EXAMPLE.-A speculator lost 34% of an investment in stocks and had $10,560 remaining; what was the original investment? SOLUTION.-The original investment, on which the 34% is computed, is the base. Since there has been a loss, or decrease, the remaining $10,560 is the difference. Applying the rule, base = $10,560 ÷ (1 − .34) = $10,560.66 = $16,000. Ans. 23. The difficulty that the student is most likely to experience in percentage is the identification of the terms or elements. When an example is given, he must first determine from it which is the base, the percentage, the amount, etc. The rate is always recognized by the words per cent. or by the sign %. The base is the most important element, and it can be identified by referring to the rate. The student asks himself: "Of what number do I wish to find the percentage?" The answer to this question is the base. The percentage is always a number of the same kind as the base. If, from the statement of the problem, it is seen that the base increases or decreases, the percentage is the increase or decrease, and the final value obtained, when the base has been increased or decreased by the percentage, is the amount or the difference. To be able to recognize the elements immediately requires much practice. As an exercise, several examples are given. 24. EXAMPLE 1.-Fifteen tons of iron are obtained from 282 tons of ore; what per cent. of the ore is iron? SOLUTION. Here the statement of the example shows that the rate is what is required. From the phrase, "what per cent. of the ore," we see that the ore is the thing of which a per cent. is taken. Therefore, the 282 T. must be the base. 15 T. is a number of the same kind as the base, and is the part of the base corresponding to the rate. It is, therefore, the percentage. = = Rate = percentage base 15 ÷ 282 = .05319 5.319%. Ans. EXAMPLE 2.-Out of a cargo of oranges, 8% spoiled and 4,600 boxes remained. How many boxes were in the cargo? How many boxes spoiled? SOLUTION. The rate is .08; .08 of what? The number of boxes in the cargo. Therefore, the base is the original number of boxes and the less number of boxes must be the difference. The number of boxes spoiled is the percentage. EXAMPLE 3.-A farmer lost 63 sheep, which was 18% of his flock; how many had he left? SOLUTION. The rate is .18. The number of sheep in the flock must be the base, since it is the number on which the 18% is computed. The decrease, or number of sheep lost, is the percentage, and the number remaining is the difference. (h) 40%. (a) What Ans. { {G_871% (h) What per cent. of 280 is 112? 1. A man's salary is $1,800 per year and he saves $225: per cent. of his salary does he save? (b) What per cent. of it does he spend? 2. A man has 32% of his money invested in stocks, 18% in grain, and the remainder, which is $7,620, in real estate; what is the total value of his property? Ans. $15,240, 3. If wool loses 32% of its weight in washing, how many pounds of unwashed wool are required to produce 35,360 pounds of washed wool? Ans. 52,000 lb. 4. In 1890, the population of a city was 85,000, which was 36% more than the population in 1880; what was the population in 1880? Ans. 62,500 5. If gunpowder contains 75% of saltpeter, 10% of sulphur, 15% of charcoal, how much of each is there in a ton of powder? 6. A man bequeathed to a charity 32% of his estate; to another charity he gave $23,100, which was 23% less than the amount given to the first charity: (a) What was the value of the estate? (b) What per cent. of the estate was given to the second charity? 7. A man owning a ship worth $225,000, sells one-fourth of it to A, 20% of the remainder to B, and 35% of what then remains, to C; how much each do A, B, and C pay for their shares? A, $56,250. 26. Profit and loss treats of the gains or losses arising in business transactions. If the price for which merchandise is sold is greater than the cost of the merchandise, the difference is profit or gain. If the selling price is less than the cost, the difference is loss. 27. The gross cost of merchandise is its first cost plus the expenses of purchase, transportation, and storage. Such expenses are commission, freight, insurance, drayage, etc. 28. The net selling price is the gross selling price, less all discounts and expenses of sale. 29. Computations in profit and loss are made according to the rules of percentage. The gross cost of the merchandise is the base, upon which the rate of profit or loss is computed. The profit or loss is the percentage. If the merchandise is sold at a profit, the net selling price is the amount; if at a loss, the net selling price is the difference. 30. Rule.-To find the profit or loss, multiply the gross cost by the rate of gain or loss. (Art. 14.) Formula. Profit or loss = cost X rate. EXAMPLE.-A house costing $3,000 is sold for 22% above cost; what is the profit? SOLUTION.-Profit = cost X rate == $3,000 × .22 = $660. Ans. 31. Rule.-To find the rate of profit or loss, divide the difference between the selling price and gross cost by the gross cost; or divide the profit or loss by the gross cost. Formula. Rate = profit or loss ÷ gross cost. (Art. 15.) EXAMPLE. A merchant sold for $768 a lot of dry goods for which he paid $900; what was the per cent. loss? = $132. = .14 or 14%. Ans. SOLUTION.- Loss = $900 - $768 - 32. Rule. To find the selling price, the cost and rate of gain or loss being given, multiply the cost by 1 plus the rate of gain, or by 1 minus the rate of loss. Formulas. Selling price = { Cost X (1+rate of gain). (Art. 19.) (Art. 20.) EXAMPLE. If hay is bought for $8 per ton, and if baling and shipping costs $5.50 per ton additional, at what price must it be sold to yield a profit of 16%? SOLUTION.- Gross cost = $8+ $5.50 $13.50. $15.66. Ans. 33. Rule. To find the cost, the selling price and rate of gain or loss being given, divide the selling price by 1 plus the rate of gain, or by 1 minus the rate of loss. EXAMPLE.-Sold drugs for $112 and gained 75%; what was the cost of the drugs, and what was the profit? EXAMPLES FOR PRACTICE. 34. What is the profit or loss (a) If the gross cost is $85 and the rate of gain is 32%? (c) If the gross cost is $240 and the rate of gain is 16% ? (a) $27.20. Ans. (b) $100.50. (c) $40.00. What is the rate of gain or loss (d) If the gross cost is $6.50 and selling price is $9.10? What is the selling price (e) Ans. { (d) 40%. 10%%. (g) If the cost is $945 and the rate of gain is 33%? (h) If the cost is $3.50 and the rate of gain is 12% ? (i) If the cost is $125 and the rate of loss is 18% ? (f) 14%. What is the cost Ans. (g) $1,260. (j) If the selling price is $575 and the rate of gain is 15%? 1. A house and lot that cost $3,250, is sold at a profit of 12%; what is: (a) the profit? (b) the selling price? Ans. { (a) $390. 2. What must be the selling price of a suit of clothes that costs $18 in order that the profit may be 33% ? Ans. $24. 3. A harvesting machine costs the hardware merchant $90 net, and $6 for freight and cartage; if sold for $108, what is the gain per cent.? Ans. 12%. 4. A carload of cattle is sold for $875, which is at a loss of 16%; what was the cost of the cattle? Ans. $1,041.67. 5. A sells a steam tug to B, gaining 14%, and B sells it to C for $4,104 and gains 20%; how much did the tug cost A? Ans. $3,000. 6. How much must hay sell for per ton, to gain 25%, if when sold for $8.40 per ton, there is a gain of 163% ? Ans. $9. 7. Six horses were sold at $125 each; three of them at a profit of 25% and the others at a loss of 25%. What was the net gain or loss? Ans. $50 loss. |