When long decimals are to be multiplied and the product is required to only a few decimal places, contracted multiplication is of great advantage. The next problem illustrates this sort of multiplication. 7. The radius of a circle is 238.36 ft. What is the length of the circumference to the second decimal place, or to the nearest .01 foot? NOTE. The length of the diameter is 476.72 feet. EXPLANATION.-In the shortened form the digits of the multiplier are written in reverse order, and the units digit is always written under that decimal place in the multiplicand which is to be the last one retained in the product. Multiply by units digit first, then by tens, and so on; in each case begin the multiplication by any digit with the digit just above it in the multiplicand. Begin the writing of each partial product in the same vertical line on the right. NOTE. It is necessary on beginning to multiply by any digit to glance at the product by the preceding digit of the multiplicand to see how many units are to be added into the product by the digit just above. Thus, the multiplication by 4 would begin with 6, but 4 times the preceding digit (7) is 28, and this being nearly 3, the product 46 would be increased by 3, giving 27. Expert computers use the shortened form altogether. 8. Find these products to the nearest .01 by the method of shortened multiplication: (1) 36.428 × 3.1416 (3) 7.8843 × 1.0863 (4) 168.7431 × 28.329 9. Find the following products to the nearest .001 by shortened multiplication: (1) 36.1872 × 6.8734 (2) 128.63 3.8629 §146. Shortened Division. CONVENIENT FORM 68.37)648.7863 615 33 33 45 27 35 6 10 6 15 (3) 629.3865 X 3.1416 (4) 1284.683 × 3.1416 68.372 to the nearest .01. EXPLANATION.-Find the units digit of the quotient in the usual way. Then cut off one digit from the right of the divisor and find the next digit of the quotient, then cut off another digit from the divisor, etc. A dot is sometimes placed over each digit in the divisor as it is set aside. 2. Find quotients for the following to the nearest .01: (2) 632.068 ÷ 8.6249=? (4) 865.28476 ÷ 361.2946=? §147. Shortened Square Root: Square Root by Subtraction. 1. Find the square root of 68432.93628 correct to the third decimal figure. 2. Find, by the shortened method, to two decimals the square roots of the following: (1) 6432.1864 (2) 38,629.72468 (3) 8764.932651 Square root may be found by subtraction, after notic It is seen that the square root of the sum of the odd numbers in order from 1 upward is equal to the number of odd numbers added. An example will illustrate the use of this principle: 3. Extract the square root of 104976. 104976(324 square root 1 9 3 3 subtractions 6 5 61 149 61 88 63 641 2576 2 subtractions 641 1935 643 1292 645 647 647 4 subtractions EXPLANATION.-Place a dot over each alternate digit beginning on the right, thus separating the number into groups of 2 digits each. From the first group on the left subtract 1, then 3, then 5, and so on as shown, until the remainder is less than the next odd number. The number of subtractions is the first root digit. In the present case it is 3. Bring down the next two-digit group. Double the root digit found and annex 1 to it, and subtract, then replace the 1 by 3 and subtract, etc. The number of subtractions indicates the next root digit. Study the remaining steps and learn how to proceed further. This method gives the exact square root and may be used to check results found in the ordinary way. Some computing machines are based on this method of obtaining the square root. ing: 4. Find the square roots, by subtraction, of the follow $148. Miscellaneous Problems for Review. 