The order in which the months occur, and their length in days, may be seen by the following: 14. How many days are there in the common year? 15. How many days in every fourth year? 16. What is this year called? 17. To which month is the day added? 18. How many days has February? 19. Into how many months is the common year divided? 20. What are these months called? 21. Why called calendar months? 22. How many lunar months are there? 23. By what is this division of the year made? 24. Repeat the months in each year, commencing with the spring. 25. Repeat the months of the year, commencing with January, and also the number of days in each month. 26. How many days in December? LESSON TWENTY-FIRST. EXPLANATIONS AND DEFINITIONS. 1. The terms of a fraction are the numerator and denominator taken together. 2. A measure of two or more numbers is a number which will divide those numbers without a remainder. 3. The greatest common measure of two or more numbers is a number which will divide those numbers without a remainder. 4. A common multiple of two or more numbers is a num ber which may be divided by those numbers without a remainder. Hence the product of two or more numbers is their common multiple. 8 is a common multiple of 4 and 2, for twice four is 8. 5. The least common multiple of two or more numbers is the least number that can be divided by those numbers without a remainder. The least common multiple of 4 and 2 is 4. 6. A common multiple of the denominators of two or more fractions, is called their common denominator. 7. The common denominator of two or more fractions expresses the denomination to which each of the fractions must be reduced. 8. When any one denominator is a multiple of each of the others, it is their least common denominator. 9. A factor is a number employed as a multiplier. 10. A prime factor is a number divisible only by itself, or a unit, as 2, 3, 5, 7. 11. All the prime factors of a number are the least factors which, multiplied together, will produce it. 2, 2, and 3 are all the prime factors of 12. 12. Two numbers are prime to each other, when no one number will divide them both without a remainder. 8 and 9 are prime to each other. 13. A composite number is the product of two or more numbers. 15 is a composite number; 3 times 5 are 15. 3 and 5 are called the component parts of 15. 14. A component factor of a number, is a composite number that will measure it. 4 and 6 are the component factors of 12; and 4, 6, and 12 are all the component factors of 12. 1. What are the terms of a fraction? 2. What is a common measure of two or more numbers ? 3. What is the greatest common measure of two or more numbers? 4. What is the least common multiple of two or more numbers? 5. What is a common multiple of the denominators of two or more fractions called? 6. How may it be found? 7. What does the common denominator of two or more fractions express? 8. What is a factor? 9. What is a prime factor? Give an example. 10. What are the prime factors of 8? of 16? of 18? 11. When are numbers prime to each other? Give an example. LESSON TWENTY-SECOND. Let it be required to find a common denominator to the fractions and, and to reduce the fraction to the denomination expressed by the common denominator. Since 8 is a multiple of 2 and 4, we may reduce them both to eighths. But it is obvious to the eye, that 4 is also a multiple of 2 and 4, and that it is the least common multiple. If we consider 8 as the common denominator, we must reduce both the fractions to eighths; if 4, then we have simply to reduce the to fourths. To reduce a fraction of a higher denomination to a fraction of a lower, we first find a common denominator; then divide the common denominator by the denominator of each of the fractions; we thus ascertain how many of the lower denomination make one of the higher. Into the quotient thus obtained, we multiply the terms of each of the fractions, and the fractions are reduced to the denomination required. 8 contains 2, four times, therefore, four eighths make a half; and 8 contains 4 twice, therefore, 2 eighths make a fourth; 4 contains 2 twice therefore, 2 fourths make a half. Their numerators added, give the answer. 1. Reduce and to a common denominator, and add them together. 2. What is the common multiple of 2 and 6? 3. What is the least common multiple ? How do you know? 4. How many twelfths make a half? 7. How many twelfths are and? is equal to ? Because 6 is half of 12, and one is half of 2. 8. How do you know that 14. John had some nuts: he gave of what he had to his mother, to his sister, and to his brother; what part of what he had did he give away, and what part had he left? 15. James had some candy: he gave to one playmate of what he had; to another; what part had he left? In subtraction of fractions, we reduce them to a common denominator, as in addition, then subtract the less numerator from the greater. 16. A man sold ton to neighbor B.; Give the process. of a ton of hay to neighbor A., } of a what part of a ton had he left? 17. A man sold of a cord of wood to one man, to another; what part of a cord did he sell, and what part of a cord had he left? Give the process. 18. How many shillings in 2 pounds? 19. How many twentieths in 2 wholes? 20. How many shillings in 2 pounds and 1 shilling? 21. How many 20ths in 2 wholes and? 22. How many fourths in 1 whole and I fourth? 23. How many fourths in 1 half and 2 fourths? 24. Do you perceive any similarity between the addition of whole numbers of different denominations, and the addition of fractions of different denominators? 25. In what does the similarity consist? LESSON TWENTY-THIRD. 1. of is how much? 2. What kind of a fraction is of ? 3. How is it reduced to a simple fraction? 4. What effect upon the value of a fraction has multiplying its denominator? 5. How does it diminish its value? 6. What effect has multiplying its numerator? 7. What effect has multiplying both numerator and denominator by the same quantity? 8. A man bestowed, in charity, a certain sum of money; to one object he gave of the whole sum, to another object }; what part of the whole sum had he then given away, and what part had he left? 9. What is the sum of 1 |