PROBLEMS GIVING RISE TO SIMULTANEOUS EQUATIONS OF THE FIRST

DEGREE.

1. What fraction is that, to the numerator of which, if 1 be added, its value will be one third, but if 1 be added to its denominator, its value will be one fourth?

Let x denote the numerator, and

2. To find two numbers such that their sum shall be equal

to a and their difference equal to b.

3. A person possessed a capital of 30000 dollars, for which he drew a certain interest per annum; but he owed the sum of 20000 dollars, for which he paid a certain interest. The interest that he received exceeded that which he paid by 800 dollars. Another person possessed 35000 dollars, for which he received interest at the second of the above rates; but he owed 24000 dollars, for which he paid interest at the first of the above rates. The interest that he received exceeded that which he paid by 310 dollars. Required the two rates of interest.

Then, the interest on $30000 at x per cent. for one year will be

The interest on $20000 at y per cent. for one year will be

Hence, from the first condition of the problemn,

In like manner from the second condition of the problem we find

35y 24x=
= 31

Combining equations (1) and (2) we find,

y=5 and x = 6.

(2).

Hence, the first rate is 6 per cent. and the second rate 5 per cent.

Verification.

$30000, placed at 6 per cent., gives $300 × 6 = $1800.

The second condition can be verified in the same manner.

4. There are three ingots formed by mixing together three metals in different proportions.

One pound of the first contains 7 ounces of silver, 3 ounces of copper, and 6 ounces of pewter.

One pound of the second contains 12 ounces of silver, 3 ounces of copper, and 1 ounce of pewter.

One pound of the third contains 4 ounces of silver, 7 ounces of copper, and 5 ounces of pewter.

It is required to form from these three, 1 pound of a fourth ingot which shall contain 8 ounces of silver, 33 ounces of copper, and 4 ounces of pewter.

Let x

y

denote the number of ounces taken from the first.

denote the number of ounces taken from the second. z denote the number of ounces taken from the third. Now, since 1 pound or 16 ounces of the first ingot contains 7

ounces of silver, one ounce will contain

1

of 7 ounces: that

16

But since 1 pound of the new ingot is to contain 8 ounces of

Whence, finding the values of x, y and z, we have x=8, the number of ounces taken from the first.

5. What two numbers are they, whose sum is 33 and whose

difference is 7?

Ans, 20 and 13.

6. Divide the number 75 into two such parts, that three times the greater may exceed seven times the less by 15.

Ans. 54 and 21.

7. In a mixture of wine and cider, of the whole plus 25 gallons was wine, and part minus 5 many gallons were there of each?

gallons, was cider; how

Ans. 85 of wine, and 35 of cider.

8. A bill of £120 was paid in guineas and moidores, and the number of pieces of both sorts that were used was just 100; if the guinea were estimated at 21s., and the moidore at 27s., how many were there of each ?

Ans. 50.

9. Two travelers set out at the same time from London and York, whose distance apart is 150 miles; they travel toward each other; one of them goes 8 miles a day, and the other 7; in what time will they meet? Ans. In 10 days.

10. At a certain election, 375 persons voted for two candidates, and the candidate chosen had a majority of 91; how many voted for each?

Ans. 233 for one, and 142 for the other.

11. A's age is double B's, and B's is triple C's, and the sum of all their ages is 140; what is the age of each?

12. A person bought a chaise, horse, and harness, for £60; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness; what did he give for each?

Ans.

£13 6s. Sd.
£ 6 13s. 4d.
£40

for the horse. for the harness. for the chaise.

13. A person has two horses, and a saddle worth £50; now, if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first • what is the value of each horse?

Ans. One £30, and the other £40.

14. Two persons, A and B, have each the same income. A saves of his yearly; but B, by spending £50 per annum more than A, at the end of 4 years finds himself £100 in debt; what is the income of each? Ans. £125.

15. To divide the number 36 into three such parts, that of the first, of the second, and of the third, may be all equal to each other. Ans.. 8, 12, and 16.

14 A footman agreed to serve his master for £8 a year and a livery, but was turned away at the end of 7 months, and received only £2 13s. 4d. and his livery; what was its value?

Ans. £4 16s.

17. To divide the number 90 into four such parts, that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product, and quotient, so obtained, will be all equal to each other. Ans. The parts are 18, 22, 10, and 40.

18. The hour and minute hands of a clock are exactly together at 12 o'clock; when are they next together?

Ans. 1 h. 5 min.

19. A man and his wife usually drank out a cask of beer in 12 days; but when the man was from home, it lasted the woman 30 days; how many days would the man be in drinking it alone? Ans. 20 days.

20. If A and B together can perform a piece of work in 8 days, A and C together in 9 days, and B and C in 10 days; how many days would it take each person to perform the same work alone? Ans. A 143 days, B 1723, and C 2331

. 21. A laborer can do a certain work expressed by a, in a time expressed by b; a second laborer, the work c in a time d; a third, the work e in a time f. Required the time it would take the three laborers, working together, to perform the work g.