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3. If the upper figure be less than the lower, increase it by ten, and then subtract the lower.
4. Whenever this is done, increase the next figure in the lower line by 1, before subtracting it from the upper figure.
5. Proceed in this manner, till all the figures in the lower line are subtracted, when, if there still remain figures in the upper line, supply (in your mind) ciphers to the lower, and proceed as before. If the last figure on the left would be a cipher, omit it.
EXAMPLES. 1. Subtract 37786 from 58998.
Def. Multiplication is the operation by which, in a shorter manner than by addition, we find the sum of several, the same numbers, which sum is called the product. The number which is repeated is called the multiplicand; that which expresses how often the number is repeated, is the multiplier.
I. TO MULTIPLY TOGETHER TWO NUMBERS LESS THAN TWELVE.
Rule. The products of all such numbers are to be found in the Multiplication Table.
11. TO MULTIPLY A NUMBER GREATER THAN TWELVE, BY ONE
LESS THAN TWELVE.
Rule-1. Write the multiplier under the units' figure of the multiplicand, and multiply this figure by it.
2. Write down the units' figure of the product under the units, and carry (in your memory) the tens.
3. Multiply the tens' figure in the same way, and add to the product the number of tens carried.
4. Write down the tens' figure of the sum under the tens, and carry the number of hundreds.
5. Proceed in this way, till the last figure of the multiplicand has been multiplied; then set down the whole of this last product.
III. TO MULTIPLY BY 10, 100, 1000, &c. Rule. Annex to the right of the multiplicand, as many ciphers as there are in the multiplier.
IV. TO MULTIPLY BY ANY NUMBER GREATER THAN TWELVE. Rule-1. Write the multiplier under the multiplicand, units under units, &c.
2. Multiply by each figure in the multiplier, setting down the first figure obtained in each product under the figure in the multiplier, from which it has been obtained.
3. Add together the several products, column by column. 4. If a cipher appear among the figures of the multiplier, write down only one cipher of the product under it, and proceed with the next figure.
5. If the multiplier is composed by the product of several numbers, multiply by any one of these first; then the product so found by another; this product by a ird, and so on.
EXAMPLES. Multiply 808072418 by 7, 1000, 5068, and 98.
Def. Division is the operation by which we find how often one number is contained in another; or into how many parts of a fixed magnitude a given quantity can be divided; or by what number we must multiply the less of two numbers to produce the greater.
TO DIVIDE ONE NUMBER BY ANOTHER.
Rule—1. Write the divisor and the dividend in one line, and place a curved line on each side of the dividend.
2. Take from the left hand of the dividend the least number of figures, which make up a number not less than the divisor.
3. See how many times the divisor is contained in this number, and write the quotient on the right of the dividend, as the first figure of the whole quotient.
4. Multiply the divisor by this figure, and subtract the product from the number, which was taken off at the left.
5. On the right of the remainder, place the next figure of the dividend, and if the remainder so increased be greater than the divisor, divide it, and place the quotient to the right of the first figure found.
6. But if the remainder be still less than the divisor, bring down the next figure, or figures of the dividend to the right of the remainder, till you have a number greater than the divisor: but take care to place a 0 in the quotient for every figure brought down except the first. Divide now as before.
7. Continue this process, till all the figures of the dividend are exhausted.
8. If the divisor be not greater than 12, each figure of the quotient may be set down under the figure of the dividend from which it has been obtained, the subtractions being performed in the mind, and the several dividends being formed by prefixing the remainder from one division to the next figure of the dividend.
9. If the divisor be composed by the product of several numbers, divide by any one of them first; then the quotient thus obtained by a second; this quotient by a third, and so on. To form the final remainder, multiply the remainder from each division by the product of all the previous divisors; add the products and the remainder from the first division.
10. If the dividend consist of the product of several factors, the quotient may be obtained by dividing one factor, or the product of any number of factors, by the divisor, and multiplying the quotient by the rest.
11. If the divisor have ciphers at the end, cut off by a comma from the dividend as many figures as there are ciphers at the end of the divisor; and cut off the ciphers from the divisor; divide the remaining figures of the dividend by those of the divisor. To the remainder affix the figures, which were cut off from the dividend.
4. Divide the product of 36, 17, and 100 by 25.
Quotient of 100 by 25=4
QUESTIONS IN THE FOUR SIMPLE RULES. 1. Of three bars of iron, one weighs 137 lbs. another 58 lbs. and the third 163 lbs., required the weight of them altogether.
1st bar weighs 137 lbs.
Whole weight.... 358 lbs. Ans.
2. A person bought 3008 yards of silk, and sold to one person 1700 yards, and to another 856 yards, how many yards has he left?
Yards bought .... 3008
3. A person began business with £2000; he cleared every year for 21 years £450, how much was he worth at the end of that time? Yearly gain..... £450
4. A person's income is £9125 yearly, how much is that per day? 1 year = 365 days. 365)9125(25
VI. MEASURES. Def. 1. A measure of a number is another number which divides the first without remainder.
Def. 2. A common measure of two or more numbers is a measure of all of them.