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in $1000 more, Y $1500, and Z takes out $500.

Sept. 1, X takes out $500, Y puts in $1000, and Z $1000. At the end of the year they settle, having gained $ 6880; what is each partner's share of the gain?

25. A, B, and C traded in company. A at first put in $1200, B $1500, and C $ 1600; in 4 months A put in $ 300 more and B $400, and C took out $500; in 8 months from the commencement of business, A withdrew all his stock but $600, B put in as much as he at first put in, and C withdrew $500. At the end of a year they found they had gained 12% on the largest total stock at any one time in trade. How many dollars ought each to take if the firm is dissolved?

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371. Write a note for bank discount, and explain its terms. 372. What is bank discount? 373. What are the avails, or proceeds? 374. What is meant by days of grace? 375. What is the Rule for finding the bank discount, and the avails of a note payable at a specified future time? 377. Give the Rule for finding the sum for which a note must be written that the avails may be a specified sum?

379. What is Equation of Payments? 380. What is the equated time? 382. Give the Rule for finding the equated time, by the interest method, for the payment of sums due at different times? By the product method? 384. Give the Rule for finding the equated time for paying the balance of an account which has both debit and credit entries?

385. What are bonds? 386. What are coupons? 387. Name the different kinds of U. S. bonds, and explain what their names mean.

391. What is Exchange? 392. What is a draft, or bill of exchange? 393. Who is the drawer? the drawee? the payee? 394. Explain the term at sight. 395. Who is the buyer? the holder? 396. What are indorsers? 397. Explain the term accepted. 398. Is there any discount in bills of exchange?

400. What is Partnership? A Company? Partners? Capital? 402. When capital is invested for unequal times, what represents each one's share? 404. How is the gain, or loss, of each partner found?

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RATIO.

406. Oral Exercises.

1. If John is 16 years old and his father 48, how do their ages compare?

2. What part of 75 is 15?

3. I have $35 and James has $5; how many times as many dollars as James have I?

4. Compare 18 with 9; with 6; with 3; with 2; with 1; with 12; with 15.

407. Ratio is the relation of one quantity to another of the same kind; or, it is the quotient obtained by dividing one quantity by another of the same kind.

408. Ratio is usually indicated by two dots; thus, 8: 4 expresses the ratio of 8 to 4.

The two quantities compared are the terms of the ratio; the first term is the antecedent, the second the consequent, and the two terms, collectively, a couplet.

409. The antecedent is a dividend, and the consequent a divisor.

Thus,
and

8:

4 =8÷ 4 =
= 2,
3:12312 & 1.
3
-
-
12

410. An inverse, or reciprocal, ratio is a ratio inverted; that is, the antecedent becomes the consequent and the consequent the antecedent. Thus, the inverse ratio of 4: 9 is 9:4.

411. The antecedent and consequent being a dividend and divisor, it follows that any change in the antecedent causes a like change in the value of the ratio, and any change

in the consequent causes an opposite change in the value of the ratio (Art. 81 and 118). Hence,

1.

Multiplying the antecedent multiplies the ratio; and dividing the antecedent divides the ratio (Art. 79, a and b).

2. Multiplying the consequent divides the ratio; and dividing the consequent multiplies the ratio (Art. 79, c and d).

3. Multiplying both antecedent and consequent by the same number, or dividing both by the same number, does not affect the ratio (Art. 80, a and b).

412. The antecedent, consequent, and ratio are so related to each other, that if two of them are given the other can be found; thus: in 12 : 3 = 4, we have

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413. When there is but one antecedent and one consequent the ratio is said to be simple; thus, 15:5=3, is a simple ratio.

414. When the corresponding terms of two or more simple ratios are multiplied together the resulting ratio is said to be compound; thus, by multiplying together the corresponding terms of the simple ratios,

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we have the compound ratio, 6 x 8 x 10 : 3 × 2 × 2 = 40.

5. What is the ratio of 30 to 6?

6. What is the ratio of 4 to 11?

7. What is the inverse ratio of 20 to 4?

Ans.

8. What is the inverse ratio of 3 to 7?

9. What is the ratio compounded of 9 to 4 and 6 to 5? 10. Which is the greater, the ratio of 8 to 7, or of 17 to 14?

11. Which is the greater, the ratio of 7 to 4, or of 18 to 13? 12. If 63 is the antecedent and 9 the consequent, what is the ratio?

13. If 72 is the antecedent and 8 the ratio, what is the consequent?

14. If 14 is the consequent and 4 the ratio, what is the antecedent?

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15. How does the ratio of 16 to 8 compare with the ratio of 24 to 12?

16. How does the ratio of 8 to 10 compare with 12 to 15?

416. Proportion is an equality of ratios. Thus, 8:6 4:3 is a proportion.

=

The equality of two ratios is indicated by the sign of equality (=), or by four dots (::). Thus,

12:38:2, or 12 : 3 :: 8:2,

reads, 12 to 3 equals 8 to 2, or 12 is to 3 as 8 is to 2.

417. The first and last terms of a proportion are called extremes; the second and third, means.

=

418. If the means are the same number, this number is called a mean proportional. Thus, in 9:6 6:4, 6 is a mean proportional between 9 and 4.

419. In a proportion the product of the extremes is equal to the product of the means.

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Reducing these fractions to a common denominator, we have

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As these fractions are equal, and their denominators the same, their numerators must be equal, or 3 x8 = 6 x 4. But 3 and 8 are the extremes, and 6 and 4 the means.

420. It follows that either extreme is equal to the product of the means divided by the other extreme; and either mean equal to the product of the extremes divided by the other mean. That is, if any three terms of a proportion are given, the remaining term can be found. Hence, the name, Rule of

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25. If 7 pounds of sugar cost 63 cents, what will 11 pounds

cost?

OPERATION.

7: 1163: Ans.

Ans. 99 cents.

The cost evidently will increase as the quantity increases, that is, 11 pounds will cost as many times 63 cents as 11 pounds is times 7 pounds,

or 7 pounds is to 11 pounds as 63 cents is to the required cost.

26. If 8 men can build a certain wall in 25 days, how long will it take 12 men to build the same wall?

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