(12) Multiply 780149326 by 3, 4, 5, 6, 7, 8, 9, and 10. (13) Multiply 123456789 by 4, 5, 6, 7, 8, and 9. (14) Multiply 987654321 by 9, 10, 11, and 12. When the multiplier is between 12 and 20, multiply by the units' figure in the multiplier, adding to each product the last figure multiplied.† (15) 5710592 × 13. | (18) 2057165 × 16. | (20) 9215324 × 18. (16) 5107252 × 14. | (19) 6251721 × 17. | (21) 2571341 × 19. (17) 7653210 × 15. When the multiplier consists of several figures, multiply by each of them separately, observing to put the first figure of every product under that figure you multiply by. Add the several products together, and their sum will be the total product.+ PROOF. Make the former multiplicand the multiplier, and the multiplier the multiplicand; and if the work is right, the products of both operations will correspond. Otherwise. A presumptive or probable proof (not a positive one) may be obtained thus: Add together the figures in each factor, casting out or rejecting the nines in the sums as you proceed; set down the remainders on each side of a cross, multiply them together, and set down the excess above the nines * To multiply by 10, annex a cipher to the multiplicand, for the product. To multiply by 100, annex two ciphers, &c. + Multiply 96048 by 15. 96048 15 1440720 EXAMPLES. Say 5 times 8 are 40, set down 0 and carry 4; 5 times 4 are 20 and 4 are 24, and 8 are 32, set down 2 and carry 3; 5 times and 3 are 3, and 4 are 7, set down 7; 5 times 6 are 30, set down 0 and carry 3; 5 times 9 are 45 and 3 are 48, and 6 are 54, set down 4 and carry 5; 5 and 9 are 14, set down 14. in their product at the top of the cross. Then cast out the nines from the product, and place the excess below the cross. If these two correspond, the work is probably right: if not, it is certainly wrong, (24) 170925164 × 7419. (22) 271041071 × 5147. (23) 62310047 x 1668. (25) 9500985742 x 61879. (26) 1701495868567 × 4768756. When ciphers are intermixed with the significant figures in the multiplier, they may be omitted; but great care must be taken to place the first figure of the next product under the figure you multiply by.* Ciphers on the right of the multiplier or multiplicand (if omitted in the work) must be placed in the total product.† (27) 571204 X 27009. (30) 1379500 × 3400. (31) 7271000 × 52600. A number produced from multiplying two numbers together, is called a compósite number; and the two numbers producing it are called the factors, or compōnent parts. When the multiplier is a compósite number, you may multiply by one of the factors; and that product multiplied by the other will give the total product.‡ (33) 771039 x 35. (34) 921563 x 32. (35) 715241 × 56. (36) 679998 x 132. (37) 7984956 × 144. (38) 8760472 × 999.§ (40) A boy can point 16000 pins in an hour. How many can five boys do in six days, supposing them to work 10 clear hours in a day? (41) If a person walks upon an average 7 miles a day, how many miles will he travel in 42 years, reckoning 365 days to a year? (42) Multiply the sum of 365, 9081, and 22048, by the difference between 9081 and 22048. (43) Required the continued product of 112, 45, 17, and 99. NOTE. Multiply all the numbers one into another. DIVISION TEACHES to find how often one number is contained in another or to divide a number into any equal parts required. The number to be divided is called the Dividend; that by which we divide is the Divisor; and the number obtained by dividing is the Quotient; which shows how many times the divisor is contained in the dividend. When it is not contained an exact number of times, there is a part of the dividend left, which is called the Remainder. RULE. When the divisor is not more than 12, find how often it is contained in the first figure (or two figures) of the dividend; set down the quotient underneath, and carry the overplus (if any) to the next in the dividend, as so many tens; find how often the divisor is contained therein, set it down, and continue in the same manner to the end. When the divisor exceeds 12, find the number of times it is contained in a sufficient part of the dividend, which may be called a dividual; place the quotient figure on the right, multiply the divisor by it, subtract the product from the dividual, and to the remainder bring down the next figure of the dividend, which will form a new dividual: proceed with this as before, and so on, till all the figures are brought down. PROOF. Multiply the divisor and quotient together, adding the remainder (if any) and the product will be the same as the dividend. (1) Divide 725107 by 2.