SECTION LX. To find the square of any number ending in 25: To the square of the figure or figures preceding the 25, add one half as many units as there are in the number that is squared, and to this sum annex 0625 (the square of 25 with a 0 prefixed). If the figure or figures preceding the 25 are odd, add to the square one half of the next smaller number, and to this sum annex 5625 (the square of 75). the square of 925 is 855625. Thus SOLUTION. The square of 9 is 81; plus one half the next smaller nnmber (8) are 85; and 85 with 5625 annexed is 855625. 3. What is the square of 425? 525? 625? 725? 4. What is the square of 5. What is the square of 6. What is the square of 7. What is the square of 8. What is the square of 925? 9. What is the square of 1025? 10. What is the square of 1125 ? 11. What is the square of 1225? 12. What is the square of 1325? 13. What is the square of 1425? 14. What is the square of 1525? 15. What is the square of 1625 ? 16. What is the square of 1725 ? 17. What is the square of 1825? 18. What is the square of 1925? 19. What is the square of 2025? 20. What is the square of 3025? 21. What is the square of 4025 ? SECTION LXI. To find the product of two numbers one of which ends in 75 and the other in 25, when the preceding figure or figures in each are alike. Multiply the preceding figure or figures of one number by the same increased by 1, and to the product 721875. annex 1875. Thus, 825 × 875 = SOLUTION. The figure preceding the 75 (8) multiplied by the same increased by 1 (9) gives 72; 72 with 1875 annexed is 721875. ANS. 721875. 225 X 275? 325 X 375? 425 × 475 ? 525 X 575? 625 × 675? 725 × 775? 825 X 875? 925 × 975? 1025 × 1075? 1125 × 1175? 1. What is the product of 2. What is the product of 3. What is the product of 4. What is the product of 5. What is the product of 6. What is the product of 7. What is the product of 8. What is the product of 9. What is the product of 10. What is the product of 11. What is the product of 1225 × 1275? 12. What is the product of 1325 X 1375? 13. What is the product of 1425 × 1475 ? 14. What is the product of 1525 × 1575 ? 15. What is the product of 1625 X 1675? OBS. In multiplying 16 by 17, the pupil must use the process given in Section XLV. 16. What is the product of 1725 X 1775? OBS. In multiplying 21 by 22, the pupil must use the process shown in Section LIII. 21. What is the product of 3125 × 3175? 22. What is the product of 4125 × 4175? 23. What is the product of 5125 × 5175? 24. What will cost 175 young fruit trees, at $1.25 apiece? 25. What is the interest of 375 dollars for 5 years and 5 months, at 6 per cent.? 26. What is the interest of 225 dollars for 4 years and 7 months, at 6 per cent. ? SECTION LXII. To find the square of any number ending in 75: Prefix the square of the figures preceding 75, plus of these figures, to 5625 (the square of 75). Thus, the square of 1275 is 1625625. SOLUTION. The square of 12 is 144; § of 12 are 18; 144+18=162; and this prefixed to 5625 is 1625625. When the figures preceding the 75. are not an even number, add to the square of the next smaller number, plus 2, and annex to the sum 0625. Thus, the square of 375 is 140625. SOLUTION. The square of 3 is 9; 2 of 2 are 3; 9+ 3+2=14; to which annex 0625, and the answer will be 140625. 1. What is the square of 275? 2. What is the square of 375? 3. What is the square of 475? 4. What is the square of 575? 5. What is the square of 675? 6. What is the square of 775? 7. What is the square of 875? 8. What is the square of 975? 9. What is the square of 1075? 10. What is the square of 1175? 11. What is the square of 1275? 12. What is the square of 1375? 13. What is the square of 1475? 14. What is the square of 1575? 15. What is the square of 2075? OBS. The pupil must fix firmly in his mind the figures to be annexed, and then the above questions can be readily performed. SECTION LXIII. MISCELLANEOUS QUESTIONS. 1. At what part of their cost must goods be sold, to gain 20 per cent.? 2. Reduce to common fractions the following: 10 per cent. per cent. ANS. 12 per cent. ANS. . 163 20 per cent. 25 per cent. 333 per cent. 37 per cent. 50 per cent. 3. At what part of their original cost must goods be sold, to gain 12 per cent.? 334 per cent.? 37 per cent.? 4. A grocer bought flour at 5 dollars, but is obliged to sell it at a loss of 20 per cent.; at what price per barrel does he sell it? 5. A man bought cloth at 6 dollars and sold it at 7 dollars each yard; what per cent. did he gain? 6. What is the interest of 75 dollars for 15 days? for 12 days? for 10 days? OBS. The interest for 60 days being as many cents as there are dollars in the principal, the interest for 15 days will be 15 or as many cents as there are dollars in the principal 7. What is the interest of $120 for 20 days? 8. What is the interest of $60 for 30 days? 9. What is the interest of $120 for 19 (15+ 4) days? 10. What is the interest of $270 for 22 days? 11. Divide 38 into two parts which shall be to each other as to 3. 12. A person being asked the hour of the day, said that the time past noon was of the time to midnight; what was the time? 13. What is the number to which if 2 and of itself be added the sum will be 43? 14. John is twice as old as Mary, and she is four times as old as George. The sum of all their ages is 39; what is the age of each? 15. A gentleman gave $350 for his horse and chaise, but the horse was worth only as much as the chaise ; what was the cost of each? 16. One fourth of a man's age, plus 5 years, is equal to one third of his age; how old is he? 17. How many times in 12 hours are the minute and hour hands of a clock together? At what time between 2 and 3 o'clock? ( of 12.) 18. A, B and C, shared $148 so that A had $12 more than B, and $20 more than C; how much had each? 19. A, B and C, shared $180 so that A had twice as much as B, and C had twice as much as A and B both; how many dollars had each? 20. Divide $159 between two persons so that of what the first receives shall be equal to of what the second receives; what will you give to each? OBS. The ratio of to = = 28, or 28 to 25. 21. Divide $190 into two such parts that of one shall be equal to of the other. 22. At what time between 7 and 8 are the hour and minute hands of the clock together? |