178. The Comprehensive Rule.

Compare the unit to which we change with the given unit; multiply the given number of units by the reciprocal of the ratio thus found

179. The Rule Illustrated.

1. Example of reduction descending: Change 3 bu. to quarts.

EXPLANATION:

3 bu. 4 times 3 pks. =12 pks. 8 times 12 qts. 96 qts.

FORMULA FOR EXPLANATION: "I am to change bushels to pecks; a peck is of a bushel, hence there must be 4 times as many pecks as bushels, or 4 times 3 pks. I am to change pecks to quarts; a quart is of a peck, hence there must be 8 times as many quarts as pecks, or 8 times 12 qts."

2. Example of reduction ascending: Change 571 in. to higher denominations.

"I am to change as

MODIFIED FORMULA FOR EXPLANATION: inches to feet; a foot is 12 times an inch, hence there are many feet as inches, or of 571 ft., etc. I am to change feet to yards; a yard is 3 times a foot, hence there are as many yards as feet, or of 47 yds., etc. I am to change yards to rods; a rod is of a yard, hence there are as many rods as yards, or of 15 rds., etc."

3. Change 4 yds. 2 ft. 54 in. to the fraction of a rod.

SOLUTION:

54 in.=4 in.=¿1⁄2 of ¥ ft.=7 ft.

2 ft. ft. of 8 yd.=1 yd.
418 yds.=17 yds. of 1 rds.=ļ rd.

Hence, 4 yds. 2 ft. 51 in.=} rd.

5. 2 pks. 3 qts. what decimal of a bushel?

.282 T. 20 times .282 cwt.=5,640 cwt.

.64 cwt. 4 times .64 qr.=2.56 qrs.

.56 qr.=25 times .56 lb. =14 lbs.

Hence, .282 T.-5 cwt. 2 qrs. 14 lbs.

These examples will show that the rule and the formula of explanation will apply to any problem in reduction. Pupils of some maturity will have no trouble in mastering and understanding the process; but this method should not be given to pupils in the elementary stage.

Note that, in the solutions given above, all mixed numbers are changed to improper fractions, and that common factors in multipliers and divisors are canceled whenever it is possible.

EXERCISES.

180. Solve all these problems by the general rule:

1. Change 34 cd. to cubic feet.

2. How many bushels, etc., in 37,184 pts.?

3. Reduce of a pound Troy to lower denominations.

4. Change .1875 of a day to lower denominations.

5. 5 days 3 hrs. is what part of a week?

6. Express

of a chain in lower denominations.

7. Express .625 gal. in lower denominations.

8. Express 8s. 10d. as the decimal of a £.

9. Find the value of of a degree.

10. Change 12 cu. ft. 486 cu. in. to the fraction of a cubic yard.

11. What decimal of a bushel are 1 pk. 5 qts. 1 pt.?

12. Find the value, in lower denominations, of 3, Ꭲ.

13. Express 1 qt. 1 pt. 1 gi. as the fraction of a gallon.

14. Express of a mile in rods, feet, and inches.

15. Change 25" to the decimal of a degree.

16. How many acres in 1,611,720 sq. ft.?

REVIEW.

ADDITION.

1. Define number. Simple number.

2. Define unit. Primary unit. Relative unit.

3.

zero?

What is the use of the decimal point? Of the

When it

4. What is the effect upon the value of a figure when it is moved one place toward the left? is moved one place toward the right?

5. What is a decimal fraction? What kind of numbers are decimal fractions? What kind of units do decimal fractions express?

6. Define addition. Sum. How are numbers written for adding? Why? Where begin to add? Why?

EXERCISES.

182. 1. Find the sum of $347, $516, and $789.

2. Find the sum of $.347, $.516, and $.789.

How shall the numbers in the second problem be written for adding? Why? Where begin to add? Why? How does addition of decimals differ from addition of integral numbers?

Addition of decimals is the same as addition of integers. The only new consideration is the decimal point.

3. A boy earned $.625 on Monday, $.75 on Tuesday, $.6 on Wednesday, $.875 on Thursday, $.85 on Friday, and $.7 on Saturday. How many dollars did he earn during the week?

4. A man sold three loads of hay. For the first, he received $18.75; for the second, $13.90; and for the third, $16.45. How many dollars did he receive for the

three loads?

5. What is the weight of five bars of silver weighing, respectively, .85 lb., .765 lb., .9 lb., .875 lb., and .78 lb.?

6. What were the annual expenses of a family that paid $236.35 for clothing, $396.78 for provisions, $200 for rent, $35.50 for the services of a physician, $63.875 for fuel, $23.88 for books, and $115.625 for all other expenses?

7. A vessel sailed, in six successive days, the following distances: 236.26 mi., 249.13 mi., 215.9 mi., 198.86 mi., 244 mi., 238.165 mi. How many miles did it sail in the six days?

8. How many cords of wood in four piles which contain, respectively, 5.437 cd., 3.8 cd., 4.75 cd., and 6.095 cd.?

9. How many rods of fence will inclose a field the four sides of which are, respectively, 44.25 rds., 32.865 rds., 42.9 rds., and 33.125 rds.?

10. A man bought three Jersey cows. For the first he paid $30.75; for the second, $60.35; and for the third, $48.65. How many dollars did he pay for the

three?