These exercises, well understood and applied, will serve as a key to unlock the mysteries of Arithmetic. The plan of attempting to teach Arithmetic without a familiar acquaintance with these primary principles has created a distaste for the study of one of the most useful branches of science.

The previous examples and exercises in Mental Arithmetic are intended to prepare the mind of the pupil for a more thorough acquaintance with the principles of WRITTEN ARITHMETIC than he would otherwise obtain.

The manner of reading equations has already been referred to in Article XIV., page 72. This practice should be insisted on by the Teacher, so that the learner may become familiar with the order of analyzing numbers. The following examples will serve to illustrate the process.

115. If a stone 3 feet long, 2 feet wide, 12 of a foot thick, cost of a dollar, what will be the cost of a stone of a foot long, of a foot wide, and of a foot thick? Here the quantities are all direct, and we have for the Given quantity, 3 feet to 6* of a foot long, 21 of a foot wide, and 12% of a foot thick.

= to

Equivalent of the given quantity, of a dollar. Required quantity, of a foot long, of a foot wide, and of a foot thick.

We will now make the given quantity (,, ), and its equivalent (4), form the first, or GIVEN EQUATION, by placing the former on the left of (=) the sign of equality, and the latter on the right, marking the terms of each, for the sake of distinguishing them, and proceed as in Article XIV. Thus:

* 3 is changed to halves here, for the sake of making all the terms fractions, instead of we might have used (3), with equal propriety.

We will now read the equation by mentioning the denomination of each particular term in the left hand members, beginning with the given equation.

Thus: If of a foot long, of a foot wide, and of a foot thick, equal of a dollar, then 1 foot long, 1 foot wide, and 1 foot thick will equal of of of of a dollar; hence, of a foot long, of a foot wide, and of a foot thick, will equal of of of of of 7 of 9 of a dollar, equal to of a dollar.

ANSWER.

116. If 3 men can do a piece of work in 4 days, by working 12 hours a day, how many hours per day must 8 men labor to do the same work, in 9 days? Thus:

Given quantity, 3 men, 4 days, inverse.

Equivalent, 12 hours.

Required quantity, 8 men, 9 days, inverse. Hence,

If 3 men for 4 days equal 12 hours a day, then, inversely, 1 man for 1 day, will equal 3 times 4 times 12 hours a day; hence, 8 men for 9 days, will, inversely, equal of of 12 hours a day, equal to 12 of an hour, equal to 2 hours per day.

EXAMPLES FOR ILLUSTRATION.

1. What will be the cost of 7 reams of paper, at 4 dollars a ream? Here 1 ream is given, equal (in cost,) to 4 dollars, and the cost of 7 reams is required. Hence, the given quantity is 1 ream, and its equivalent, 4 dollars, and the required quantity is 7 reams. We, therefore, have for the simple, or

In the preceding examples the given quantity is a unit, (1 ream;) hence, in this and all similar examples, the given and simple equations become one, and the solution requires but two equations.

1. If 7 reams of paper cost 28 dollars, what will 9 reams cost? Here the given quantity is 7 reams, its equivalent value 28 dollars, and the required quantity is 9 reams; hence, we have for the

Given equation,

Reams. Dols,

Simple equation, 1=7

Dols,

9 9.28 9.4 36 Dols.

Required equation, 1-17-11-1

= 36. Answer.

1.

2. If 1 pound of butter cost of a dollar, what will 6 pounds cost?

Given quantity is 1 pound, and its

Here the Equivalent value of a dollar.
Required quantity is 6 pounds.

hence,

Pounds. Dols.
1 2

Given equation, 1=3

6 6.2

2.2 4 Dols.

Required equa'n. 1=1-3-1-1-1-4. Answer.

3. If of a pound of butter cost of a dollar, what will 6 pounds cost?

Here the

Given quantity is of a pound, and its
Equivalent value of a dollar.

Required quantity is 6 pounds. Hence,

4. If of a yard of cloth cost 5 dollars, what will of

a yard cost?

Given quantity is of a yard;

Thus: Equivalent of the given quantity is 5

dollars;

Required quantity is of a yard. Hence, we have

5. If 1 yards of cloth cost 1 dollars, what will of a yard cost?

Given quantity, 13

of a yard.

10

Thus: Equivalent (price,) 1=1o of a dollar.

6. If of a pound of Indigo cost 13 dollars, what will 7

pounds cost?

Given quantity, of a pound.

Thus: Equivalent (price,) 13-1 dollars:

Required quantity, 7 pounds. Hence,

Pound. Dols.
7 7

Given equation, 8 4

7 7.8.7 7.2 14 Dols.

Required equation, 1-1..4-11-1

14. Answer.

7. If 11 yards of ribbon cost of a dollar, what will 52 yards cost?

Given quantity, 11= of a yard.

Thus: Equivalent (cost), of a dollar.

Required quantity, 53=23 of a yard. Hence,

8. If 7 men can build 36 rods of wall in 28 days, how many rods will 84 men build in 14 days?

Thus:

Given quantity is 7 men and 28 days, i. e. 7 × 28.
Equivalent is 36 rods (result of their labor.)
Required quantity is 84 men and 14 days, i. e.,
84 x 14. Hence,

9. If a piece of cloth of a yard long, and wide, cost 4 dollars, what will be the cost of a same kind of cloth which is of a yard long, and wide?

Given quantity is of a yard long, and

wide, i. e.,

3.

14

Thus Equivalent (price,) 43, 1 dollars.

of a yard piece of the

of a yard

of a yard

Required quantity is of a yard long, and of a yard wide, i. e., . Hence,