Equations of the First Degree with one unknown quantity.

Ans. At 5 minutes past 1, at 1010 minutes past 2, at 16 minutes past 3, and so on, in each successive hour, 55 minutes later.

31. Two bodies move after one another in the circum

ference of a circle, which measures p feet. At first they are distant from each other by an arc measuring a feet; the first moves c feet, the second C feet, in a second. When will those two bodies meet for the first time, second time, and so on, supposing that they do not disturb each other's motion ?

α

p+a 2p+a Ans. In CCC-c'

&c., seconds.

32. When will they meet if the first begins to move t seconds sooner than the second?

Ans. In

a+ct p+a+ct 2p+a+ct
C-CC-c

&c., seconds.

C-c

33. But when will they meet, if the first begins to move t seconds later than the second?

34. When will they meet, if the first, instead of running in the same direction with the second, runs in the opposite direction, and starts at the same time?

α p+a 2p+a 3p+a
Ans. In
C+c2 C+c' C+c' C+c'

&c., seconds.

35. When will they meet, if, moving in an opposite direction to the second, the first starts t seconds sooner than the second?

Equations of the First Degree with one unknown quantity.

36. But when will they meet, if, moving in an opposite direction to the second, the first starts t seconds later than the second?

Ans. In

a+ct p+a+ct 2p+a+ct
C+c' C+c C + c

&c., seconds.

37. A wine merchant has two kinds of wine; the one costs 9 shillings per gallon, the other 5. He wishes to mix both wines together, in such quantities, that he may have 50 gallons, and each gallon, without profit or loss, may be sold for 8 shillings. How must he mix them?

Ans. 371 gallons of the wine at 9 shillings, with 12 gallons of that at 5 shillings.

38. A wine merchant has two kinds of wine; the one costs a shillings per gallon, the other b shillings. How must he mix both these wines together, in order to have n gallons, at a price of c shillings per gallon?

Ans.

(a—c)n gallons of the wine at b shillings, and

a

(c-b)n a -b

gallons of that at a shillings.

39. To divide the number a into two such parts, that, if the first is multiplied by m and the second by n, the sum of the products is b.

Ans.

b. -na

ma- b

and

40. One of my acquaintances is now 30, his younger brother 20; and consequently 3:2 is the ratio of his age to his brother's. In how many years will their ages be as 5:4? Ans. In 20 years.

a: b; ?

41. What two numbers are those, whose ratio but, if c is added to both of them the resulting ratio = m : n

Equations of the First Degree with one unknown quantity.

42. Find a number such that 5 times the number is as much above 20, as the number itself is below 20.

Ans. 63.

43. A person wished to buy a house, and in order to raise the requisite capital, he draws the same sum from each of his debtors. He tried, whether, if he obtained $250 from each, it would be sufficient for the purpose; he found, however, that he should then still lack $2000. He tried it, therefore, with $340; but this gave him $880 more than he required. How many debtors had he?

Ans. 32.

44. A father leaves a number of children, and a certain sum, which they are to divide amongst them as follows: The first is to receive $ 100, and then the 10th part of the remainder; after this, the second has $200, and the 10th part of the remainder; again, the third receives $300, and the 10th part of the remainder; and so on, each succeeding child is to receive $100 more than the one preceding, and then the 10th part of that which still remains. But it is found that all the children have received the same What was the fortune left? and what was the number of children?

sum.

Ans. The fortune was $8100, and the number of children 9.

45. Divide the number 10 into two difference of their squares may be 20.

such parts, that the

Ans. 6 and 4.

46. Divide the number a into two such parts, that the difference of their squares may be b.

a2+b a2 b and

2 a

2 a

47. What two numbers are they whose difference is 5,

and the difference of whose squares is 45?

Ans. 7 and 2.

Examples of unknown quantity equal to Zero.

48. What two numbers are they whose difference is a, and the difference of whose squares is b?

127. Corollary. When the solution of a problem gives zero for the value of either of the unknown quantities, this value is sometimes a true solution; and sometimes it indicates an impossibility in the proposed question. In any such case, therefore, it is necessary to return to the data of the problem and investigate the signification of this result.

128. EXAMPLES.

1. In what cases would the value of the unknown quantity in example 25 of art. 126 become. zero? and what would this value signify?

Solution. As the value of the unknown quantity of the example is the fraction, which is its answer; it is zero, when

and, in either case, this value signifies that the couriers are together at the outset; and zero must, therefore, be regarded as a real solution.

2. In what cases would the value of the unknown quantity in example 35 of art. 126 become zero? and what would this value signify?

and either of these equations signifies that the bodies are together when the second body starts, the first body having just arrived at the point of departure of the second, and zero is, therefore, to be regarded as a real solution.

3. In what cases would the value of one of the unknown quantities in example 38 of art. 126 become zero? and what would this value signify?

Ans. When either

a = c, or b = c;

and, in either case, these equations indicate that the price of one of the wines is just that of the required mixture, and, of course, needs none of the other wine added to it to make it of the required value; and zero, must, therefore, be regarded as a true solution.

4. In what cases would the value of one of the unknown quantities in example 39 of art. 126 become zero? and what would this value signify?

Ans. When

b=na, or = ma;

1

and these equations indicate that a is itself such that, multiplied either by m or by n, it gives a product = b; and zero may be regarded as a true solution, expressing that one of the parts is zero, while the other is the number a itself.

5. In what cases would the value of one of the unknown quantities in example 41 of art. 126 become zero? and what would this value signify?

and, in this case, zero is a true solution by regarding all numbers as having the same ratio to zero.