EXAMPLES FOR MENTAL PRACTICE. 532. 1. If the rate be 10%, what will be the amount per cent. ? 2. If the rate be 75%, what will be the amount per cent. ? 3. If the rate be 110%, what will be the amount per cent. ? 4. Find the amount per cent., if the rate per cent. be 163? 5. Find the amount per cent., if the rate per cent. be 874 ? EXAMPLES FOR WRITTEN PRACTICE. 533. 1. Goods costing $1400 were sold for $1470. Find the amount per cent. of the selling price ? 2. Last month I sold $2750 worth of coffee, while the previous month I sold $3000 worth. What was the amount per cent. of my sales for the previous month as compared with those of the last month ? 3. If tea costing 621€ per pound sell at 8714, what amount per cent. do the sales show as compared with the cost ? 534. To find the Difference Per Cent., the Rate being given. EXAMPLE.-If the rate be 5%, what is the difference per cent. ? OPERATION. EXPLANATION.-Since the difference per cent. '(definition, 100% 1.00 = a unit. page 160), is equal to 100 per cent., or 1, less the rate, if we take 5% .05 = rate. the given rate, 5 per cent., from 100 per cent., the remainder, 95% = .95 = dif.%. 95 per cent., will be the answer required. Rule.-Subtract the rate from the unit 1. Formula.—Difference Per Cent. = 1 – Rate. EXAMPLES FOR MENTAL PRACTICE. 535. 1. If the rate be 15%, what is the difference per cent. ? EXAMPLES FOR WRITTEN PRACTICE. 536. 1. The pupils of a school are reduced in number from 112 to 80. What per cent. is the present of the former attendance ? 2. Walter, having 48 marbles, gave Henry 15. What per cent. had he left ? 3. Find the difference per cent., if the rate equals cf . 537. To find the Amount, the Base and Rate being given, EXAMPLE.— What is the amount of 550 increased by 8% of itself ? OPERATION. EXPLANATION.—The amount equals base plus percentage (de550 = base. finition, page 160). The base is 550 and 8 per cent. of 550 cquals .08 = rate. 44, the percentage; therefore the amount must equal 550 plus 44, 44.00 = per cent. or 594; or, since 550 equals 100 per cent. of itself, an increase of 8 550 base. per cent. would give 108 per cent. of the original number; and 108 per cent. of 550, or 1.08 times 550 equals 594. 594 = amount Rules.-1. Find the percentage and add it to the base. Or, 2. Multiply the base by 1 plus the rate. Formula.-Amount = Base + Percentage. EXAMPLES FOR MENTAL PRACTICE. 538. 1. If the base is 1500, and the rate 10%, what is the amount ? EXAMPLES FOR WRITTEN PRACTICE. 539. 1. What amount will be received for a house costing $13500, if it is sold at a gain of 7%? 2. A bought two horses for $180 each, and sold one at a gain of 20% and the other at a gain of 331%. How much did he receive for both ? 3. A section of Kansas prairie was bought at $12.50 per acre, and sold at advance of 40%. How much was received for it ? 4. What is the amount of 768 increased by 25% of 4 of itself? 6. If the base is $864.88 and the rate 33% of of itself, what is the amount ? 540. To find the Difference, the Base and Rate being given. EXAMPLE. — What remains after diminishing 450 by 10% of itself ? OPERATION. 100% 450 base. . 10% :.10 EXPLANATION.-Since 100 per cent. of the number equals 450, 10 per cent. of it will equal 90% dif. % 45.00 = percentage. 45; and 450 minus 45 equals 405. Or, since 100 450 base. per cent. equals 450, 10 per cent. less than 100 45 percentage. per cent., or 90 per cent. will equal 405. 405 difference. Rules.-1. Find the percentage and subtract it from the base. Or, 2. Multiply the base by 1 minus the rate. Formula.—Difference = Base - Percentage. EXAMPLES FOR MENTAL PRACTICE. 541. 1. If from a brood of 15 chickens 20% are lost, how many will remain? 2. What number will remain if 225 is diminished by 33% of itself ? 3. If the base is 1050 and the rate 10%, what is the difference ? 4. 816, less 25% of itself, equals what number? 8. A boy having 648 ft. of kite string, lost 12% of it. How many feet had he remaining ? EXAMPLES FOR WRITTEN PRACTICE. 542. 1. A speculator lost 35% of 1 of $16250. How much did he lose? 2. A planter having 616 acres in rice, lost t of 33% of his planting by flood. How many acres had he left for harvest ? 3. Brown deposited $1147 in a savings bank, and his son deposited 21% less. How much was deposited by both ? 4. An agent earned $250 in May, 15% less in June, and 20% less in July than in June. What was the amount earned for the three months ? 543. To find the Base, the amount, or Difference, and the Rate being given. EXAMPLE (first illustration).-—What number, increased by 15% of itself, amounts to 345 ? OPERATION. EXPLANATION.-Since the number must be 100 100% - 1.00 per cent. of itself, if it has been increased 15 per 15% = .15 Amount. Base. cent., 345 must be 115 per cent. of that number; 115% amt. % 1.15) 345.00 (300 if 115 per cent. is 345, 1 per cent. must be tio of 345 345, or 3; and 100 per cent. will be 100 times 3, or 300. 00 EXAMPLE (second illustration).-What number, diminished by 35% of itself, equals 975 ? OPERATION. 100% = 1.00 35% EXPLANATION.-If the number be diminished Diff. by 35 per cent. of itself, there will be remaining but 65% dif. %= .65 ) 975.00 ( 1500 65 per cent, of itself; and if 65 per cent. of the 65 number be 975, 1 per cent, must be as of 975, or 325 15; and if 1 per cent. be 15, 100 per cent. must be 325 1500. = .35 Base. 00 Rules.-1. Divide the amount by 1 plus the rate. Or, 2. Divide the difference by 1 minus the rate. Formulas.