4. When and where will the express train in Exercise 3 be 30 miles behind the freight? 5. Two trains leave a city at the same time, traveling at the rates of 30 and 45 miles an hour respectively. One arrives at another city 3 hours ahead of the first. Find the distance between the cities and the time in which each train made the trip. 6. How much vinegar must be added to a barrel of vinegar 60% pure to make it an 80% solution? 7. A cough medicine is 50% paregoric. How much paregoric should be added to 4 ounces of the medicine to make it 75% paregoric? 8. Brass is an alloy of copper and zinc. How many cubic centimeters of zinc, density 6.9, must be combined with 100 cubic centimeters of copper, density 8.8, to form brass whose density is 8.3? 9. Coinage silver is an alloy of copper and silver. How many cubic centimeters of copper, density 8.8, must be added to 10 cubic centimeters of silver, density 10.6, to form coinage silver whose density is 10.4? 10. At what time between 2 and 3 are the hands of a clock together? opposite? 11. At what time between 7 and 8 are the hands of a clock together? opposite? 12. In a clock which is not keeping true time, it is observed that the interval between successive coincidences of the hour and minute hands is 66 minutes. What is the error of the clock? 13. The planet Mercury makes a circuit around the sun in 3 months, and Venus in 7 months. Find the time between two successive times when Mercury is between Venus and the sun. 14. A shoe dealer buys 100 pairs of shoes at $2.00 a pair, and sells 75 pairs at $2.50 a pair. To sell the remaining shoes he marks them down so as to make 20% profit on the whole. What price per pair will give this result? 15. A and B can do a piece of work in 10 days, but at the end of 7 days A falls sick and B finishes the work in 5 days. How long would it take each man to do the work? 16. One man can do a piece of work in 5 days which it takes a second man 7 days to do. How long will it take the two men working together? 17. A piece of work can be done by A in 10 days, and by B in 12 days. If A starts the work and works alone for 4 days, how long will it take A and B working together to complete it? 18. A pound of a certain alloy contains two parts of silver to three parts of copper. How much copper must be melted with this alloy to obtain one which contains three parts of silver to seven of copper? 19. When weighed in water, silver loses 0.09 of its weight, and copper 0.11 of its weight. If a 12-pound mass of silver and copper loses 1.17 pounds, find the weight of the silver and of the copper in the mass. 20. The crown of Hiero of Syracuse, which was part gold and part silver, weighed 20 pounds, and lost 14 pounds when weighed in water. How much gold and how much silver did it contain, if 19 pounds of gold and 10 pounds of silver each lose 1 pound in water? 21. The admission to an entertainment was 50¢ for adults and 25¢ for children. The proceeds from 125 tickets were $51.25. How many adults and how many children were admitted? 22. A tank can be filled by one pipe in 20 minutes, and by another pipe in 30 minutes. How long will it take to fill the tank if both pipes are opened? 23. A tank can be filled by one pipe in 5 hours and emptied by another in 8 hours. If the tank is half full, and both pipes are opened, how long will it take to fill the tank? 25. Remarks on Measurements. This section and the one following contain considerations which are of value to all who make measurements and computations based upon them. In particular, they are important for the statistician. Some of them find application in connection with tables of values of several functions which we shall study. The operation of making a measurement is counting. It is the determination of the number of units of a certain kind required to equal a magnitude of the same kind. In measuring a length with a scale graduated to tenths of an inch, the length is recorded to the nearest tenth. For instance, if a line appears to be 7.8 inches, the length is recorded as 7.8 inches; if it is about 7.8 inches it is recorded as 7.9 inches. If the length is so near the mid point of a subdivision that it is impossible for the observer to decide which point of division is nearer it is customary to choose the submultiple which is even. . Thus 7.8 inches is recorded as 7.8 inches, and 7.3 as 7.4 inches. An experienced observer determines the length to the hundredth part of an inch by making a mental subdivision of the tenth of an inch. If a higher degree of accuracy is desired, instruments which measure more precisely are employed, and various indirect methods of measurement are used. If a change of units