.005 = 100 2. Multiply .005 by .05 .05 = 1 ΙοσσXTδο -Ιστόσο Toiboy as a decimal =.00025. How does the number of decimal places in the product 0.00025) compare with the number of decimal places in both the multiplier (.05) and the multiplicand 0.005)? 3. Multiply 1.5 by 3.75 1.5 =1or 15 3.75 -3166 or 175 15x175 =1685 or 5190256 51000 6.25 in the decimal form =5.625 How does the number of decimal places in the product (5.625) compare with the number of decimal places in both the multiplier (3.75) and the multiplicand (1.5)? In the above examples we see the followingł: PRINCIPLE: In multiplying decimals, the number of deci mal places in the product is equal to the sum of the decimal In multiplication of decimals it is not necessary to place units under units, tenths under tenths, etc. There are 4. Multiply .025 by 1.5 .025 Multiply as if the numbers were whole numbers. 1.5 3 decimal places in the multiplicand and one decimal place in the multiplier. Therefore, there must be 3+1, or 4 decimal 125 places in the product. We must prefix a zero before the 3 25 and place the decimal point in front of this zero. Then .0375 .025 X 1.5=.0375 To the teacher: All new principles should be developed inductively as shown above. If the pupils are not able to see the principle from the three illustrations that are given, additional examples can readily be supplied by the teacher until the pupils are able to see the principle involved. 21. 27. Multiply the following: 5. .5 X.27 17. 7.75 X.075 29. 16.5 X 16.5 6. 1.2 X.75 18. .038 X 26.5 30. 5.5 X 5.5 7. .05 X $175 19. .48 X 1250. 31. 1.15 X 2.3 8. .06 X $142.50 20. 25.5 X3.2 32. 7.5 X3.1416 9. .055 X $275 79X.025 33. .375 X 4.8 10. 27.5 X 10.9 22. 12X3.1416 34. 2150.42X3.5 11. 2.5 X79 23. .06 X 355.25 35. 30.25 X 5.75 12. 6X3.1416 24. .075 X.062 36. .06 X $850.50 13. 8X.866 25. .1x.001 37. .05 X $2000 14. 16 X.7854 26. .0525 X 875 38. .042 X 275 lb. 15. .0025 X 6.5 7.5 X 6.25 39. .038 X 156 lb. 16. 396.09X1.5 28. .075 X 4.25 40. .037 X 280 lb. 41. An owner of a factory employs 21 men at $2.50 per day, 14 men at $3.25 per day and 8 men at $3.75 per day. Find the total amount of wages paid per day. 42. A farm of 83.5 acres was sold for $122.75 per acre. Find the amount received for the farm. 43. Mr. Adams bought 8 tons of hard coal at $10.75 per ton. What was the cost of that amount of coal? 44. Philip went to visit his grandparents who lived 85 miles away. He traveled .3 of the distance by boat and the rest by train. How many miles did he travel each way? 45. A farmer had a drove of 64 hogs and sold .75 of them. How many did he sell? 46. A gardener sold a grocer 16 bushels of rutabagas at 85 cents a bushel. He received in exchange 125 pounds of sugar at 11 cents per pound, and the balance in cash. How much cash did he receive? 47. Find the cost of 1.75 cwt. of freight at $.24 per cwt. (Cwt. is an abbreviation of hundred weight.) Exercise 7. Multiplying by 10, 100, 1000, Etc. 1. Multiply .5 by 10; 3.5 by 10; .08 by 10; 6.005 by 10; 12.35 by 10. 2. Compare the positions of the decimal points in the products with the positions of the decimal point in the multiplicands. If you have multiplied correctly, your results should show that: Multiplying a decimal by 10 moves the decimal point one place to the right. 3. Multiply .5 by 100; 3.5 by 100; .08 by 100; 6.005 by 100; 12.35 by 100. 4. Compare the positions of the decimal points in the products with the positions of the decimal points in the multiplicands. 5. Multiplying a decimal by 100 moves the decimal point how many places to the right? 6. Multiply each of the multiplicands in the first problem by 1000 and determine how many places the decimal point is moved to the right in pointing off the products. From these examples we see that in multiplying by ten or any number which is a product of tens (as 100, 1000, etc.), we move the decimal point as many places to the right as there are zeros in the multiplier. Multiply each of the following numbers by 10, by 100, by 1000, without using a pencil: 7. 1.2 13. 2150.42 19. .06 25. .042 8. .055 14. 16.5 20. 1.01 26. 1.035 9. .0025 15. 30.25 21. 4.125 27. 1.4142 10. 6.25 16. .866 22. .0525 28. 1.732 11. .075 17. .7854 23. 25.5 29. $850.50 12. 3.1416 18. 5.25 24. .625 30. $2.75 Exercise 8. Division of a Decimal by a Whole Number. 1. Divide .8 by 4 Express .8 as the common fraction To %+4= * o expressed as a decimal = 2 2. Divide .08 by 4. Express .08 as the common fraction 180 180+4=1 To expressed as a decimal =.02 3. Divide .12 by 4 .12= 10% 10:4=187 To as a decimal=.03 4. Divide .025 by 5 .025=10 6 1360+5=100 0 10% 0%.005 We may now put these examples in the regular division form, pointing off the quotients by the answers obtained by using common fractions: .2 .02 .03 .005 4).8 4).08 4).12 5).025 PRINCIPLE: In dividing a decimal by a whole number the decimal point is placed in the quotient directly above the Divide: 6. .125 by 5. 6. .84 by 7. 7. 6.25 by 25. 8. .96 by 8. 9. 15.75 by 15. 10. .875 by 35. 11. 9.315 by 9. 12. 81.25 by 13. 13. .0025 by 5. 21. 138.161 by 23. 22. 28.576 by 19. 23. .705 by 94. 24. 669.6 by 72 25. 10.392 by 12. 26. 50.2656 by 16. 27. 28.3544 by 36. 28. 127.278 by 90. .05=100 5 5 10 1 .25= 25 Exercise 9. Division of a Decimal by a Decimal .025=1857 -=.5 Therefore .025:.5=.5 %+100=&*100=40. 100 Therefore 2.5+.25=10 4. Divide .375 by 12.5. 12.5=125. X125 =100 18=.03. Therefore .375+12.5=.03. Let us next put these examples in a long division form so that the principle of division of decimals may be more clearly The quotients are pointed off as shown above: 40. 10. .03 1 10 .375=375 1 3 7 5 1000 1000 3 125 3 7 5 10 1000 100 seen. .05).025 25 .02).80 8 .25) 2.50 25 12.5).375 375 0 0 6. Answer the following questions for each of the above problems: (a) How many decimal places are there in the divisor? (b) The decimal point in the quotient is how many places to the right of the decimal point in the dividend? |