III. SUBTRACTION. EACHETH to take a leffer number from a greater, to find their difference or remainder. ΤΕ RULE. 1. Place the greater number uppermoft, and the other under it, so that units may be under units, tens under tens, &c. and draw a line under them. 2. Begin at the right hand or place of units, and fubtract the lower figure from the upper, and fet down the difference underneath them; do the fame with the reft of the figures. 3. When the under figure exceeds that which stands over it, you must borrow ten, and add it to the upper number, from which subtract the lower, and fet down the remainder; carry one to be added to the next lower figure, and fubtract the fun from the upper, fetting down the remainder; and fo on from one row to another. PROOF. The way to prove fubtraction is no more than, to the leffer number add the remainder; if the fum be like the greater, the work is right. EXAMPLE 1. E. 5. E. 3. From 869426017 unfold 8663 Rem. 654444253 Rem. 58524 242295 64638 94165 368419 86459 Proof 9876 Proof 869426017 To work example firft, fay, 1 from 5 and there remains 4, write down four in the place of units, and fay 4 from 6 and there remains 2, which write down in the place of tens; then fay 6 from 1 I cannot, but 10 that I borrow added to 1 is 11, 6 from 11 and there remains 5; then that I borrowed and 5 is 6, 6 from 4 I cannot, but 6 from 14, and there remains 8; 1 that I borrowed and 3 is 4, 4 from 9 and there remains 5, which fet down, and the work is done. But as these things are fo eafy, I think any farther explanation of the reft would be looked upon as prolixity only. Questions to exercife Subtraction of Whole Numbers. Queft. 5. The difference between two numbers is 36842, the greater is 864952, what is the leffer? 864952 36842 Queft. 6. Suppofe this prefent year 1800, you were 14 years old, What year was you born in ? 1800 14 Queft. 8. When the air preffes with its full weight, in very fair Weather, it may be demonstrated, that there preffes upon a human Body about 33905 Pounds of that fluid matter; and in foul Weather, when the Air is moft light, but 30624 Pounds. What Difference of Weight lies on fuch a Body, in the two greatest Alterations of the Weather? In fair Weather, lbs. 33905 Answer 3281 IV. MULTIPLICATION. EACHETH how to increase any one number by another, so often TEA as there are units in that number by which the one is increased; and ferves instead of many additions. There are three principal members belonging to this rule, viz. 1. The multiplicand, or number to be multiplied. 2. The multiplier, or number by which the multiplicand is multiplied. 3. The product, or number produced in multiplying. Note. For the ready performance of this, and all the following rules, it is abfolutely neceffary the following table fhould be got by heart. PROOF. The best way to prove multiplication is by divifion, but as the learner is fuppofed not yet to know that rule, he cannot prove by it; he must therefore make the multiplicand the multiplier; then if the product is the fame as the other, the work is right. Note. There is a way of proving multiplication by cafting away the nines, which is moftly taught in fchools, and is very expeditious, but liable to error; but for the exercise and improvement of the learner, I hall fhew him both ways in its proper place. 12|24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 1441 CASE 1. To multiply by a fingle figure. RULE 1. Place the multiplier underneath the units place of the multiplicand. 2. Multiply from the right hand to the left, thus; begin with the units or lowest figure of the multiplier, by which multiply the lowest figure of the multiplicand, and fet down the overplus, above the tens; and carry the tens: Then multiply the second figure of the multiplicand by the fame, adding fo many units, as you had tens to cary; fet down the overplus, and carry the tens as before. Do thus until you come to the last figure, whofe product must be fet down entire. EXAMPLE 1. Multiplicand 6874 Multiplier Product 27496 I 7X4 To work this example you muft fay, 4 times 4 is 16, fet down 6 and carry one; then 4 times 7 is 28, and 1 that I carried is 29, set down 9 and carry 2; then fay 4 times 8 is 32; and 2 that I carried is 34, fet down 4 and carry 3 ; laftly, 4 times 6 is 24, and 3 that I carried is 27, which fet down, and the product will be 27496, as by the work. To prove the foregoing work by cafting away the nines, make a crofs, and add all the figures of the multiplicand together, as units, thus 6+8+7+4=25, throw away the nines, and fet the remainder 7 on the fide of the cross; do the fame with the multiplier 4, but as there are no nines to throw away, I fet down 4 on the other fide of the crofs. Do the like with the product, 2+7+4+9+6=28, throw away the nines, and there remains 1 to be fet at the top of the crofs. Laftly, multiply the figures on the fides, thus, 4 times 7 is 28, throw away the nines and fet the remainder i at the bottom of the crofs, which is the same as the top, and proves the work to be right. The other way to prove multiplication is, by making the multiplicand the multiplier. E. 2. 32142352 2 64284704 E. 3. 53124564 E. 4. 3. 21684218 E. 5. 16841509 84207545 E. 9. 6802124 3214568 3456102 8 6 7 1406178 9 40812744 22501976 27648816 12655602 CASE 2. When the multiplier confifts of feveral figures; RULE. Multiply each figure through the line, and write down the products according to their places, that is each refpective product underneath that figure of the multiplier, by which you multiply, then add them up, and you will have the true product in one line. CASE 3. When cyphers are intermixed with the figures in the multiplier. RULE. Omit the cyphers, and place the firft figures of each particular product under its refpective multiplier; the following examples, wrought at full length, will fufficiently explain this rule. CASE. 4. When there are cyphers at the right hand of either, or both the multiplier and multiplicand. RULE. Multiply as before, neglecting the cyphers until all the particular products are added together, and to that fun place the number of cyphers that are on the right hand of both factors, T. 24 |