CHAPTER XII. PROBLEMS INVOLVING TWO OR MORE UNKNOWN NUMBERS. 197. It is often necessary in the solution of problems to employ two or more letters to represent the numbers to be found. In all cases the conditions must be sufficient to give just as many equations as there are unknown numbers employed. If there are more equations than unknown numbers, some of them are superfluous or inconsistent; if there are fewer equations than unknown numbers, the problem is indeterminate. EXERCISE 77. 1. If A gave B $10, B would have three times as much money as A. If B gave A $10, A would have twice as much money as B. How much has each? y+10= the number of dollars B would have. .. y + 10 = 3 (x − 10). (1) If B gave A $10, x+10= the number of dollars A would have, y 10 the number of dollars B would have. ..x+10=2(y — 10). (2) From the solution of equations (1) and (2), x = 22, and y = 26. Therefore, A has $22, and B has $26. 2. If the smaller of two numbers is divided by the greater, the quotient is 0.21, and the remainder 0.0057; but if the greater is divided by the smaller, the quotient is 4 and the remainder 0.742. Find the numbers. Therefore, the numbers are 4.78 and 1.0095. 3. If A gave B $100, A would then have half as much money as B; but if B gave A $100, B would have one third as much as A. How much has each ? 4. If the greater of two numbers is divided by the smaller, the quotient is 7 and the remainder 4; but if three times the greater number is divided by twice the smaller, the quotient is 11 and the remainder 4. Find the numbers. 5. If the greater of two numbers is divided by the smaller, the quotient is 4 and the remainder 0.37; but if the smaller is divided by the greater, the quotient is 0.23 and the remainder 0.0149. Find the numbers. 6. If A gave B $5, he would have $6 less than B; but if he received $5 from B, three times his money would be $20 more than four times B's. How much has each? 7. If the numerator of a fraction is doubled and its denominator diminished by 1, its value will be. If its denominator is doubled and its numerator increased by 1, its value will be . Find the fraction. The solution of equations (1) and (2) gives 5 for x and 21 for y. Therefore, the required fraction is 8. A certain fraction becomes equal to if 3 is added to its numerator and 1 to its denominator, and equal to if 3 is subtracted from its numerator and from its denominator. Find the fraction. 9. A certain fraction becomes equal to if 1 is added Find the to double its numerator, and equal to if 3 is subtracted from its numerator and from its denominator. fraction. 10. There are two fractions with numerators 11 and 5, respectively, whose sum is 13; but if their denominators were interchanged their sum would be 23. 12 11. A certain fraction becomes equal to when 7 is added to its denominator, and equal to 2 when 13 is added to its numerator. Find the fraction. 12. A certain fraction becomes equal to when the denominator is increased by 4, and equal to 2 when the numerator is diminished by 15. Find the fraction. 13. A certain fraction becomes equal to if 7 is added to the numerator, and equal to if 7 is subtracted from the denominator. Find the fraction. 14. A certain number is expressed by three digits. The sum of the digits is 21. The sum of the first and last digits is twice the middle digit. If the hundreds' and tens' digits are interchanged, the number is diminished by 90. Find the number. Solving these equations, x = 8, y = 7, z = 6. 15. The sum of the two digits of a number is 9, and if 27 is subtracted from the number, the digits will be reversed. Find the number. 16. The sum of the two digits of a number is 9, and if the number is divided by the sum of the digits, the quotient is 5. Find the number. 17. A certain number is expressed by two digits. The sum of the digits is 11. If the digits are reversed, the new number exceeds the given number by 27. Find the number. 18. A certain number is expressed by three digits, the units' digit being zero. If the hundreds' and tens' digits are interchanged, the number is diminished by 180. If the hundreds' digit is halved, and the tens' and units' digits are interchanged, the number is diminished by 336. Find the number. 19. A number is expressed by three digits. If the digits are reversed, the new number exceeds the given number by 99. If the number is divided by nine times the sum of its digits, the quotient is 3. The sum of the hundreds' and units' digits exceeds the tens' digit by 1. Find the number. |