44

$6.12

$5.125

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1. If a man owes $97.66 and pays $70.89, how much does he then owe?

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Process. $97.66 70.89

$26.77

Explanation.

Since he owes $97.66, and pays $70.89, he still owes the difference between $97.66 and $70.89.

Since the numbers have the same unit, one dollar, they are like numbers, and their difference can be found. It is $26.77.

NOTE. In each problem let the process be indicated first, and then performed and explained.

2. A man bought some land for $8765, and sold it for $10,890. What was his gain?

3. The first line of telegraph was established in the United States in 1844. How long ago?

4. In 1890 the population of the United States was 62,622,250, and in 1840 it was 17,063,353. How much did it increase in the 50 years?

5. A man was born in 1785: what was his age in 1830? 6. How old was George Washington at the time of his death? He was born in 1732, and died in 1799.

7. 29,400 feet is the greatest depth of water measured. 37,000 feet is the greatest height reached by a balloon. Find by how much the greatest height reached exceeds the greatest depth reached.

8. The displacement of the battle-ship "Alabama" is 11,525; of the cruiser "Charleston," 3730. How much does the displacement of the "Alabama" exceed that of the "Charleston"?

9. The estimated population of the United States in 1800 was 5,308,483; in 1898 it was 74,500,000. Find the growth in population in the 98 years.

10. How many dollars must be added to $4872 to make $8021?

11. How many dollars increased by $74,015 make a million dollars?

12. What number must be taken from $6412 to leave $5366?

13. Find the value of $8.052

$3.687.

14. John has $20.19 and James has $40. How much more money has James than John?

15. A bankrupt has $6456 assets, and owes $33,860. How much more does he owe than he can pay?

16. Mt. Everest is 29,062 feet high; Mt. Whitney is 14,900 feet high. How much higher is the former than the latter?

17. The height of Mt. Cenis, an Alpine peak, is 11,792 feet; the height of the pass over it is 6884 feet. How much higher is the mountain than the pass?

18. At an election 3245 persons voted, and the candidate elected received 1808 votes. How many did the defeated candidate receive?

19. Benjamin Franklin died in 1790, and was 84 years old at his death. When was he born?

20. Subtract MMMIX. from LXVIIXI.

ADDITION AND SUBTRACTION IN COMBINATION.

EXERCISES.

1. What is the value of 20,324 + 4756 13,186?

Explanation.

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ANALYTIC AND SYNTHETIC.

signifies that I must add 20,324 and 4756.

Adding, the sum is 25,080.

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25,080

13,186

11,894

The sign — signifies that I must subtract from that sum 13,186. Subtracting, the remainder is 11,894.

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3. 20,201 9022 + 2002 ?
4. 132,57190,79878,318?

5. $238.70 $53.36+ $22.27 ? A Parenthesis, (), or Vinculum,

indicates that

all the quantities it incloses are to be considered as a single quantity; as, (2 + 5 + 10 + 13), or 2 + 5 + 10 + 13.

2. What is an Equation?

3. Prove the following equations to be correct:

First perform the operations indicated within the parentheses.

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5. (300+ 100 +95 +60 + 125) — 125 + 25+ 40

= 490.

4. Complete the following partial equations:

1. (350,000 + 225,100 + 4000 + 96,000) — 450,120

= ?

2. 23,191,876 3,204,313 + 434,495 = ?

3. (367 + 875 + 1012) — 423 + 912 = ?
4. (36 200 + 150).- 331 = ?

PROBLEMS.

NOTE.—The indicating of a solution often facilitates the completing of it. The pupil should be faithfully drilled in the use of signs to indicate the actual solution to be made.

1. Mr. A. gave his note for $6000. He paid at one time $3586 and at another time $2000. How much remained to be paid?

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