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RULE.-1. Multiply each payment by its term of credit, and divide the sum of the products by the sum of the payments; the quotient is the average term of credit.

2. To find the equated time of payment.-Add the average term of credit to the date at which the several credits begin.

3. On the first day of December, 1880, a man gave 3 notes, the first for $500 payable in 3 mo.; the second for $750, payable in 6 mo.; and the third for $1200, payable in 9 mo. What was the average term of credit, and the equated time of payment?

4. Bought merchandise Jan. 1, 1883, as follows: $350 on 2 mo., $500 on 3 mo., $700 on 6 mo. What is the equated time of payment?

5. A person owes a debt of $1680 due in 8 months, of which he pays in 3 mo., in 5 mo., † in 6 mo., and † in When is the remainder due?

7 mo.

6. Bought a bill of goods, amounting to $1500 on 6 mo. credit. At the end of 2 mo., I paid $300 on account, and 2 mo. afterward, paid $400 on account, giving my note for the balance. For what time was the note drawn?

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7. On a debt of $2500 due in 8 mo. from Feb. 1, the following payments were made: May 1, $250; July 1, $300; and Sept. 1, $500. When is the balance due?

8. Find the average term of credit, and the equated time of payment from Dec. 15, of $225 due in 35 da., $350 due in 60 da., and $750 due in 90 da.

9. Dec. 1, 1884, bought goods to the amount of $1200, on terms as follows: 25% in cash, 30% in 3 mo., 20% in 4 mo., and the balance in 6 mo. Find the equated time of payment, and the cash value of the goods, computing discount at 7%.

357. To find the equated time and the average of terms of credit beginning at different dates.

1. L. C. Hill bought goods of John Beach as follows: June 1, 1882, amounting to $350 on 2 mo. credit; July 15, 1882, $400, on 3 mo.; Aug. 10, $450, on 4 mo. ; and Sept. 12, $600, on o mo. What is the equated time of payment?

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Hence the equated time is 124 da. from Aug. 1, or Dec. 3.

EXPLANATION.-Computing the terms of credit from Aug. 1, the earliest date at which any of the debts become due, we find the terms of credit to be from Aug. 1 to Oct. 15, 75 da. ; to Dec. 10, 131 da., and to March 12, 223 da. The average term of credit is therefore 124 da. from Aug. 1, and the equated time Dec. 3.

PROOF.-Assume as the standard time the latest date, March 12. The operation will then be as follows:

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Hence, the equated time is 99 da. previous to March 12, or Dec. 3.

2. Tefft, Weller, & Co. sell to H. P. Griffing the following bills of goods: March 1, 1885, on 60 da., $800; April 15, on 30 da., $350; May 20, on 4 mo, $3800.

What is the equated time for settlement?

RULE.-1. Find the date at which each debt becomes due.

2. From the earliest of these dates as a standard, compute the time to each of the others.

3. Then find the average term of credit and equated time as in (356).

PROOF.-Compute the terms of credit backward from the latest date, and subtract the average time from that date for the equated time.

If the earliest date is not the first of the month, it is more convenient to assume the first of the month as the standard date.

3. Bought mdse. as follows: Jan. 15, 1884, on 4 mo., $375; Feb. 3, on 60 da., $550; March 25, on 4 mo., $1100; April 2, on 30 da., $250. Find the equated time.

4. M. H. Decker bought of Halstead & Haines the following bills of goods on 4 months' credit:

Jan. 1, 1882, $650; Feb. 10, $380; Mar. 12, $900; Mar. 18, $350; April 3, $600.

April 5, he discounted his bills at 2% per month. Find the equated time of payment, and the discount.

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Allowing 30 days' credit on each of the bills, what is the

equated time of payment?

RATIO AND PROPORTION.

358. Ratio is the relation between two numbers of the same denomination, expressed by the quotient of the first divided by the second.

Thus, the ratio of 12 to 4 is 12÷4 = 3; of 6 to 9 is 6÷9 = :.

359. The sign of ratio is the colon (:).

Thus, the ratio of 9 to 3 is expressed 9:3, or 9÷3, or in the form of a fraction, and is read, the ratio of 9 to 3, or 9 divided by 3.

The terms of a ratio are the numbers compared.

The antecedent is the first term, or the dividend.
The consequent is the second term, or the divisor.
The two terms together form a couplet.

360. A simple ratio is the ratio of two numbers.
Thus, 6: 8, $12 : $3, and 9 mi. : 27 mi. are simple ratios.

361. A compound ratio is the ratio of the product of the corresponding terms of two or more simple ratios. Thus, the ratio compounded of the simple ratios,

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When the multiplication is performed, the result is a simple ratio. 362. The reciprocal of a ratio is 1 divided by the ratio. Thus, the ratio of 4 to 5 is 4: 5, or , and its reciprocal is 1÷‡ = {. Reciprocal ratio is sometimes called inverse ratio.

The ratio of two fractions is obtained by reducing them to a common denominator, when they are to each other as their numerators (125)

If the terms of a ratio are denominate numbers, they must be reduced to the same unit value.

363. Since the antecedent is a dividend, and the consequent a divisor, any change in either or both of the terms of a ratio will affect its value according to the laws of division, or of fractions (72, 102). Hence, the following

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The relations of ratio, antecedent, and consequent to each other are such, that if any two are given, the other may be found. Hence, the

1. The antecedent ÷ consequent = ratio. FORMULAS: 2. The antecedent ratio consequent. 3. The consequent × ratio = antecedent.

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364. Express in the lowest terms the following ratios:

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10. The consequent is 16 and the ratio 24; find the ante

cedent.

11. Antecedent 14.5, ratio 3; find the consequent.

12. Consequent 3, ratio; find the antecedent.

13. Antecedent of, consequent .75; find the ratio.

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