16. Show the application of the statements in 15 in multiplying by. 17. If a divisor is used that is 4 times the correct divisor, the quotient obtained is of the correct quotient. What must be done to the first quotient to get the correct quotient? 18. In dividing by a fraction, if we divide by its numerator, the quotient obtained is such a part of the correct quotient as 1 is a part of the denominator of the fraction. Therefore the first quotient must be multiplied by the denominator of the fraction to secure the correct quotient. 19. Show the application of the statements in 18 in dividing by, and that it is the same as inverting the divisor and proceeding as in multiplication. 20. Give an example in which it is desired to find a fractional part of a number. Give the rule for finding a fractional part of a number. Change your example to an example in percentage and analyze it. 21. Give an example in which it is desired to find what fraction one number is of another. Give the rule for finding what fraction one number is of another. Change your example to an example in percentage and analyze it. 22. 8 is what fraction of 20? State the general problem that this example illustrates. Give the rule for solving examples coming under this general problem. Change the problem to a question in percentage and analyze it. DECIMAL FRACTIONS 87. A decimal fraction is a fraction whose denominator is 10 or some power of ten, expressed or merely indicated; thus, 1.3, 14, 14, are decimal fractions. Decimal fractions are usually called decimals. Each decimal whose denominator is not expressed is preceded by a period called the decimal point; thus, .3, .025. A mixed decimal is one having an integral and a decimal part; as, 12.75. By decimal places we mean the number of figures to the right of the decimal point; thus, 2.06 has two decimal places; .0764, four. SYSTEM OF DECIMAL NOTATION Study Recitation 88. In integral or whole numbers, the value of a figure in any order is the value of the same figure in the next higher order, and 10 times the value of the same figure in the next lower order. In the following numbers, show that the value of any figure is the value of the figure in the next higher order, and 10 times the value of the figure in the next lower order : 1111 2222 3333 4444 5555 9999 In whole numbers, the name of the lowest order is units, or ones. In the number 1111, of the value of the figure in the fourth order is 1 hundred, or the value of the figure in the third order; of the value of the figure in the third order is 1 ten, or the value of the figure in the second order; of the value of the figure in the second order is 1 unit, or the value of the figure in the first order; of the value of the figure in the first order is. As a decimal it is written without the denominator by placing a decimal point at the right of the units' order and then writing 1 at its right; thus, 1111.1. The name of the first order at the right of the decimal point is tenths. The value of .1 is of the value of 1 10. in tenths' order is of, or 100, and is written at the right of the 1 in tenths' order; thus, .11. The name of the second order at the right of the decimal point is hundredths. 1 the value of 1 in hundredths' order is of do, or too, and is written at the right of the 1 in hundredths' order; thus, .111. The name of the third order at the right of the decimal point is thousandths. In the same way it may be shown that of 1000 is 10000, that it is written at the right of the thousandths' order, and that the fourth order at the right of the decimal point is tenthousandths. The name of the fifth order is hundred thousandths. The name of the sixth order is millionths. The name of the seventh order is ten millionths. TO THE TEACHER. The names of still lower orders may be given if the teacher so desires. It will be seen that in writing decimal fractions, by using the decimal point, the fractions may be written in the same manner as whole numbers are written, the denominator being indicated by the number of places at the right of the decimal point. Hundredths Ten-thousandths Tenths Millionths .1.. 1 place or order is required to write tenths. Too=.01. 1000.001. 3 places or orders are required to write thousandths. Toooo=.0001. 4 places or orders are required to write ten-thousandths. 100000=.00001 5 places or orders are required to write hundred-thousandths. 1000000=.000001 6 places or orders are required to write millionths. 2 places or orders are required to write hundredths. 10000000000000 places or orders are required to write ten-millionths. The number of places required to write a decimal is the same as the number of zeros in its denominator. It is rarely necessary to use decimal orders beyond millionths. 89. To read decimals. RULE: Read the decimal as though it were a whole number and give it the name of the right-hand order. 1. Read .21567. The name of the right-hand order is hundredthousandths. As a whole number, we read : Twenty-one thousand five hundred sixty-seven. Adding the name of the right-hand order, we read: Twenty-one thousand five hundred sixtyseven hundred-thousandths. 2. Read .0032. The name of the right-hand order is tenthousandths. The decimal is read: Thirty-two ten-thousandths. In reading a whole number and decimal combined, as, 156.27, read the whole number first; add the word "and," and then read the decimal, thus: One hundred fifty-six and twenty-seven hundredths. Read the following. In examples 3-6 read down; in examples 7-18 read from left to right. RULE: Find the difference between the number of figures required to write the numerator and the number of zeros in the denominator. Write that number of zeros at the right of the decimal point and follow with the numerator. 1. Write 56 thousandths. The numerator is 56; the denominator is 1000. How many figures are there in the numerator? How many zeros in the denominator? How many zeros must be placed at the right of the decimal point before the numerator?. At the right of the decimal point write the required number of zeros and the numerator; thus, .056. * Read 2 and 8 tenths. ↑ Read 124 and 4 thousandths. |