2. Begin with the units and subtract each figure in the lower line from the figure over it; and set down the remainder. 3. When any figure in the lower line is greater than the figure above it, add 10 to the upper figure and from that amount subtract the lower figure and set down the remainder; observing when you so increase the upper number by 10 always to add 1 to the next lower figure before subtracting, and thus proceed until the whole is finished. Proof. Add the remainder to the less number, and if the sum be equal to the greater number, the work is right. Note. The greater number is sometimes called the minuend, and the smaller the subtrahend. EXAMPLES. 1. What is the difference between 1856 and 642? Operation. Minuend 1856 We place the less number under the Subtrahend 642 greater, units under units, tens under tens, &c. then take 2 units from 6 units Difference 1214 leave 4 units; and 4 tens from 5 tens leave 1 ten; and 6 hundreds from 8 hundreds leave 2 hundreds, and nothing from 1 thousand leaves 1 thousand; and the answer is 1214. 2. Subtract 528 from 873 and how many remains? Greater num. 873 Less number 528 Remainder Proof It is evident we cannot take 8 units from 3 units, but will suppose the upper number 3 to be increased by 10, 345 that is, we will take 1 ten off of the 7 tens, and add it to the 3 units, making 873 13, and then subtract 8 from 13, leaves 5. (We have now borrowed 1 ten from the 7 tens, and in reality it is only 6 tens from which we are to subtract the 2 tens in the lower number.) We will now carry or add 1 to the next lower figure 2 tens, makes 3 tens; and 3 from 7 leaves 4. Then subtract 5 (hundreds) from 8 (hundreds) leaves 3 (hundreds) and the work is done. Note. It will be seen that carrying 1 to the lower figure 2 tens and subtracting it from the 7 tens produces the same effect; and is more convenient in practice than it would be to consider the upper figure (7 tens) from which we borrowed 1 ten; to be only 6 tens, and subtracting the lower figure 2 tens from it; thus, 2 from 6 leaves 4, the same as above. Hence is derived the Rule. "When the lower figure is greater than the figure above it add 10 to the upper figure," &c. Proof. The remainder and less number added together are equal to the greater number, therefore the work is right. 22. If the minuend be 95657 and the subtrahend 84465 what is the difference or remainder ? 23. If the minuend be 1002243 and 667788 what is the remainder? Ans. 11192. the subtrahend Ans. 334455, 24. From fifty-five thousand and thirty-six take five thousand and twenty-one. Ans. 50015. 25. Can you tell me the difference between one milljon and nine hundred and ninety-nine thousand? Ans. One thousand. EXERCISES IN ADDITION AND SUBTRACTION. 1. A man bought a ship for 25311 dollars, and sold her for 34728 dollars, what did he gain by the purchase? Ans. 9417 dollars. 2. How much is 1200 greater than 365 and 721 added together? Ans. 114, 3. If I have 3541 dollars, and pay A 1200 dollars and B 975 dollars out of it, how many dollars shall I have left? Ans. 1366. 4. What number added to 9709 will make 10000 ? Ans. 291. 5. General Washington was born in the year 1732 and died in the year 1799, how old was he when he died? Ans. 67. 6. The whole population of the six New-England States in the year 1800 was 1,232,454; in 1830 it was 1,954,220 ; what was the increase in 30 years ? Ans. 721766. 7. What other numbers with these four, viz: 21, 32, 16, and 12, will make 100? Ans. 19. 8. A gentleman's whole estate amounted to eleven thousand five hundred dollars; and he gave his two sons Charles and Edward each four thousand five hundred dollars, and the remainder to his daughter Eliza; what was Eliza's share? Ans. $2500, 9. From Hartford to New York is 123 miles, thence to Philadelphia 90 miles, thence to Washington 138 miles; now if a man travel 6 days from Hartford towards Washington, at the rate of 47 miles a day, how far will he then be from Washington ? Ans. 69 miles. 10. A wine merchant bought 843 pipes of wine for 97530 dollars, and sold 684 pipes thereof for 89631 dollars; how many pipes had he left, and what do they stand him in? Ans. 159 pipes left, and they stand him in $7899, SUBTRACTION OF FEDERAL MONEY, RULE. Write the less number below the greater, dollars under dollars, dimes under dimes, cents under cents, &c. and subtract as in Simple Subtraction. 8. From 135 dollars take 8 dollars 8 cents. Ans. $126,92, 9. From 154 dollars 1 cent take 51 dollars 10 cents. Ans. $102,91, 10. From 348 dollars 90 cents take 158 dollars 99 cents. Ans. $189,91. 11. From 100 dollars take 99 cents. 13. From $50,10 take $19,01. 14. From 783 dollars 48 cents subtract 15. From $51, take 42 cents. 16. From 25 dollars subtract 25 cents. Ans, $89,19. Ans. $131,48. Ans. $24,75. Ans. $39,59,9. Ans. $1538,10. Ans. $99,99,9. 20. From 30 dollars 75 cents take 74 cents 1 mill. Ans. $30,00,9, multiplicand, and the less number the multiplier; or the less number the multiplicand and the greater number the multiplier; for the product of any two factors will be the same in either case: (thus 5 taken 3 times is the same as 3 taken 5 times.) But in practice it is most convenient to make the greater number the multiplicand and the less, the multiplier. CASE I. When the multiplier is not greater than 12, RULE. Place the multiplier under the multiplicand, and multiply each figure in the multiplicand by the multiplier, setting down and carrying as in Simple Addition. EXAMPLES. 1. There are 365 days in one year; how many days are there in three years ? Illustration-It is evident that if one year contains 365 days, three years will contain 3 times 365 days. Operation. We write down the multiplicand and 365 multiplicand place the multiplier under it, making the greater number the multiplicand; the less the multiplier. 3 multiplier 1095 Product We then say 3 times 5 are 15 setting down the 5 (units) and carrying 1 (ten,) as in Simple Addition; then 3 times 6 are 18 and 1 to carry makes 19, setting down 9 and carrying 1 to the next, we say 3 times 3 are 9 and 1 makes 10, setting down the whole product. Hence we find that 3 times 365 is equal to 1095. We might have obtained this same answer by setting down 365 the multiplicand 3 times and adding it up. |