62. Solutions of problems involving two unknowns. The same principle of translation of the problem into algebraic symbols should be followed here as in the solution of problems leading to simple equations (p. 37).

PROBLEMS

1. The difference between two numbers is 33. Their sum is 9. What are the numbers?

2. What are the numbers whose sum is a and whose difference is b?

3. A man bought a pig and a cow for $100. If he had given $10 more for the pig and $20 less for the cow, they would have cost him equal amounts. What did he pay for each ?

4. Two baskets contain apples. There are 51 more in the first basket than in the second. But if there were 3 times as many in the first and 7 times as many in the second, there would be only 5 more in the first than in the second. How many apples are there in each basket?

5. A says to B, "Give me $49 and we shall then have equal amounts.' B replied, "If you give me $49, I shall have 3 times as much as you. How much had each?

6. A man had a silver and a gold watch and two chains, the value of the chains being $9 and $25. The gold watch and the better chain are together twice and a half as valuable as the silver watch and cheaper chain. The gold watch and cheaper chain are worth $2 more than the silver watch and the better chain. What is the value of each watch?

7. What fraction is changed into when both numerator and denominator are diminished by 7, and into its reciprocal when the numerator is increased by 12 and the denominator decreased by 12?

8. A man bought 2 carriage horses and 5 work horses, paying in all $1200. If he had paid $5 more for each work horse, a carriage horse would have been only more expensive than a work horse. How much did each cost?

9. A man's money at interest yields him $540 yearly. 1% more interest, he would have had $60 more income. has he at interest?

If he had received How much money

10. A man has two sums of money at interest, one at 4%, the other at 5%. Together they yield $750. If both yielded 1% more interest, he would have $165 more income. How large are the sums of money?

11. A man has two sums of money at interest, the first at 4%, the second at 31%. The first yields as much in 21 months as the second does in 18 months. If he should receive 1% less from the first and 1% more from the second, he would receive yearly $7 more interest from both sums. What are the sums at interest?

12. What values have a mark and a ruble in our money if 38 rubles are worth 14 cents less than 75 marks, and if a dollar and a ruble together make 6 marks?

13. A chemist has two kinds of acid. He finds that 23 parts of one kind mixed with 47 parts of the other give an acid of 84% strength and that 43 parts of the first with 17 parts of the second give an 805% pure mixture. What per cent pure are the two acids?

14. Two cities are 15 miles apart. If A leaves one city 2 hours earlier than B leaves the other, they meet 2 hours after B starts. Had B started 2 hours earlier, they would have met 3 hours after he started. How many miles per hour do they walk?

15. The crown of Hiero of Syracuse, which was part gold and part silver, weighed 20 pounds, and lost 11⁄2 pounds when weighed in water. How much gold and how much silver did it contain if 19 pounds of gold and 10 pounds of silver each lose one pound in water?

16. Two numbers which are written with the same two digits differ by 36. If we add to the lesser the sum of its tens digit and 4 times its units digit, we obtain 100. What are the numbers?

17. In a company of 14 persons, men and women, the men spend $24 and the women spend an equal amount. If each man spends $1 more than each woman, how many men and how many women are in the company?

63. Solution of linear equations in several variables. This process is performed as follows:

RULE. Eliminate one variable from the equations taken in pairs, thus giving a system of one less equation than at first in one less variable.

Continue the process until the value of one variable is found. The remaining variables may be found by substitution.

Special cases occur, as in the case of two variables, where an infinite number of solutions or no solutions exist. Where no solution exists one is led to a selfcontradictory equation on application of the rule. See exercise 17, p. 48.

HINT. Divide the equations by xy, xz, yz respectively.

CHAPTER V

RATIO AND PROPORTION

64. Ratio. The ratio of one of two numbers to the other is the result of dividing one of them by the other.

The dividend in this implied division is called the antecedent, the divisor is called the consequent.

65. Proportion. Four numbers, a, b, c, d, are in proportion when the ratio of the first pair equals the ratio of the second pair.

The letters a and d are called the extremes, b and c the means, of the proportion.

66. Theorems concerning proportion. If a, b, c, d are in proportion, that is, if

then

a

с

a: b c d or

=

(I)

b

Equation (III) is said to be derived from (I) by inversion. Equation (IV) is said to be derived from (I) by alternation. Equation (V) is said to be derived from (I) by composition. Equation (VI) is said to be derived from (I) by division. Equation (VII) is said to be derived from (I) by composition

and division.