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10. Benjamin Franklin was born in the year 1706. He died when he was 84 years of age. In what year did he die ?
11. Three merchants entered into partnership. The first advanced $5500 dollars towards the capital; the second advanced $1000 more than the first did ; and the third advanced $1500 more than the second. What was the whole amount of their capital?
Ans. $20,000. 12. The first man was created 4004 years before Christ. How long is it from his creation to the present year?
13. There are two numbers, of which the smaller is 4520, and the difference between them 540. What is the greater number, and what is their sum ?
Ans. to the last question, 9580. 14. A man left by will to his widow $5000, and to an adopted daughter $2000. The rest of his estate, after the payment of his debts, he directed to be equally divided among his four sons. The debts amounted to $3426, and each son received $1550. What was the value of the whole property ?
Ans. $16,626. 15. Two brothers set out on a journey in different directions. The one travelled 167 miles, the other 134. How many miles were they then apart ?
16. A man sold a house and lot for $6254, which was $1746 less than they cost. What was their cost ?
SECTION II.- Subtraction.
[For an explanation of the terms and signs used in subtrac-. tion, see p. 56, 3; 58, 8.)
Exercises for the Slate and Black-board. 1. Name the difference between each of the following pairs of figures :
2 1 6 1 7 1 8 1 2 2 4 2 8 2 7 2 1 3 1 4 1 5 1 9 2 3 2 6 2 5 2 9 4 3 8 3 5 3 5 4 6 4 7 5 8 5 9 6 3 6 3 7 3 9 4 8 4 9 4 6 5 7 5 6 7 6 8 9 7 7 4 9 6 8 2 9 3 4 2 9 6 9 6 7 8 7 9 5 7 2 5 9 6 5 8 9
[These figures may be studied on the book, or transferred to the slate for that purpose. The pupil should continue to practise them till he can recite the differences rapidly from the black-board, taken regularly as well as in irregular order, without naming the subtrahend or minuend. Pupils should be accustomed to name the differences of numbers placed horizontally, and also with the smaller number above as well as below. The former is required in balancing accounts; the latter frequently occurs in long calculations.]
2. What is the difference between two heaps of apples, one of which contains 12, the other 16 ? If 10 more apples be added to each heap, will their difference be changed, or will it remain the same? Will the difference be unchanged by adding 20, 30, 40, or any other number to each ? Will the difference between any two numbers whatever be changed by adding an equal number to each ? May not the following, then, be considered the sixth principle of Arithmetic ? VI. If equal numbers be added to unequal numbers, their
difference remains unchanged. 3. What is the difference between 10 and 6 ? If this difference be added to the smaller number, to what will it be equal ?' If the difference be taken from the larger, to what will it then be equal ? Will the same principle hold in any two numbers ? [Give examples with other numbers on the black-board, when necessary.) May not the following, then, be considered the seventh principle of Arithmetic ?
VII. (1.) If the difference between two numbers be added to
the smaller, their sum is equal to the greater. (2.) If the difference be taken from the greater, the remainder will be equal to the smaller.
EXEMPLIFICATION OF SUBTRACTION,
Where some of the figures in the subtrahend are greater than
those of the same rank in the minuend.
[For the Black-board.]
From 52364878 Minuend.
Leaves 30928584 Difference, or Remainder.
Subtrahend+Difference=52364878=Minuend. Proof No. 1.
Minuend_Difference =21436294=Subtrah’d. Proof No. 2.
4. Commencing at the right, for a reason that the student will presently discover : Four from 8, how many ? Can 9 be taken from 7? Adding 10 to the second rank of minuend (see Principle VI. above) 9 from 17 ? Then adding 10 to second rank of subtrahend, also, will change the 2 in third rank to what? Three from 8, then? Can 6 be taken from 4 ? Adding 10 to fourth rank of minuend, 6 from 14 ? Adding 10 to fourth rank of subtrahend, also, how many from 6? Can 4 be taken from 3? Adding 10 to sixth rank of subtrahend, what does the 1 become? And so forth. If the figures in the minuend were always greater than the corresponding ones in the subtrahend, would it be of any consequence where the process of subtraction commenced ? [Give an example on the black-board.] Why, then, is it generally necessary to commence on the right?
Proof 1. By the seventh principle of arithmetic, to what is the sum of the subtrahend and the difference or remainder equal ? Proof 2. If the difference be subtracted from the minuend, to what will the remainder be equal ?
[All the above can be changed to new exercises, by making a slight change in the left hand figures, and substituting the subtrahend for the minuend. If necessary, others can be added by the teacher, or, still better, by the pupil
. Subtraction should be practised till the class can perform it as rapidly as the remainder can be written. But this can only
be done by reading without spelling; that is, by thinking of or writing the difference between numbers without naming those numbers either orally or mentally; and in like manner increasing them by ten when necessary, without mentioning that circumstance. Thus, in the following example, all the words are superfluous except the three words in italic, namely, four eight, two:
Three from seven leaves four ; four from two, add ten, leaves eight; one to two makes three from five leaves two.
Subtraction by Addition.
Definition. — The complement of a number is the difference between that number and 1 of the next higher rank or order; that is, a number and its complement amount to 1 of the next higher rank or order of figures. Thus, 8 is the complement of 2, because 2 and 8 together make 10. For the same reason 2 is the complement of 8; 30, also, is the complement of 70, and 70 of 30, because together they make 100, 1 of the next higher rank. Thus, also, 28 and 72 are mutually complements. The complement of 0, of course, is 10.
1. What is the complement of 6 ? Of 5 ? 3? 7 ? 4? 2 ? 9? 1? 6? 8?' 20 ? 50 ? 70 ? 40 ? 60 ?