Exercise 68.

*Subtract from the sum of the three equations each equation separately. † Multiply the equations by a, b, and c, respectively, and from the sum of the results subtract the double of each equation separately.

CHAPTER XII.

PROBLEMS INVOLVING TWO UNKNOWN

NUMBERS.

168. 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 to be found.

169. If there are more equations than unknown numbers, some of them are superfluous or inconsistent; if there are less equations than unknown numbers, the problem is indeterminate.

(1) If A gives B $10, B will have three times as much money as A. If B gives A $10, A will have twice as much. How much has each?

money as B.

Let and

X= number of dollars A has,
y = number of dollars B has.

Then, after A gives B $10,

x10 the number of dollars A has,

=

From the solution of equations (1) and (2), x = 22, and y

Therefore A has $22, and B has $26.

Exercise 69.

1. The sum of two numbers divided by 2 gives as a quo

tient 24, and the difference between them divided

by 2 gives as a quotient 17. What are the numbers?

2. The number 144 is divided into three numbers. When the first is divided by the second, the quotient is 3 and the remainder 2; and when the third is divided by the sum of the other two numbers, the quotient is 2 and the remainder 6. Find the numbers.

3. Three times the greater of two numbers exceeds twice the less by 10; and twice the greater together with three times the less is 24. Find the numbers.

4. 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. What are the numbers?

5. Seven years ago the age of a father was four times that of his son; seven years hence the age of the father will be double that of the son. What are their ages?

6. The sum of the ages of a father and son is half what it will be in 25 years; the difference between their ages is one-third of what the sum will be in 20 years. What are their ages?

7. If B gives A $25, they will have equal sums of money; but if A gives B $22, B's money will be double that of A. How much has each?

8. A farmer sold to one person 30 bushels of wheat and 40 bushels of barley for $67.50; to another person

he sold 50 bushels of wheat and 30 bushels of barley for $85. What was the price of the wheat and of the barley per bushel?

9. If A gives B $5, he will then have $6 less than B; but if he receives $5 from B, three times his money

will be $20 more than four times B's. How much has each?

10. The cost of 12 horses and 14 cows is $1900; the cost of 5 horses and 3 cows is $650. What is the cost of a horse and a cow respectively?

NOTE. A fraction of which the terms are unknown may be rep

Ex. A certain fraction becomes equal to if 3 is added to its numerator, and equal to if 3 is added to its denominator. Determine the fraction.

From the solution of these equations it is found that

11. A certain fraction becomes equal to 2 when 7 is added to its numerator, and equal to 1 when 1 is subtracted from its denominator. Determine the fraction.

12. A certain fraction becomes equal to when 7 is added to its denominator, and equal to 2 when 13 is added. to its numerator. Determine the fraction.