1. If one breathes 18 times a minute and at each breath consumes 2 cubic inches of oxygen more when moving about than when sitting, how many gallons less oxygen does one get in 8 hours if he sits than if he is active? 2. If the total tonnage of merchandise shipped through the Detroit river in 1907 was 67,296,477 tons and it was carried in 23,721 vessels, what was the average tonnage of one vessel? 3. If the total tonnage of merchandise shipped through the Detroit river in 1906 was 60,577,732 tons and it was carried in 24,077 vessels, what was the average tonnage of one vessel? 4. What change is there in the average tonnage of a vessel from 1906 to 1907? 5. Duluth, Superior, and Two Harbors form a single community. From the three ports there were shipped in 1906, 47,726,778 bushels of wheat and 43,531,540 bushels of corn, and in 1907, 63,349,585 bushels of wheat and 44,355,990 bushels of corn. What were the total shipments of wheat and corn together in 1906? In 1907? In both years? 6. What was the difference of the total shipments of wheat and corn together in 1906 and 1907 from this community? 7. The freight sent through the Soo was 1,567,741 tons in 1881; 4,527,759 tons in 1886; 8,888,759 tons in 1891; 16,239,000 tons in 1896; 28,403,065 tons in 1901; 44,270,680 tons in 1905; 51,751,080 tons in 1906; and 58,217,214 tons in 1907. What was the increase in tons from 1891 to 1896? From 1896 to 1901? From 1901 to 1905? From 1905 to 1906? From 1906 to 1907? From 1881 to 1907? 8. The total tonnage of all the commerce of the Great Lakes for 1907 was 83,387,919 tons, which was 20% larger than the tonnage for 1905. What was the tonnage for 1905? 9. The total tonnage of the commerce of the Great Lakes for 1907 was 10% larger than for 1906. What was the total tonnage for 1906? See prob. 8. The following table is an extract from the milk record of a herd of milk-cows: WEIGHT OF MORNING (A.M.) AND EVENING (P.M.) MILKINGS IN POUNDS TOTAL A. M. P. M. A.M. P.M. A.M. P.M. A.M. P.M. A.M. P.M. A.M. 1901 QUEEN Nov. 1 2 4 6 7 P.M. 7.3 5.7 12.0 11.511.0 9.6 6.5 6.5 11.0 10.4 12.5 10.2 5.6 6.0 11.0 8.411.0 11.0 7.2 5.611.5 10.511.3 8.8 3 6.4 6.0 10.8 9.511.711.5 7.3 5.7 12.0 10.4 11.2 9.8 6.0 6.5 12.5 10.212.6 13.0 7.2 6.011.511.511.0 9.8 5 6.0 7.511.410.212.011.4 6.7 5.811.611.3 11.0 10.2 6.6 6.8 12.0 10.0 8.0 11.6 8.0 5.0 12.5 11.811.7 10.0 7.4 8.5 10.8 12.4 12.0 11.7 7.0 7.212.0 12.5 11.6 12.0 8 7.87.311.211.613.212.2 7.2 5.711.511.311.5 10.1 9 7.5 6.4 11.6 10.4 12.011.6 6.6 5.4 12.0 12.210.7 10.0 10 7.8 7.211.0 10.4 12.0 12.0 6.7 4.711.5 8.511.810.6 11 6.87.3 12.211.0 13.611.5 5.3 9.0 10.0 12.3 10.0 12 7.8 7.7 12.5 12.0 13.0 11.6 5.5 12.0 12.6 12.2 10.6 13 7.6 8.0 13.4 11.5 13.5 13.0 5.811.113.0 11.811.2 14 7.0 8.0 12.0 11.814.0 12.7 15 8.3 7.5 13.0 10.715.0 12.0 16 9.0 7.0 12.5 11.0 15.213.7 17 8.0 7.0 12.5 10.5 14.0 13.0 18 8.17.7 13.4 10.9 14.0 11.5 19 7.0 7.0 11.511.4 13.611.0 20 7.5 6.5 12.6 10.5 12.7 12.0 21 7.0 7.6 12.0 10.213.711.3 22 8.0 7.5 12.211.914.7 11.2 23 7.5 9.0 12.211.0 14.412.0 24 7.0 7.1 13.5 10.6 13.811.5 25 7.4 8.0 12.5 11.214.0 10.5 26 6.5 8.4 12.0 11.8 12.0 12.0 27 7.5 8.5 13.711.814.0 13.0 7.29.212.0 12.0 13.6 13.8 29 6.5 9.0 13.5 12.6 15.0 13.5 6.7 30 7.2 8.0 11.712.714.2 12.2 6.0 28 7.7 5.7 12.5 12.711.5 10.3 5.0 14.0 12.4 12.3 9.0 6.6 5.5 12.7 12.3 14.1 8.5 6.4 10.0 9.0 12.0 10.5 5.0 12.5 11.511.0 9.1 5.3 12.2 11.711.110.0 6.5 4.911.811.411.5 9.5 6.1 5.213.0 11.0 10.6 9.6 6.7 6.0 12.4 10.711.3 9.5 7.0 5.5 12.8 10.811.3 10.4 7.2 5.113.0 10.7 12.0 10.4 7.4 5.7 12.7 11.7 12.011.0 6.1 5.813.2 11.5 11.5 10.5 6.1 5.813.0 12.7 11.410.5 4.9 13.112.5 11.7 10.5 5.0 10.212.6 12.512.6 4.3 12.4 11.5 12.7 10.3 |