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Equations of the First Degree with one unknown quantity.

24. Find a number such that if you multiply it by 5, subtract 24 from the product, divide the remainder by 6, and add 13 to the quotient, you will obtain this number. Ans. 54.

25. A courier left this place n days ago, and makes a miles daily. He is pursued by another making 6 miles daily. In how many days will the second overtake the first ?

Ans.

na

a

days.

26. A courier started from a certain place 12 days ago, and is pursued by another, whose speed is to that of the first as 8:3. In how many days will the second overtake the first? Ans. 7 days.

27. A courier started from this place n days ago, and is pursued by another whose speed is to that of the first as p is to q. In how many days will the second overtake the first?

Ans.

nq p-q

28. Two bodies move in opposite directions; one moves c feet in a second, the other C feet. The two places, from which they start at the same time, are distant a feet from one another. When will they meet?

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29. Two bodies move in the same direction from two places at a distance of a feet apart; the one at the rate of c feet in a second, the other pursuing it at the rate of C feet in a second. When will they meet?

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30. At 12 o'clock, both hands of a clock are together. When and how often will these hands be together in the next 12 hours?

Equations of the First Degree with one unknown quantity.

Ans. At 5 minutes past 1, at 101 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 circumference 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?

a

Ans. In-c'
p+a 2p+
ccca, &c., seconds.

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

a+ct p+a+ct 2p+a+ct

Ans. In

&c., seconds.

C-CC-C

C-c

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

a-ct p+a-ct 2p+a-ct

Ans. In

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&c., seconds.

C-c

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?

a p+a 2p+a 3p+a Ans. In C+c' C+c' C + c' 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?

Ans. In

a-ct pact 2p+act
C+c' C+c ' C + c

&c., seconds.

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. 37 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

-b)n

a -b

gallons of the wine at b shillings, and 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

m-n

and

ma b

m -n

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.

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

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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 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 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.

Ans.

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?

b-a2 b+a2

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Ans.

and

2 a

2 a

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

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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?

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