REDUCTION OF DECIMAL FRACTIONS.

167. To reduce a compound number to the decimal of a higher unit.

OPERATION.

12) 6.0 pence.

Thus, reduce 14s. 6d. to the decimal of a £. Since 1 penny is of a shilling, 6 pence is f of a shilling, which reduced to a decimal (Art. 97), is .58. The given number then becomes 14.5 shillings. Dividing this by 20 will bring it to pounds. Hence .725 £ is the decimal required.

We may therefore deduce the following

RULE.

.5 shillings. 20) 14.5 shillings.

Ans. .725 £.

Divide the lowest number corresponding to the lowest unit expressed, by as many as make one of the next higher units. Place the quotient as the decimal part of the number corresponding to the next higher unit. Divide again by as many as make 1 of the next higher unit still, and continue in this way until you have reached the unit to which the reduction has to be made; the last quotient will be the an

swer.

Q. How do you reduce a compound number to a decimal of a higher unit? By what would you divide pence to bring them to the decimal of a shilling? The decimal of a pound? By what would you divide shillings to bring them to the decimal of a pound?

EXAMPLES.

1. Reduce 9 pence to the decimal of a £.

Ans. £.0375.

2. Reduce 1 pwt. to the decimal of a pound troy.

Ans. .004166+ lb.

3. Reduce 7 drachms to the decimal of a pound avoirdu

pois.

Ans. .02734375 lb.

Ans. £.10833+.

4. Reduce 26 pence to the decimal of a £.

5. Reduce .056 poles to the decimal of an acre.

Ans. .00035 A.

6. Reduce 17s. 9 d. to the decimal of a pound.

7. Reduce £19 17s. 3 d.

Ans. £.890625.

Ans. £19.863+.

8. Reduce 15s. 6d. to the decimal of a £.

Ans. £.775.

9. Reduce 3 R. 35 P. to the decimal of an acre.

Ans.

10. Reduce 23 hrs. 49 m. to the decimal of a day.

Ans.

A.

d.

11. Reduce 5 oz. 12 pwts. 16 grs. to the decimal of a lb.

Ans. .46944+ lb.

12. Reduce 4° 37' 21" to the decimal of a degree.

Ans.

168. To reduce decimals of a higher unit to whole numbers of a lower unit.

Thus, reduce £.275 to its proper value in whole num

bers.

Multiply the given decimal by as many of the next lower units as make 1 of the given units, and point off from the right of the product as many decimals as there are in the multiplicand. Multiply the decimal part of this product by as many of the next lower units still as make 1 of the units corresponding to the new multiplicand, and cut off as before.

G

Continue this operation through all the different units. The whole numbers in each separate product connected together will be the answer.

Q. How may a decimal of a given unit be reduced to whole numbers of a lower unit? How may .5 of a shilling be brought to pence? How many pence will it be equal to? 2 of a brought to shillings? .5 of a $to cents? 5 of a ct. brought to mills? What is the value of .75 of a $? Of .4 of a $?

EXAMPLES.

1. What is the value of .333 dollars? 2. What is the value of £.775?

3. What is the value of £.4765?

Ans. $3.33 cts.
Ans. 15s. 6d.
Ans. 9s. 6d. +

4. What is the value of .3376 lb. avoirdupois ?

5. What is the value of .104376 bush.?

6. What is the value of .946 yds.?

Ans. 5 oz. 6 dr.+

7. What is the value of .04967 A.?
8. What is the value of .0046987 miles?
9. What is the value of .00946 cwt.?

PROPORTION.

169. Two quantities of the same kind may be compared together in two ways:

1st. By considering how much one exceeds the other, and their relation will be shown by their difference.

Thus, by comparing the numbers 7 and 5 with respect to their difference, we find that 7 exceeds 5 by 2.

2dly. By considering how many times one contains the other, and their relation will be shown by their quotient.

Thus, by comparing 12 and 3 to ascertain how many times 12 contains 3, the quotient 12-4 shows this relation.

Q. In how many ways may quantities of the same kind be compared? What is the first method? What shows the relation between the two quantities in this case? Give an example. What is the second method? What shows the relation between the two quantities in this case? Give an example.

170. RATIO is the relation which one quantity bears to another of the same kind.

When the quantities are compared by considering how much one exceeds the other, the ratio, which is expressed by their difference, is called their arithmetical ratio, or simply their difference.

Thus, the arithmetical ratio of 7 to 5 is 7-5=2; of 10 to 8 is 10-8-2.

When the quantities are compared by considering how many times one contains the other, the ratio is called their geometrical ratio, and is expressed by dividing the first by the second.

Thus, the geometrical ratio of 12 to 3 is 12 or 4; that of 3 to 12 is. A geometrical ratio may therefore be entire or fractional.

Q. What is meant by ratio? What is an arithmetical ratio? What is the arithmetical ratio of 7 to 5? 5 to 3? 3 to 1? What is a geometrical ratio? What is it expressed by? What is the geometrical ratio of 6 to 3 Of 8 to 4? Of 10 to 5? Of 3 to 4? Of 4 to 3? Is a geometrical ratio ever fractional? Is it in the ratio of 4 to 2? Of 2 to 4?

171. The numbers which enter into a ratio are called its terms. Every ratio then has two terms. The first term of a ratio is called the antecedent; the one with which it is compared the consequent.

In the arithmetical ratio 5-2, 5 is the antecedent, 2 the consequent.

In the geometrical ratio §, 5 is the antecedent, 4 the consequent.

Q. What are the numbers called which enter into a ratio? Which is the antecedent? Which the consequent? In the ratio 5-2, which is the antecedent? Which the consequent? In the ratio §, which is the antecedent? Which the consequent ?

172. An arithmetical ratio is not changed in value when its terms are increased or diminished by the same number, because their difference, which expresses this ratio, remains the same.

Thus, the arithmetical ratio of 12 to 5 is the same as that of 14 to 7, and of 10 to 3, although in the first case the

terms were increased by the number 2, and in the second diminished by the same number.

Q. Is an arithmetical ratio changed by increasing or diminishing equally its terms? Why not? What is the arithmetical ratio of 12 to 5? Of 14 to 7? Of 10 to 3?

173. A geometrical ratio is not changed in value by multiplying or dividing its terms by the same number. For this ratio is expressed by placing the first term over the second, as a fraction, and a fraction is not changed in value "by multiplying or dividing its terms by the same number (Art. 69 and 70).

Thus, the geometrical ratio of 4 to 2 is the same as that of 8 to 4, or of 2 to 1, since 2, &=2, 2=2.

Q. Is a geometrical ratio changed by the same number? Why not? is the geometrical ratio of 4 to 2? these ratios expressed?

by multiplying or dividing its terms How is this ratio expressed? What Of 8 to 4? Of 2 to 1? How are

174. Four numbers are said to be in arithmetical proportion, when the difference between the first and second is equal to the difference between the third and fourth. This is also called a proportion by differences.

Thus, the four numbers 4, 2, 10, 8, are in arithmetical proportion, since 4-2-10-8.

Four numbers are said to be in geometrical proportion, when the quotient arising from the division of the first by the second is equal to that arising fron the division of the third by the fourth. This is also called a proportion by quotients. Thus, the four numbers, 4, 2, 16 and 8, are in geometrical proportion, since.

PROPORTION, therefore, is an equality of two ratios.

Q. When are four numbers in arithmetical proportion? What is this also called? Give an example of four numbers in arithmetical proportion. When are four numbers in geometrical proportion? What is this also called? Give an example. What is meant by proportion?

175. There are four terms in every proportion. The first and fourth terms are called the extremes; the second and third the means.