7. The multiplier is 24; the multiplicand is 691,344; if the multiplicand be diminished 4 times, what will be the product? 8. The multiplier is 36; the product is 91884; if the multiplicand is diminished 3 times, what will be the product? 9. The divisor is 9; the quotient is 6847; if the divisor be diminished 3 times, what will be the quotient? 10. The divisor is 28; the quotient is 418,374; if the divisor be increased 8 times, what will be the quotient? 11. The quotient is 9186; what will be the quotient if the divisor is increased 8 times? 13 times? If decreased 9 times? 17 times? 12. The divisor is 34; the quotient is 7641; the remainder is 11; what is the dividend? 13. The divisor is 78; the quotient is 3746; what is the dividend? 14. The product of two factors is 39746; the multiplier is 374; what is the multiplicand? 15. The sum of two numbers is 3974; their difference is 1389; what are the numbers? 16. The sum of two numbers is 49683; their difference is 3841; what are the numbers? LESSON VIII. 67. CONTRACTED OPERATIONS IN THE FUNDAMENTAL RULES. (a.) To multiply by 10, 100, 1000, &c. Annexing a cipher to any number changes the local value of the unit figure to tens, and of tens to hundreds, &c.; and therefore increases the number ten times; thus, 53 by an hundreds. nexing a cipher, thus: 530, the number is increased ten times. According to the same principle two ciphers increases the value one hundred times, three ciphers one thousand times, &c. (b.) To multiply by 25. RULE.-Annex two ciphers to the multiplicand and divide the result by 4. ILLUSTRATION.-If we annex two ciphers to the multiplicand we multiply it by 100 (67 a); but as 100 is 4 times larger than 25, the product is 4 times too large (66 d); therefore one fourth of this product is the product required. (c.) To multiply by 124. RULE.-Annex two ciphers to the multiplicand and divide the result by 8. NOTE-Let the pupil be required to illustrate each of these rules after the model given. (d.) To multiply by 331. RULE.-Annex two ciphers to the multiplicand and divide the result by 3. (e.) To multiply by 125. RULE.-Annex three ciphers to the multiplicand and divide the result by 8. (f) To divide by 10, 100, 1000, &c. By removing a figure from the right of any number the local value of ten's figure is changed to units, and hundreds to tens, &c., therefore the number is decreased ten times, hundreds. tens. units. tens. Ounits. thus: 637, by removing the right hand figure, 7, thus: 63, the number is decreased ten times with a remainder of 7. According to the same principle removing two figures decreases the value a hundred times, three figures a thousand times, &c. (g.) To divide by 25. RULE.-Multiply the number by 4, and divide the product by 100. ILLUSTRATION.-As 100 is 4 times larger than 25, if we divide any number by 100, the quotient will be 4 times too small, (see 66, h); therefore, 4 times this quotient will be the quotient required. It is most convenient first to multiply the number by the 4, and then divide the product by 100. (h.) To divide by 121. RULE.-Multiply the number by 8, and divide the product by 100. (i.) To divide by 331. RULE.-Multiply the number by 3, and divide the product by 100. (j) To divide by 125. RULE.-Multiply the number by 8, and divide the product by 1000. SECTION VII. LESSON I. UNITED STATES MONEY. 68. The Currency of the United States is a decimal currency, and is sometimes called FEDERAL MONEY. (a.) TABLE OF DENOMINATIONS. 69. (a.) COINS.-The Gold coins are the doubleeagle, eagle, half-eagle, quarter-eagle, three-dollar piece, and dollar. (b.) The Silver coins are the half-dollar, quarter-dollar, dime, half-dime, and three-cent piece. . (c.) The Copper coins are the two-cent piece and cent. 70. The decimal point, (.), is used to separate dollars from cents and mills; thus, $46.873 equals 46 dollars 87 cents and 3 mills. NOTES.-1. Since the places in Federal Money increase in value from the right to the left in a ten-fold ratio like simple numbers, it may be added, subtracted, multiplied, and divided, according to the same rules. 2. In business transactions, eagles are read as tens of dollars, and dimes as tens of cents; thus, 4 eagles 9 dollars 5 dimes 6 cents and 3 mills, are read, 49 dollars 56 cents and 3 mills. 3. The fifty-dollar piece is not a legal coin. 4. The Copper half-cent is no longer coined. The mill is not a coin. 5. The dollar sign ($) is by some supposed to be a contraction of U. S. 6. All other coins are composed in part of alloy, about nine-tenths only being pure metal. LESSON II. 71. REDUCTION. 1. How many cents in 75 dollars? MODEL OPERATION. $75 100 Cents. 7500 Cents. (a.) ANALYSIS.-Since there are 100 cents in one dollar, in 75 dollars there are 75 times 100 cents, which are 7500 cents. (b.) Therefore, there are 7500 cents in 75 dollars. (c.) RULES.-To change dollars to cents, multiply by 100; that is, annex Two ciphers. (d.) To change dollars to mills, annex THREE ciphers. (e.) To change cents to mills, annex ONE cipher. EXAMPLES FOR PRACTICE. 2. Reduce $24 to cents. 3. Reduce $36 to cents. 4. Reduce $14 to mills. 5. Reduce $102 to mills. 6. Reduce $100 to cents. NOTE.-To change dollars and cents to cents, or dollars, cents and mills to mills, remove the decimal point and the dollar sign. 10. Change $4.36 to cents. 12. Change $6.374 to mills. 11. Change $3.73 to cents. 13. Change $487.411 to mills. 14. How many dollars in 3746 cents? MODEL OPERATION. Cents, 1'00)37'46 cents. $37.46 ANALYSIS.-Since in 100 cents there is 1 dollar, in 3746 cents |