CHAPTER XIX.

PROBLEMS.

194. We now give examples of problems in which the relations between the known and unknown quantities are expressed algebraically by means of quadratic equations.

In the solution of problems it often happens that by solving the equations which are the algebraical statements of the relations between the magnitudes of the known and unknown quantities, we obtain results which do not all satisfy the conditions of the problem.

The reason of this is that the roots of the equation are the numbers, whether positive or negative, integral or fractional, which satisfy it; but in the problem itself there may be restrictions, expressed or implied, on the numbers, and these restrictions cannot be retained in the equation. For example, in a problem which refers to a number of men, it is clear that this number must be integral, but this condition cannot be expressed in the equations.

Thus there are three steps in the solution of a problem. We first find the equations which are the algebraical expressions of the relations between the magnitudes of the known and unknown quantities; we then find the values of the unknown quantities which satisfy these equations; and finally we examine whether any or all of the values we have found violate any conditions

which are expressed or implied in the problem, but which are not contained in the equations.

The following are examples of problems which lead to quadratic equations.

Ex. 1. How many children are there in a family, when eleven times the number is greater by five than twice the square of the number?

Let x be the number of children; then we have

Thus there are 5 children, the value being inadmissible.

Ex. 2. Eleven times the number of rod is greater by five than twice the yards.

How long is the rod ?

yards in the length of a square of the number of

This leads to the same equation as before, only in this case we cannot reject the fractional result. Thus the rod may be five yards long, or it may be half a yard long.

Ex. 3. A number of two digits is equal to twice the product of the digits, and the digit in the ten's place is less by 3 than the digit in the unit's place. What is the number?

Let x be the digit in the ten's place; then x + 3 will be the digit in the unit's place. The number is therefore equal to

Now the digits of a number must be positive integers; hence the second value is inadmissible.

Therefore the digits are 3 and 6, and the required number is 36.

Ex. 4. The square of the number of dollars a man possesses is greater by 1000 than thirty times the number.

the man worth?

How much is

Let x be the number of dollars the man is worth; then, by the conditions of the problem

Both of these values are admissible, provided a debt is considered as a negative possession.

Hence the man may have $50, or he may owe $20.

Ex. 5. The sum of a certain number and its square root is 42: what is the number?

Square both sides; then, after transposition, we have

x285x+1764 = 0,

the roots of which are 36 and 49.

The value 49 will not however satisfy the condition of the problem, if by a square root of a number is meant only the arithmetical square root.

Ex. 6. The sum of the ages of a father and his son is 100 years; also one-tenth of the product of their ages, in years, exceeds the father's age by 180. How old are they?

Let the father be x years old; then the son will be 100 — x years old. Hence by the conditions of the problem,

that is

Hence

x (100x)=x+180;

.. x2 - 90 x + 1800 = 0,

(x-60)(x — 30) = 0.

x=60, or x = 30.

The second value is inadmissible, although it is a positive integer, for it would make the son older than his father. Hence the father must be 60, and the son 40 years old.

EXAMPLES LX.

1. Find two numbers one of which is three times the other and whose product is 243.

2. Find two numbers whose sum is 18 and whose product is 77. 3. Find two numbers whose difference is 20, and the sum of whose squares is 650.

4. Divide 25 into two parts whose product is 156.

5. Divide 80 into two parts the sum of the squares of which is 3208.

6. A certain number is subtracted from 36, and the same number is also subtracted from 30; and the product of the remainders is 891. What is the number?

7. A rectangular court is ten yards longer than it is broad; its area is 1131 square yards. What is its length and breadth ?

8. The product of the sum and difference of a number and its reciprocal is 34: find the number.

9. The number of tennis balls which can be bought for a pound is equal to the number of shillings in the cost of 125 of them. How many can be bought for a pound?

10. The number of eggs which can be bought for 25 cents is equal to twice the number of cents which 8 eggs cost. How many eggs can be bought for 25 cents?

11. A cask contains a certain number of gallons of water, and another cask contains half as many gallons of wine; six gallons are drawn from each, and what is drawn from the one cask is put into the other. If the mixture in each cask be now of the same strength, find the amount of water and wine.

12. The cost of an entertainment was $20, which was to have been divided equally among the party, but four of them leave without paying, and the rest have each to pay 25 cents extra in consequence. Of how many persons did the party consist?

13. A man buys a certain number of articles for $5, and makes $3.82 by selling all but two at 4 cents each more than they cost. How many did he buy ?

14. A man bought a certain number of railway-shares for $9375; he sold all but 15 of them for $10,450, gaining $20 per share on their cost price; how many shares did he buy?

15. A crew can row a certain course up stream in 84 minutes, and if there were no stream they could row it in 7 minutes less than it takes them to drift down the stream; how long would they take to row down with the stream?

16. A boat's crew can row 8 miles an hour in still water. What is the speed of a river's current if it take them 2 hours and 40 minutes to row 8 miles up and 8 miles down?

17. Two trains run without stopping over the same 36 miles of rail. One of them travels 15 miles an hour faster than the other, and accomplishes the distance in 12 minutes less. Find the speed of the two trains.

18. A person having 7 miles to walk increases his speed one mile an hour after the first mile, and is half an hour less on the road than he would have been had he not altered his rate. How long did he take?

19. A and B together can do a piece of work in a certain time. If they each did one half of the work separately, A would have to work one day less, and B two days more than before. Find the time in which A and B together do the work.