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dollars, and ten dimes at ten cents each, make one dollar, therefore, if instead of setting down the whole number of dimes, you take as many of them as it takes to make a dollar, and call it one dollar, and all there are over this number call dimes, you get your sum in dollars and dimes, expressing the same value as if you called them all dimes ; and as you have dollars to add, you can add this one dollar, to them, and express their sum in one number. In this example you have 10 dollars given, and 13 dimes, and by taking the one dollar from the dimes, and adding it to the dollars, you express all the dollars possible, and all the dimes there are over.
Pup. From what you have said, it appears very plain to me that carrying for every ten gives a true sum of all the numbers added, and that it would not do to carry for any other number. I should like now to have some examples, that I may make what you have said familiar.
Tut. I will give you some examples, and you must perform them with care, and endeavour to understand every part, as you go along, for your progress in the succeeding part depends greatly on understanding well the preceding part of arithmetic. Add 84371, 6250, 10, 3842, and 631 together.
Answer, 95104. Add 3004, 523, 8710, 6345, and 784 together.
Ans. 19366. Add 66947, 46742 and 132684 together.* Ans. 246373. .
I will now instruct you in the next part of this conversation. And can you tell what Simple Subtraction is ?
Pup. Simple Subtraction, I think must relate to taking one number from another.
Tut. Simple Subtraction does indeed relate to taking one number from another in such a manner as to show
* Addition may be proved as follows ; draw a line underneath the top line of figures ; add together all the figures below the line, and write their sum beneath the sum of the whole ; then add together the top line and the sum of the other lines, and if this sum agrees with the sum of the whole, the work is right ; otherwise it
wrong, and must be looked over again,
the true difference between them. Addition, you will recollect, is a rule by which numbers are put together, so as to find their whole amount in one sum ; but Subtraction shows how to take one number from another, and consequently is the reverse of addition. In addition I told you, that all small numbers were added in the mind, and their sums committed to memory ; (4. Con. I.) so in subtraction, the most natural way to find the difference between small numbers is to count to the full expression of one of those numbers, which must be the smallest, and then begin with unity, and add one at a time till you have expressed the amount of the largest number; the number of units which you added will be the difference between the two numbers. In this way the difference between small numbers is obtained and committed to memory. But, as in addition, this method of subtracting is extremely tedious and impracticable for large numbers ; therefore to avoid this, we have the following rule for performing questions in
Write the greater number, and directly under it write the less, so that units may stand under units, tens under tens, hundreds under hundreds, &c. and draw a line underneath.
Begin with the right hand figure of the lower number, and find the difference between it and the figure above it, setting the result underneath.
If the figure in the lower number or subtrahend* is larger than the figure over it, borrow ten from the next figure towards the left in the minuend, and add to the figure which is too small ; then write this sum and the lower figure as before, and carry one to the next figure of the subtrahend for the ten which you borrowed.
Proceed in this manner with all the figures, and the several dif. ferences taken together will be the difference of the two numbers.
If you wish to find the difference between 464 and 342, you will place them as follows.
4 6 4
1 2 2
* The larger number is called the minuend, and the smaller the subtrahend.
Here you begin on the right, and take the difference between 2 and 4, which is 2, which set underneath ; then take the difference between 4 and 6, and between 3 and 4, all of which, read together, is the difference of the two numbers.
If it be required to find the difference between 86032, and 65264, you have another part of the rule to consider.
8 6 0 3 2 minuend.
20 mg 6 8 remainder.
In this example you begin as before, at the right, and seek the difference between 4 and 2 ; but here the lower figure is larger than the upper figure, and it is for such cases that the rule directs to borrow ten. In this case you find that you cannot take 4 from 2 ; but as 1 in the place of tens, where 3 is, is equal to 10 in the place of units, you must take 1 from 3, and reduce it to units, or divide it into ten parts, which, with the 2 make 12, so that the 2 is changed into 12, from which 4 can be subtracted, leaving 8 for a remainder or difference. Now as you borrowed 1 from the place of tens, viz. from 3, you must consider this one less than it is; for, baving taken 1 from it, it must be 1 less. But as it is more convenient to consider the figure under it 1 larger than it really is, you may carry 1 to that figure, and subtract as before. In this example, when you come to subtract 6 from 3, you find that you must borrow ten again, which you are to do from the next figare. But here the next figure is a cipher, and ten cannot be borrowed from it ; in this case you
borrow from the next figure towards the left, viz. 6, but 6 is in the place of thousands, and 1 from that is equal to 1000, and you do not want but 100, for 0 is in the place of hundreds, and had you borrowed from that place it would have been 100.
Pup. You say that I orrow ten, and then carry one to the next figure, for the ten which I borrowed; and that if I had borrowed from 0 it would have been 100, the sum wanted ; I do not understand it.
Tut. When you borrow ten, you do not in reality borrow but one,
you take one from a higher, and re
duce it to the next lower denomination, which makes ten of it ; and as you borrowed but one from the higher denomination, you have but one to carry to it. As to borrowing 100, as in the example, you must consider that 6 is really more than 6; it is in the place of tens, and is 60, and the l' which is to carry to it makes it 70. The three above is thirty, and you cannot subtract 70 from 30, but must borrow 100 and add to it; yet because the numbers increase in a ten-fold proportion, it is called borrowing ten, however large the number really is. And here, as O is in the place of hundreds, the 100 must be borrowed from 6, and 1 from 6 being equal to 1000, 100 must be taken from the 1000, and the 900 remaining are considered as placed at the right of 6, so that the minuend would read thus, 86,900 ; but as 6 was made less by 1,
it must be called 5, so that setting aside the two right hand figures of each given number, they will stand thus.
8 5 9 6 5 2
2 O 7
You have now seen the whole process of Subtraction, which you must make perfectly familiar.
Pup. I think I understand the principles of this rule, and shall be able to answer any question you may ask me.
Tut. You may, in order to render what I have said, familiar, perform the following questions.*
What is the difference between 16844 and 9786 ?
Ans. 7058. What is the difference between 124682 and 113465 ?
I will now examine you in the preceding rules, and see if you understand them; if you do not, you must review them, and make every part perfectly familiar.
* Subtraction is proved by adding the remainder and subtrahend together ; if their sum agrees with the minuend, the work is right.
What is arithmetic ?
you place at the right of the significant figure ? When you wish to express any number between any
number of hundreds and tens, how do you do it? What is simple addition ? How do you place the numbers to be added ? Which column of figures do you add first? What is the answer called ? How do you write down the sum of each column ? Why do you carry for ten rather than for any other num
ber? How do you prove addition What is simple subtraction ? How many numbers are required to perform operations
in this rule ? What are these numbers called! How are they placed for performing the work? When the figure in the subtrabend is larger than the fig
ure above it, how do you proceed ? How does it appear that borrowing ten and carrying one
to the next figure does not give a wrong answer ? How is subtraction proved ?
Examples for Practice.
1. A man bought a coach and 4 horses ; for his coach he gave $200, for the two first horses he gave $300, and for the other two $250; what did they all cost him?
Ans. $750. 2. From Boston to Worcester is thirty-eight miles ; from Worcester to Hartford is sixty-two miles ; from Hartford to New Haven thirty-four miles ; from New-Haven to New-York seventy-six miles ; how far is it from Boston to New York ?
Ans. 210 miles. 3. There are five numbers ; the first is 2617; the second 893; the third 1702 ; the fourth as much as the three first; the fifth twice as much as the third and fourth ; what is the whole sum ?