TWO UNKNOWN QUANTITIES. 550. Preliminary Problems. 1. I paid a dollar for two 25¢ balls and five bats. How much did I pay apiece for the latter ? 2. When three times one number is added to five times another, the sum is 84. If the second number is 12, what is the first number? 3. A girl paid 75¢ for pound of tea and 2 pounds of coffee. The coffee cost 20¢ per pound. What was the price of the tea per pound? 4. A man sold pigs at $5 each and lambs at $8 each, receiving $42. He sold 4 lambs. How many pigs did he sell? 5. Four times a father's age added to twice his daughter's age amounts to 180 years. The girl is 10 years old. What is the father's age? 6. Eight peaches and seven pears cost 44. The peaches cost 2 each. What is the cost of a pear? 7. Two pieces of cloth and eleven pieces of silk contain 152 yards. There are 10 yards in each piece of cloth. How many yards in each piece of silk? 8. Two-thirds of a yard of linen and three-fourths of a yard of lace cost 40%. The price of the lace is 32¢ a yard. Find the price of the linen. 9. Three and one-half times one number added to four and one-third times a second number equals 60. The second number is 9. What is the first number? 551. Written Exercises. Find the value of the unknown quantity: 1. The cost of 3 apples and 2 peaches is 7 cents. The cost of 2 apples and 2 peaches is 6 cents. Subtracting the second lot of fruit from the first lot we have 1 apple. Subtracting the price of the second lot from the price of the first lot we have 1 cent. 1 apple costs 1 cent. If equals are subtracted from equals, the remainders are equal. 2. A boy gave 25¢ for 3 lemons and 8 oranges, another boy paid 17 for 3 lemons and 4 oranges. How much did 3. If 3 coats and 14 vests cost $78, and 2 coats and 14 vests, at the same rate, cost $66, how much does 1 coat cost? What is the price of a vest? 4. Given 4x + 7y=53, to find the value of y. (1) (2) First multiply (2) by 2, making it 4x+6x= 50. Why? 5. What is the value of x in equation (1) in above example, when the value found for y is substituted therein? Substitute the same value for y in equation (2) and find the value of x. 553. Written Exercises. Find the values of x and y in the following equations: Multiply second equation by 2, 6x+4y = 70. 5. 7x+5y=82, 2x+3y=36. 7. 3x+5y=17, 8x+2y=17. 6. 5x+9y=14, 9x+5y=14. 8. 2x-3y=18, 3x+5y=65. Given x 3y = } To find values of x and y. (1) x+3y= 46, Multiply (1) by 7, (2) 7x+ 21 y = 322 10. 4x+3y=17, (1) Multiply (2) by 2 and subtract. 12. 3x+5y=13, (1) Multiply (1) by 7 and (2) by 3. 7x+3y=13. (2) Subtract. 554. Written Problems. 1. The sum of two numbers is 37. Twice the first added to three times the second is 96. What are the numbers? = second number. Let x = first number; y: 2. The difference between two numbers is 28. Five times the first less twice the second is 197. What are the numbers? x - y = 28; 5x-2y= 197. 3. The product of the first of two numbers by 5, added to the product of the second by 3, gives 37. The product of the first by 6, diminished by five times the second, equals 10. Find the numbers. 4. Divide 65 into two parts whose difference shall be 19. Let x and y = parts. Solve also by one unknown quantity. 5. A person pays $103 with 32 bills, some of them $2 bills, the others $5 bills. How many of each does he use? 6. For 25 head of pigs and sheep, a farmer received $145. How many of each did he sell, if he sold the former at $7 each, the latter at $5 each? 7. 10 oranges and 4 peaches cost 384; 6 oranges and 7 peaches cost 324. Find the cost of an orange. Of a peach. 8. 5 pounds of tea and 3 pounds of coffee cost $3.75; 8 pounds of tea and 1 pound of coffee cost $5.05. What is each worth per pound? 9. A farmer buys a certain number of horses at $125 each and a certain number of cows at $40 each. They cost together $740. If he had bought half as many horses and twice as many cows they would have cost $730. How many of each did he buy? 10. A man paid 75 for 2 pounds of raisins and 3 pounds of cheese. 5 pounds of raisins and 2 pounds of cheese at the same prices would have cost 94%. What did each cost per pound? |