MULTIPLICATION OF COMPOUND QUANTITIES BY SIMPLE QUANTITIES. §61. Suppose you purchase 8 melons at 7 cents apiece, and afterwards find that you must give 5 cents apiece more for thein. In this case you pay 8 times 7 cents, and also 8 times 5 cents; that is, first, 56 cents, and afterwards 40 cents. §62. Let us apply this principle to Algebra. You pay in all, 8 times 7+5, which =56+40. Which shows that in multiplying a compound quantity, you multiply each term by itself. We can easily see that this operation will give the right answer; for in the case of the melons, they cost 12 cents apiece, and therefore their whole cost was 8 times 12 cents which 96 cents. But the answer just obtained, 56+40 =96. §63. But suppose that after you had paid 7 cents apiece, a deduction of 5 cents apiece was made. The whole cost would then be 8 times 7-5, which =56—40. And this agrees with the truth; for you first paid 56 cents, and afterwards 40 cents were deducted. $64. This shows that + multiplied by +, produces +; and multiplied by +, produces —. 6. Multiply 40+x, by 10. 7. Multiply x-32, by 9. 8. Multiply 52-x, by 12. 9. Multiply 2x+14, by 7. 10. Multiply 27+3x, by 14. 11. Multiply 3x-62, by 15. 12. Multiply 97-4x, by 12. 13. Multiply x+7—y, by 7. 14. Multiply 3x+y-12, by 8. 15. Multiply 2x-3y-6, by 6. 16. Multiply 3x-12+y, by 5. $65. Franciscus Vieta, a Frenchman, introduced about the year 1600, the vinculum or a straight line drawn over the top of two or more quantities when it is wished to connect them together. Thus, x+4x3, signifies that both x and 4 are to be multiplied by 3. $66. In 1629, Albert Girard, a Dutchman, introduced the parenthesis as a convenient substitute, in many cases, for the vinculum. Thus, (x+4)×3, is the same as x+4x3; and is read, x+4, both ×3. If there are more than two terms under the vinculum, we say, after repeating those terms, all, &c. Thus, (x+y) x (a-b+c), is read x+y both into a-b+c all. See also §100., EQUATIONS.-SECTION 5. 1. x-9×11=121, to find x. Solution, x-9x11=121 Performing the multiplication, 11a-99—121 Transposing and uniting, 11x=220 Dividing by 11, x=20. 6. Given (8+x)×2+14=72, to find x. 7. Given (15+x)×3—27=48, to find x. 8. (112—2x)×3=(2x-7)×4, to find x. 9. (3x+14)x4 (78-x)×5, to find x. 10. 2x+8x5=(32+x)×3, to find x. 11. (3x-14)x7=(17-x)×6, to find x. 12. 120-3x-2=(4x-6)×9, to find x. PROBLEMS. Ans. 21. Ans. 8. Ans. 7. 1. Two persons, A and B, lay out equal sums of money in trade; A gains $126, and B loses $87; and now A's money is double of B's. What did each lay out? Stating the question, Forming the equation, Transposing and uniting, x, the sum for each. x+126=A's sum now. x-87-B's sum now. 2x-174- the double of B's. x+126-2x-174 2. A person, at the time he was married, was 3 times as old as his wife; but after they had lived together 15 years, he was only twice as old. What were their ages on their wedding day? Stating the question, x= the wife's age. 3x the man's age. = x+15= the wife's after 15 years. 3x+15= the man's after 15 years. 2x+30 twice the wife's age. Forming the equation, 3x+15=2x+30 x=15 the wife's age. Transposing and uniting, 3x=45 the man's age. 3. A man having some calves and some sheep, and being asked how many he had of each sort; answered that he had twenty more sheep than calves, and that three times the number of sheep was equal to seven times the number of calves. How many were there of each? If x= number of calves, there x+20= number of sheep. Ans. 15 calves, and 35 sheep. 4. Two persons, A and B, having received equal sums of money, A paid away $25, and B paid away $60; and then it appeared that A had just twice as much money as B. What was the sum that each received? Ans. $95. 5. Divide the number 75 into two such parts, so that three times the greater may exceed 7 times the less by 15. If x = the greater, thus 75-x: will 7 times the less +15. = = the less; and 3r Ans. 54 and 21. What is the 6. The garrison of a certain town consists of 125 men, partly cavalry and partly infantry. The monthly pay of a horse soldier is $20, and that of a foot soldier is $15; and the whole garrison receives $2050 a month. number of cavalry, and what of infantry? Ifx number of cavalry, then 20x Ans. 35 cavalry, and 90 infantry. pay, &c. their whole 7. A grocer sold his brandy for 25 cents a gallon more than he asked for his wine; and 37 gallons of his wine came to as much as 32 gallons of his brandy. What was each per gallon? Ans. $1.60 for wine; and $1.85 for brandy. 8. A wine merchant has two kinds of wine; the one costs 9 shillings a gallon, the other 5. He wishes to mix both wines together, so that he may have 50 gallons that may be sold without profit or loss for 8 shillings a gallon. How much must he take of each sort? The whole mixture will be worth 50 times 8 shillings. Ans. 37 gallons of the best; and 12 of the poorer. 9. A gentleman is now 40 years old, and his son is 9 years old. In how many years, if they both live, will the father be only twice as old as his son? In x years he will be 40+x, and his son 9+x. 10. A man bought 20 oranges and 25 lemons for $1.95. For each of the oranges he gave 3 cents more than for a lemon. What did he give apiece for each ? Ans. 3 cents for lemons, 6 cents for oranges. 11. A man sold 45 barrels of flour for $279; some at $5 a barrel, and some at $8. How many barrels were there of each sort? Ans. 27 at $5; and 18 at $8. 12. Says John to William, I have three times as many marbles as you. Yes, says William; but if you will give me 20, I shall have 7 times as many as you. How many has each? P Let x= William's and 3x-John's. Then after the change, x+20= William's and 3x-20- John's. Ans. John 24; William 8. 13. A person bought a chaise, horse, and harness, for $440. The horse cost him the price of the harness, and |