REVIEW AND TEST QUESTIONS.

665. 1. Define Simple, Compound, and Annual Interest. 2. Illustrate by an example every step in the six per cent method.

3. Show that 12% may be used as conveniently as 6%, and write a rule for finding the interest for months by this method.

4. Explain the method of finding the exact interest of any sum for any given time. Give reasons for each step in the process.

5. Show by an example the difference between true and bank discount. Give reasons for your answer.

6. Explain the method of finding the present worth.

7. Explain how the face of a note is found when the proceeds are given. Illustrate each step in the process.

8. Define Exchange, and state the difference between Domestic and Foreign Exchange.

9. State the difference in the three bills in a Set of Exchange.

10. What is meant by Par of Exchange?

11. State the various methods of Domestic Exchange, and illustrate each by an example.

12. Illustrate the method of finding the cost of a draft when exchange is at a discount and brokerage allowed. Give reasons for each step.

13. State the methods of Foreign Exchange.

14. Illustrate by an example the difference between Direct and Indirect exchange.

15. Define Equation of Payments, an Account, Equated Time, and Term of Credit.

16. Illustrate the Interest Method of finding the Equated Time when there are but debit items.

17. State when and why you count forward from the assumed date of settlement to find the equated time.

RATIO

PREPARATORY PROPOSITIONS.

666. Two numbers are compared and their relation determined by dividing the first by the second.

For example, the relation of $8 to $4 is determined thus, $8÷$4 = 2. Observe, the quotient 2 indicates that for every one dollar in the $4, there are two dollars in the $8.

Be particular to observe the following:

1. When the greater of two numbers is compared with the less, the relation of the numbers is expressed either by the relation of an integer or of a mixed number to the unit 1, that is, by an improper fraction whose denominator is 1.

Thus, 20 compared with 4 gives 20 ÷ 4 = 5; that is, for every 1 in the 4 there are 5 in the 20. Hence the relation of 20 to 4 is that of the integer 5 to the unit 1, expressed fractionally thus, §.

Again, 29 compared with 4 gives 29÷4=71; that is, for every 1 in the 4 there are 7 in 29. Hence, the relation of 29 to 4 is that of the mixed number 74 to the unit 1.

2. When the less of two numbers is compared with the greater, the relation is expressed by a proper fraction.

Thus, 6 compared with 14 gives 6 ÷ 14 = √& = ‡ (255); that is, for every 3 in the 6 there is a 7 in the 14. Hence, the relation of 6 to 14 is that of 3 to 7, expressed fractionally thus, .

Observe, that the relation in this case may be expressed, if desired,

as that of the unit 1 to a mixed number. Thus, 6÷ 14 =

(255); that is, the relation of 6 to 14 is that of the unit 1 to 21.

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668. PROP. II.-No numbers can be compared but those which are of the same denomination.

Thus we can compare $8 with $2, and 7 inches with 2 inches, but we cannot compare $8 with 2 inches (155—I).

Observe carefully the following:

1. Denominate numbers must be reduced to the lowest denomination named, before they can be compared.

=

: 15 in.,

For example, to compare 1 yd. 2 ft. with 1 ft. 3 in., both numbers must be reduced to inches. Thus, 1 yd. 2 ft. 60 in., 1 ft. 3 in. and 60 in. ÷ 15 in. = 4; hence, 1 yd. 2 ft. are 4 times 1 ft. 3 in.

2. Fractions must be reduced to the same fractional denomination before they can be compared.

For example, to compare 3 lb. with oz. we must first reduce the 3 lb. to oz., then reduce both members to the same fractional unit. Thus, (1) 31 lb. = 56 oz.; (2) 56 oz. = 280 oz.; (3) 280 oz. ÷ oz. 280 70 (290); hence, the relation 3 lb. to oz. is that

DEFINITIONS.

670. A Ratio is a fraction which expresses the relation which the first of two numbers of the same denomination has to the second.

Thus the relation of $6 to $15 is expressed by ; that is, $6 is % of $15, or for every $2 in $6 there are $5 in $15. In like manner the relation of $12 to $10 is expressed by §.

671. The Special Sign of Ratio is a colon (:).

Thus 4:7 denotes that 4 and 7 express the ratio ; hence, 4:7 and are two ways of expressing the same thing. The fractional form being the more convenient, should be used in preference to the form with the colon.

672. The Terms of a Ratio are the numerator and denominator of the fraction which expresses the relation between the quantities compared.

The first term or numerator is called the Antecedent, the second term or denominator is called the Consequent.

673. A Simple Ratio is a ratio in which each term is a single integer. Thus 9:3, or, is a simple ratio.

674. A Compound Ratio is a ratio whose terms are formed by multiplying together the corresponding terms of two or more simple ratios.

Thus, multiplying together the corresponding terms of the simple ratios 7:3 and 5:2, we have the compound ratio 5 × 7:3 × 2 =

35: 6,

Observe, that when the multiplication of the corresponding terms is performed, the compound ratio is reduced to a simple ratio.

675. The Reciprocal of a number is 1 divided by that number. Thus, the reciprocal of 8 is 1÷8

}.

676. The Reciprocal of a Ratio is 1 divided by the ratio.

Thus, the ratio of 7 to 4 is 7:4 or, and its reciprocal is 1÷ 7 = 4, according to (291). Hence the reciprocal of a ratio is the ratio inverted, or the consequent divided by the antecedent.

677. A Ratio is in its Simplest Terms when the antecedent and consequent are prime to each other.

678. The Reduction of a Ratio is the process of changing its terms without changing the relation they express. Thus 1, 4, †, each express the same relation.

PROBLEMS ON RATIO.

679. Since every ratio is either a proper or improper fraction, the principles of reduction discussed in (235) apply to the reduction of ratios. The wording of the principles must be slightly modified thus:

PRIN. I.-The terms of a ratio must each represent units of the same kind.

PRIN. II.-Multiplying both terms of a ratio by the same number does not change the value of the ratio.

PRIN. III.-Dividing both terms of a ratio by the same number does not change the value of the ratio.

For the illustration of these principles refer to (235).

680. PROB. I.—To find the ratio between two given numbers.

Ex. 1. Find the ratio of $56 to $84.

SOLUTION.-Since, according to (666), two numbers are compared by dividing the first by the second, we divide $56 by $84, giving $56 ÷ $84 = 5; that is, $56 is 5 of $84. Hence the ratio of $56 to $84 is 4.

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