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?

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?

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?

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

b. na

ma b

Ans.

and

40. One of my acquaintances is now 30, his elder 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.

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

=

a: b;

=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 would 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 sum. What was the fortune left? and what was the number of children? Ans. The fortune was $8100,

and the number of children 9.

45. Divide the number 10 into two such parts, that the difference of their squares may be 20.

Ans. 6 and 4.

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

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

95. 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.

EXAMPLES.

1. In what cases would the value of the unknown quantity in example 25 of art. 94 become zero? and what would this

value signify?

Ans. When
or when

and, in either case, this value

n = = 0,

α = 0; 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. 94 become zero? and what would this

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.

Examples of unknown quantity equal to Zero.

3. In what cases would the value of one of the unknown quantities in example 38 of art. 94 become zero? and what would this value signify?

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 re-
quired value; and zero must, therefore, be re-
garded as a true solution.

4. In what cases would the value of one of the unknown quantities in example 39 of art. 94 become zero? and what would this value signify? Ans. When

b = na, or = ma;

and these equations indicate that a is itself such
that, multiplied either by m or by n, it gives a pro-
duct= 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. 94 become zero? and what would this value signify?

Ans. First. When a = 0, or b = 0,

and, in this case, zero is a true solution by re-
garding all numbers as having the same ratio to

and, in this case, the problem is impossible, for
the ratio of no two numbers can be equal to
each of the unequal ratios a : b and m : n.

and, in this case, the problem is impossible, for

Cases in which the value of an unknown quantity is infinite.

no two numbers, whose ratio a: b, and which
are therefore unequal, can, by the addition of
c to each of them, become equal to each other,
as required by the ratio m: n = m ; m = 1.

96. When the solution of a problem gives, for the values of one of its unknown quantities, any fractions, the denominators of which are zero, while the numerators are not zero; such values are, generally, to be regarded as indicating an absurdity in the enunciation of the problem.

EXAMPLES.

1. In what case does the denominator of the fractional value of the unknown quantity in example 25 of art. 94 become zero? and what is the corresponding absurdity in the enunciation of the problem?

and the absurdity is that, while the couriers are
travelling at the same rate, it is required to de-
termine the time in which one will overtake the
other.

2. In what case do the denominators of the fractional values of the unknown quantity in example 38 of art. 94 become zero? and what is the corresponding absurdity in the enunciation of the problem?

and the absurdity is that, while both the wines
are of the same value, they should give a mix-
ture of a value different from their common
value.