Original Problems.

Suggestions to Pupils.

1. Take measurements of the school-room for finding the cost of joists, flooring, plastering, etc. A comparison of the measurements taken will suggest corrections of errors.

2. Take the actual measurement of some rectangular room, give the dimensions with such other information as is necessary to find the cost of carpeting or papering it.

3. Give the dimensions of a bin or box to find the quantity of grain, or potatoes, or apples it will contain.

4. Give the dimensions of a lot, and such details of information as may be needed to find the cost of fencing it. An inspection of a fence already constructed and inquiries made of workmen will suggest the points needed.

5. Tell where some sidewalk about your school-house is needed, and ask the members of the class to find what its cost would be. The pupils may determine what kind of walk they would have, and learn what it would cost per square yard or foot.

6. Take the measurements of a load or pile of cord wood; report the same to the class and ask how many cords and what it would cost at prevailing prices.

7. Find how many cubic feet of coal, such as is commonly used in your neighborhood, are estimated to weigh a ton; report the same to the class with the size of some coal bin, and ask how many tons of coal may be put into it.

8. Give the thickness of ice formed at some place near by, and ask how many tons can be taken from any given space. (The weight of 1 cubic foot of ice, at 32° Farenheit, is 57.5 pounds.)

Note. It is not designed that all the pupils shall prepare questions on all the topics suggested, nor that they should be restricted to them alone. No pupil should present a question which he is not ready to answer if required.

CHAPTER XIV.

STANDARD.

Percentage.

Illustrations.-1. A farm hand, who works "on shares," receives from one farmer an offer of 7 bushels of corn out of every 16 bushels raised, another offers him 2 out of 5, another 5 out of 12. Which is the best offer?

Here it is difficult to determine which is the best offer, because the standards of comparison, 16, 5, and 12, are different.

Expressing the shares in the form of common fractions, we have 7/16, 2/5, and 5/12, and reducing these to fractions having a common denominator, we obtain 105/240, 96/240, and 100/240. Here 240 is the common standard of comparison, and the several offers are respectively equivalent to 105, 96, and 100 out of 240 bushels produced, whence we see that the first offer is the best.

But the offer of 7 bushels out of 16, or 7/16 of a crop, is equivalent to the offer of 7/16 of each 100 bu., or at the rate of 43 3/4 bu. out of 100. Comparing all the offers with this standard, we find that they are equivalent to 43 3/4, 40, and 41/3 bushels per hundred, respectively; whence we readily see how the several offers compare with each other.

2. In like manner compare the value of two iron ores, one of which produces 52 tons of metal from 65 tons of ore, and the other 42 tons of metal from 56 tons of ore.

Suggestion. What common fractional part of each ore is metal? How many tons of metal can be produced from 100 tons of each ore?

Note. Because of its simplicity and convenience, 100 has been adopted as a standard of comparison in almost every department of business and by all civilized nations; hence we hear of a boy's spelling a certain per cent. of the words dictated, that is, at the rate of so many in a hundred, and in like manner of the merchant gaining or losing a certain per cent. of the money he lays out for his goods, of an increasing per cent. of children who are near-sighted, etc., etc.

Definitions.

273. Per Cent. is an abbreviation of the phrase per centum, and signifies by the hundred.

Caution. The abbreviation cent. in the phrase per cent. has no reference to the cent of our decimal currency.

274. A Rate per cent. is a rate per hundred.

275. The sign % is annexed to the rate, and stands for the phrase per cent.

Thus, 7% is read seven per cent., .07% is read seven hundredths of one per cent., 13% is read 1/3 of 1 per cent., .001/3 % is read 1/3 of one hundredth of 1 per cent.

276. Any per cent. of a number is equivalent to so many hundredths of it.

Illustration. If 100 marks be made, 4 in line and 25 in column, the following questions may be asked:

What common fractional part, what decimal part, what per cent. of the whole number, are in the top line? etc.

and 22 more lines

One mark is what common fractional part, what decimal part, and what per cent., of 25 marks?

Additional marks being made at the foot of a column, it may be asked: What per cent. is added to the marks of the column? that is, How many additional marks would be made in all if 1 were added to each 25 in the hundred?

What fractional part, and what per cent. of all the letters in the italicized lines below, are contained in the first word, in the first two? etc. In each part of eighty-four? In watchful? etc.

Manuscript, importance, regulation, house-plant, county maps, blind-mouse, eighty-four, be watchful, cash profit, spring-halt.

What per cent. of all the letters in either line are a's, b's? etc. What per cent. of the letters in the word Oconomowoc are o's? What per cent. are c's? etc. What per cent. of the letters in Ohio are vowels? Are consonants? What % of the numbers from 1 to 100 are primes? From 101 to 200? etc.

SLATE EXERCISES.

The pupil who desires to become expert in computations of percentage should be able to convert per cents. into corresponding decimal and common fractions almost at sight.

Write the decimals and the common fractions that are equivalent to the following expressions (all common fractions should be given in lowest terms):

alent to 12 per cent. of it?

Example.-What common fractional part of a number is equiv

Or, since the numerator, 121/2, is an aliquot part of the denominator 100, the reduction can be made directly by dividing both numerator and denominator by 121/2, thus:

In like manner the following rates per cent. are all reducible to simple common fractions.

What decimal and what common fractions are equivalent to

9.12 1/3 2/3

What per cents. are equivalent to the following fractions:

1/4 3/4 14. .04 .40 .41/3 .14

10. 1/5 12/ 2/5

1/6

Applications. Example.-1. I bought a horse for $280 and sold it so as to gain 25%. How much did I gain?

Written Work. 2.80= 1%

25

1400

25 ×

560

Analysis. At 1% (1 to a hundred) I would gain $2.80, and at 25% I would gain 25 times as much. $2.80 $70 Ans.

Or, since 25% = .25 or 1/4 of a number, to obtain 25% of 280 we may take 1/4 of it, 1/4 of $280 = $70. For exercises, see Case I, pages 276–279. Example.-2. If the horse was bought for $280 and sold so as to gain $70, what per cent. was gained?

Analysis. At 1% I would gain $2.80, and to gain $70 the rate of gain would have to be as many times 1% as $70 is times $2.80, which is 25. 25 × 1% = 25% Ans. Or, the gain $70 is 70/280 or 1/4 of the price paid. 1/4 of any number = 25% of it.

For exercises, see Case II, pages 280, 281.

Example.-3. If the horse was sold so as to gain $70 at the rate of 25%, what was the price paid?

$70

$2.8

Analysis. If $70 is 25% of the price, 1% is 1/25 of $2.80, and 100% or the whole price = 100 times $280 Ans.

Or, if 70 is 25% or 1/4 of the cost, the cost must be 4 times $70 = $280.

282.

For exercises, see Case III, page 281; and Case IV, page

277. The three principal cases of percentage are presented in the foregoing examples. They are:

I. To find a required per cent. of a number.
II. To find what per cent. one number is of another.
III. To find a number from a given per cent. of it.

A fourth case is added, which differs from the third in nothing except that the rate per cent. to be operated with is derived from the given rate by adding it to or subtracting it from 100. (See Case IV.)