is made the figures of a measurement may be preceded or followed by ciphers which are not deter mined by observation. Thus the same length may be recorded as 0.054 meter, 54 millimeters, or 54,000 microns. The digits 5, 4 are called the significant figures. The ciphers are non-significant. Ciphers in a number are not always non-significant. A length between the limits 7.795 inches and 7.805 would be recorded as 7.80 inches. The distance to the sun is sometimes given as 93,000,000 miles. This statement is ambiguous as there is nothing to indicate how many of the figures are significant and how many of the ciphers merely serve to locate the decimal point with reference to the unit employed. If all the ciphers were significant then the statement would mean that the distance was between the limits 92,999,999.5 and 93,000,000.5 miles. If the figures 9, 3 alone were significant, then the limits would be 92,500,000 and 93,500,000 miles. In a notation used to remove such ambiguities, known as the standard form, the sun's distance given to two significant figures would be written 9.3 × 107 miles. If the first three figures were significant the distance in standard form would be 9.30 × 107, which would indicate that the true distance is between 92,950,000 and 93,050,000 miles. A number like π or √2 can be calculated to any degree of accuracy desired, but there is a limit to the number of significant figures which can be obtained by measurement. The relative error in a measurement is the ratio of the possible error to the measurement. It is usually expressed as a percentage. The relative errors in the two distances 9.3 × 107 and 9.30 × 107 are respectively This illustrates the fact that the relative error depends upon the number of significant figures and not upon the position of the decimal point. Let x be a number obtained by measurement, expressed in terms of the smallest unit used. Then x differs from the true value by not more than 1, and the possible ferent numbers of significant figures is given in the following The illustrations following show how great an accuracy has been obtained in some measurements. The ratio of the mean solar day to the sidereal day has been ascertained to the eighth place of decimals. Balances have been constructed which respond to one part in a million. The accuracy attained in measuring a base line of a survey is usually about one part in 60,000, or an inch in a mile, but accuracy to one part in a million is claimed for some surveys. Three significant figures are sufficient for most engineering and commercial calculations, four significant figures for most physical and chemical computations, while some astronomical and geodetic calculations require measurements to six or seven significant figures. 26. Possible Errors in Arithmetic Calculations. Abridged Multiplication and Division. As magnitudes determined by measurements are not exact, it is important to be able to estimate the possible error, or the limit of error, in a result calculated from such measurements. Theorem 1. The possible error of the sum or difference of two measurements is equal to the sum of the possible errors of the individual measurements. + Δα If a and b are two measurements with possible errors ± and Ab, then the correct value of the sum of the measurements lies between the limits and (a + Aa) + (b + Ab) = (a + b) + (Aa + Ab), (a - Aa) + (b Ab) = (a + b) — (Aa + Ab). (Δα Hence the possible error of the sum a + b is Aa + Ab. The case for the difference of two measurements follows from the fact that the correct value of the difference lies between the Notice that in either case the possible error is half the difference of the limits. If several numbers representing measurements of different degrees of precision are added together, the significant figures are retained in the numbers and the sum is rounded off to the number of significant figures in the least accurate of the individual numbers. EXAMPLE 1. The value of the imports of Alaska from the United States in 1901, 1903, 1904 are reported as in the table. Find the sum. The first number is rounded off to thousands so that the sum should be rounded off to thousands. This may be done as the numbers are added. $13,457,000 9,509,701 10,165,110 $33,132,000 Theorem 2. The possible relative error of the product or quotient of two measurements is approximately equal to the sum of the relative errors of the individual measurements. The product will lie between the limits and (a + Aa) (b + Ab) = ab + Aa Ab + a Ab + b▲a, The possible error in the product is assumed to be, approximately, half the difference between these limits, so that the error in the product is Aaba Ab+b▲a. |