15. How many two sixths in four ? Division of Fractions by Whole Numbers. In dividing fractions by whole numbers, we do not find how many times a whole thing is contained in a part of the same thing, for that would be absurd ; but we find what part of once, a whole number is contained in a fraction. Thus if we wish to divide one half by one, we say, one unit is contained in one half, not once, but one half of once. 1. One is contained in one fourth, what part of once ? 2. One is contained in one fifth, what part of once ? 3. One is contained in one sixth, what part of once ? 4. One is contained in one seventh, what part of once ? 5. One is contained in one eighth, what part of or.ce ? 6. One is contained in one ninth, what part of once ? 7. One is contained in one tenth, what part of once ? 8. One is contained in one eleventh, what part of once ? 9. One is contained in one twelfth, what part of once ? 10. If you divide one fourth by one, which is the divi. sor? The dividend? What is the quotient? 11. If you divide one sixth by one, what is the quotieni ? The divisor? The dividend ? 12. If you divide one third by one, what is the quotient ? The divisor? The dividend ? 13. If one fourth contains one, a fourth of once, what part of once does two fourths contain it ? Ans. Twice as much, or two fourths of once. 14. If two sixths is divided by one, what is the answer ? Ans. Two sixths of once. 15. Two eighths contain one, what part of once ? Six eighths contain one, what part of once? 16. Two twelfths contain one, what part of once ? Four twelfths contain one, what part of once ? In dividing fractions by whole numbers, do we find how many times a whole thing is contained in a part of the same thing? What do we / 17. Eight twelfths contain one, what part of once ! 18. Six twelfths contain one, what part of once ? If six twelfths contain one, six twelfths of once, it would contain two, only half as often, or three twelfths of once. 19. Four eighths contain one, what part of once ? Contain two, what part of once ? Two will be contained only half as often, or two eighths of once. 20. Six tenths contain one, what part of once ? Con. tain two, what part of once ? 21. Eight tenths contain one, what part of once ? Con. tain two, what part of once ? 22. Four eighths contain one, what part of once? Contain two, what part of once? 23. Sic elevenths contain one, what part of once ! Contain two, what part of once ? 24. Eight twelfths contain one, what part of once ? Contain two, what part of once ? REDUCTION. One dime is how many cents? How many mills ? One unit of the order of dollars, is how many units of the order of dimes? How many of the order of cents ? How many of the order of mills ? One eagle is how many dollars ? Ilow many dimes? Cents ? One unit of the order of dimes is how many units of the order of cents ? Reduction is changing units of one order, to those of another. A unit of the order of eagles is how many units of the order of dollars ? Of dimes? Two eagles are how many dollars? How many dimes ? many dollars in twenty dimes ? Thirty units of the order of dimes, are how many units of the order of dollars ? What is reduction ? Two pints are one quart. Eight quarts are one peck. Four pecks are one bushel. Two units of the order of quarts, are how many units of the order of pints? Eight pints are how many quarts? Three barley-corns are one inch. Three feet are one yard. One inch is how many barley-corns? Two inches are how many? Twelve barley-corns are how many inches? One foot is how many inches? Three feet how many ? One yard is how many feet? How many barley-corns? Two yards are how many feet? How many barley-corns? Three yards are how many feet? How many barley-corns? How many inches? How many inches? How many inches? How many feet are there in five yards? How many inches in five yards? How many barley-corns? How many barley-corns are there in seven yards? From the preceding exercises, you learn that a unit of one order may contain several units of another order. What do you learn from the preceding exercises? How many units of the order of cents, are there in one unit of the order of dimes? How many units of the order of dollars, are there in one unit of the order of eagles? How many units of the order of mills, are there in one unit of the order of cents? How many units of the order of pints, are there in one unit of the order of quarts? How many units of the order of pecks, are there in one unit of the order of bushels? How many units of the order of barley-corns, are there in one unit of the order of inches? How many units of the order of feet, are there in one unit of the order of yards? How many units of the order of days, are there in one unit of the order of weeks? How many units of the order of weeks, in one unit of the order of months ? Change two units of the order of dimes, to units of the order of cents. Change twenty units of the order of cents, to units of the order of dimes. Change three units of the order of yards, to units of the order of feet. Change nine units of the order of feet, to units of the order of yards. Change ten units of the order of pints, to units of the or der of quarts. a a Change five units of the order of quarts, to units of the order of pints. Change twenty-one units of the order of days, to units of the order of weeks, &c. When units of one order are changed to units of a higher order, the process is called Reduction ascending; and when units of one order are changed to those of a lower order, the process is called Reduction descending. If twenty cents are changed to dimes, which kind of reduction is used ? If twenty cents are changed to mills, which kind of re. duction is used ? If four gallons are changed to pints, which reduction is used ? If eight feet are changed to inches, which kind of re. duction is used ? In changing twelve barley-corns to inches, which kind of reduction is used ? In changing fourteen days to weeks, which reduction is used ? In changing five hours to minutes, which reduction is used ? In changing one hundred and twenty minutes to hours, which reduction is used ? What is reduction ascending? What is reduction descending ? a a Reduce three dimes to cents; to mills. Which kind of reduction is it? Reduce three hundred mills to cents; to dimes; and which kind of reduction is it ? Reduce three hundred mills to dollars, and which kind of reduction is it? Reduce two halves to quarters, and which kind of reduction is it? Ans. As a half is of more value, it is a higher order than a quarter, therefore it is reduction descending. In performing this last exercise, the pupil will find the necessity for the following distinction in regard to units. A unit has been defined as any whole thing of a kind,” and a fraction is defined as “ a part of a thing." But it is very often the case, that fractions are consid. ered as units. Thus when we reduce quarters to halves, and halves to quarters, we change units of the order called quarter, to units of the order called half. When we say a whole quarter of an apple, and a half a quarter of an apple, we think of a quarter as a whole thing of its kind. The difference between the two kinds of units is this : When we think of a whole quarter, we think of another thing of which the quarter is a part. We think of it as a whole thing in one respect, and as a part of a thing in another respect. But when we think of a whole apple, we do not necessarily think of another ng of which it is a part. When we think of a half of a loaf of bread, do we think of something of which the half is a part? When we think of a biscuit, do we necessarily think of something of which it is a part ? When we think of a third of an orange, do we necessa. rily think of something of which it is a part? When we think of a house, do we necessarily think of any thing of which it is a part ? Those units which do not require us to think of any other thing of which they are parts, are called whole numbers, Are fractions ever considered as units ? Give an example. What is the difference between these two kinds of units ? a |