* * EXAMPLE. Divide 7828105 by 4. Divisor 4)7328105 Dividend. Quotient 1832026-1 Rem. 7328105 Proof. in 10, twice 4 are 8, and 2 over; 1 over. (2) Divide 7210472 by 3. Say the fours in 7, once and 3 over; the fours in 33, 8 times 4 are 32 and 1 over; the fours in 12, 3 times; the fours in 8, twice; the fours in 1, 0 and 1 over; the fours the fours in 25, six fours are 24 ard (3) Divide 7210416 by 4. (4) Divide 7203287 by 5. (5) Divide 5231037 by 6. (6) Divide 2532701 by 7. (7) Divide 2547325 by 8. (8) Divide 25047306 by 9. (9) Divide 70312645 by 10. (10) Divide 12804763 by 11. (11) Divide 79043260 by 12. (12) Divide 37000421 by 3, 5, 7, and 9. (14) 7210473 ÷ 37.* (21) 25473221 +27100. ÷572100. +373000. ÷215000. (13) Divide 111111111 by 6, 9, 11, and 12. When the divisor is a compósite number, you may divide the dividend by one of the component parts, and that quotient by the other; which will give the quotient required. the true remainder must be found by the following But RULE. Multiply the second remainder by the first divisor: to that product add the first remainder, which will give the + When the divisor is large, the quotient figures are most easily found by trials of the first figure (or two) in the leading figures of the dividend. Ciphers at the right of the divisor may be cut off, and as many figures from the right of the dividend; but these must be annexed to the remainder at last. § EXAMPLE. Divide 314659 by 21. 21=7X3)314659 7)104886-1 14983-5 A number may be divided by 10, 100, 1000, &c. by merely cutting off one, two, three, &c. figures on the right: the other figures are the quotient, those cut off are the remainder. Thus 76390-÷÷10=7639; 238457÷10=23845 and 7 rem. And 4598653÷÷1000=4598 and 653 rem. (29) 65941089 10. (31) 18043329 ÷ 10000. (32) 7406572 ÷ 1200. (33) What is the difference between the 12th part of 107724, and the 23rd part of 346610? (34) If a ship bound to Jamaica set sail from Liverpool on the 25th of January, 1828, and arrived at that island on the 8th of March, what was the velocity of her sailing per day and per hour; the distance being 4558 miles? NOTE. This is the direct distance. ship would be considerably more. The circuitous course of the (35) The period of Jupiter's revolution in his orbit round the sun, which is the year of that planet, is 4330 of our days. How many of our years, reckoning 365 days to the year, are equal to five years of Jupiter? (36) I would plant 2072 elms in 14 rows, the trees in each row 17 feet asunder: what length will the grove be? (37) If a chest of oranges, 1292 in number, be distributed, one moiety among 19 boys, the other among 17 girls: how many will fall to the share of each ? (38) The circumference of the earth's orbit, or annual path round the sun, is about 596440000 miles. Supposing the year to be exactly 365 days, or 8766 hours, how many miles in an hour, and how many in a minute, are we carried by this motion? (39) Required the sum, the difference, the product, and the quotient, of 3679 and 283: and also the quotient of the product divided by the sum. (40) The sum of two numbers is 4290; the less number is 143 what is their difference, product, and quotient; and the quotient of the product divided by the difference? (41) The product of a certain number multiplied by 694, when 320 are added, is equal to 500000: what is that number? (42) Allowing the earth to revolve on its axis in exactly 24 hours, and the circumference at the equator to be 24864 miles; at what rate per hour and per minute are the inhabitants of that part carried round by the revolution? Also, at what rate are the inhabitants of London carried round, the circumference in that latitude being 15480 miles? |