LV. MISCELLANEOUS PROBLEMS.

We have already given in previous Chapters collections of problems which lead to simple or quadratic equations; we add here a few examples of somewhat greater difficulty with their solutions.

1. Each of three cubical vessels A, B, C, whose capacities are as 18 27 respectively, is partially filled with water, the quantities of water in them being as 1: 2: 3 respectively. So much water is now poured from A into B and so much from B into C as to make the depth of water the same in each vessel. After this 1284 cubic feet of water is poured from C into B, and then so much from B into A as to leave the depth of water in A twice as great as the depth of water in B. The quantity of water in A is now less by 100 cubic feet than it was originally. much water did each of the vessels originally contain?

Let x= number of cubic feet in A originally;

How

Now when the depth of the fluid is the same in all, it is clear that the quantities vary as the areas of the bases of the vessels, that is, are as 1 : 4 : 9.

.. (since 6x is the total quantity) the quantity in A=

6x
9+4+1
27x

7

7'

Again, when the depth in A is twice that in B, A contains half

as much as B.

Now A contains x-100; .. B contains 2 (x-100), and C

the quantities in A, B, C at first were

500, 1000, 1500 cubic feet respectively.

2. Three horses A, B, C start for a race on a course a mile and a half long. When B has gone half a mile, he is three times as far ahead of A as he is of C. The horses now going at uniform speeds till B is within a quarter of a mile of the winning post, C is at that time as much behind A as A is behind B, but of what it was after B

the distance between A and B is only

1 th 11

had gone the first half mile. C now increases his pace by

what it was before, and passes B 176 yards from the winning post, the respective speeds of A and B remaining unaltered. What was the distance between A and C at the end of the race?

Let

11x=distance (in yards) between B and C at end of first mile, B and A

33x

=

Hence, after C increases his pace, the speeds of A, B, C will be

54 53

proportional to 1320 + 30x, 1320, and (1320+5x) respectively.

Now since C passes B when he is 176 yards from the post;

.. while B was going 440 - 176 = 264 yards,

also it will be found that C's increased pace is equal to A's; therefore there will be the same distance between them at the end of the race as there is when Bismile from winning post, viz. 3x or 3 yards.

3. A fraudulent tradesman contrives to employ his false balance both in buying and selling a certain article, thereby gaining at the rate of 11 per cent. more on his outlay than he would gain were the balance true. If, however, the scale-pans in which the article is weighed when bought and sold respectively, were interchanged, he would neither gain nor lose by the article. Determine the legitimate gain per cent. on the article.

Let w and w, be the apparent weights of the same article when bought and when sold.

Let p prime cost of a unit of weight,

=

Again in the supposed case cost of article=pw, and selling

х

From (1), (1+1)=(1+100);

4. A person buys a quantity of corn, which he intends to sell at a certain price; after he has sold half his stock the price of corn suddenly falls 20 per cent., and by selling the remainder at this reduced price, his gain on the whole is diminished 30 per cent.; if he had sold ths of his stock before the price fell, and the diminution in the price had been in the proportion of £20 on the prime cost of what he before sold for £100, he would have gained by the whole as many shillings as he had bushels of corn at first. Find what the corn cost him per bushel, and what he hoped to gain per cent.

Let x cost price (in pounds) per bushel,

y= gain per cent. he expected;

y

x (1 + 100)
;) = price per bushel for which he sold half his corn;

.. (1

y

1+ ¿) = price.....

100

.the other half;

9x

10

Now had he sold the whole as he sold the first half, the gain

Now the prime cost of what he at first sold for 100

У 1 500

=

if he were to lose £20 on this, the loss per cent. would be

Now in the supposed case the average selling price of a bushel is

5. A and B having a single horse travel between two milestones, distant an even number of miles, in 23 hours, riding alternately mile and mile, and each leaving the horse tied to a mile-stone until the other comes up. The horse's rate is twice that of B; B rides first, and they come together to the seventh mile-stone. Finding it necessary to increase their speed, each man after this walks half a mile per hour faster than before, and the horse's rate is now twice that of A, and B again rides first.