—1. Base = Amount • Amount Per Cent. 2. Base = Difference • Difference Per Cent. EXAMPLES FOR MENTAL PRACTICE. 544. 1. If the amount is 750 and the rate 25%, what is the base ? 2. What number, increased by 10% of itself, amounts to 440 ? 3. After 75% of a number had been added to it, the amount was 525. What was the number? 4. After selling 30% of his apples, a boy had 70 left. How many had he at first ? 5. I lost $600 by a bankrupt, who paid only 85% of his indebtedness. What was the full amount of my claim ? EXAMPLES FOR WRITTEN PRACTICE. 545. 1. A builder gained 15% by selling a house for $1150. What was its cost ? 2. Sold 945 tubs of butter for $5113, and thereby gained 20%. How much did the butter cost pcr tub ? 3. The income from a tenement house is $6042 this year, which is 24% less than it was last year. How much was it last year ? 4. A liveryman paid $180 for a horse, which was 40% less than he paid for a carriage. How much did he pay for both ? 5. A drover gained 16% on 33 head of cattle sold for $4081. What was the average cost per head ? 6. Smith sold two horses for $1500 each, gaining 25% on one, and losing 25% on the other. What did the horses cost him ? 7. After paying 35% of his debts, a man finds that the remainder can be paid with $19500. What was his entire indebtedness ? 8. A boat load of wheat was so damaged that it was sold for $8500, which was 15% less than its original value. What was its value before it was damaged? 9. The attendance of pupils at a school during May was 954, which was 6% more than attended during April, and this was 80% more than attended during February. What was the attendance for February ? 10. Which is better, to invest in a house that will rent for $30 per month, at 6% on its value, or to invest the same amount in a farm that in two years will bring $7000 ? How much better in the two years ? REVIEW OF THE PRINCIPLES OF PERCENTAGE. 546. 1. To find the percentage, the base and rate being given. RULE. Multiply the base by the rate expressed decimally. 2. To find the base, the percentage and rate being given. RULE.—Divide the percentage by the rate expressed decimally. 3. To find the rate, the percentage and base being given. RULE.—Divide the percentage by the base, carrying the quotient to two decimal places. 4. To find the amount per cent., the rate being given. RULE.—Add the rate to the unit 1. 5. To find the difference per cent., the rate being given. RULE.-Subtract the rate from the unit 1. 6. To find the amount, the base and rate being given. RULES.–1. Multiply the base by the rate, and to the product add the base. Or, 2. Multiply the base by 100 per cent. plus the rate. 7. To find the difference, the base and rate being given. RULES.—Multiply the base by the rate, and subtract the product from the base. Or, Multiply the base by 100 per cent. minus the rate. 8. To find the base, the amount and rate being given. RULE.—Divide the amount by 100 per cent. plus the rate. 9. To find the base, the difference and rate being given. RULE.—Divide the difference by 100 per cent. minus the rate. 547. Percentage is applied to two classes of problems: First, to those in which time is not an element; as, Profit and Loss, Commission, Brokerage, Insurance, Taxes, Customs or Duties, and Trade Discounts. Second, to those in which time enters as an element; as, Interest, Bank Dis. count, True Discount, Equation of Accounts, and Exchange. REMARK.—The pupil should be drilled in the formulas and rules of simple or abstract Percentage as above, and in their application to problems in applied Percentage to follow. MISCELLANEOUS EXAMPLES FOR PRACTICE. 548. 1. At the battle of Waterloo, of the 145000 combatants, 51000 were either killed or wounded. What per cent. were uninjured ? 2. The pressure on a steam boiler was 61.2 lb., after it had been reduced 10%. What was it before the reduction ? 3. A pupil in examination answered correctly 56 questions, which was 20% less than the number asked him. What should be his average, on a basis of 100? 4. By assessing a tax of 3%, $175000 was raised in a county. What amount of property was taxed ? 5. A benevolent lady gave $10500 to three charities; to the first she gave $2500, to the second $4500, and to the third the remainder. What per cent. did each receive ? 6. On attaining his majority, a son finds his age is 62% less than the age of his father. Find the sum of their ages ? 7. If 8% of B’s money equals 24% of C's, how much has C, if B has $324 ? 8. A farmer bought a horse, a mule, and a cow, for $385. The mule cost 15% less than the horse, and the cost of the cow was 72% of that of the horse. What was the cost of each ? 9. A creditor, after collecting 21% of a claim, lost the remainder, which was $3918.75. What was the sum collected ? 10. A woman weaving a rag carpet used 185% more weight of rags than of warp. How many pounds of each in a bale of carpet weighing 964 pounds ? 11. The sum paid for two watches was $384, and 75% of the sum paid for one equalled 105% of the sum paid for the other. Find the price of each. 12. If A has 35% more money than B, and B has 25% more than C, how much has C, if A has $192 ? 13. If a gain of $4755 was taken out of a business at the end of the first year, and a loss of $3566.25 was sustained the second year, what was the per cent. of net gain or loss, the investment having been $63400 ? |