3 A owes B a certain sum; one third is due in 6 months, one fourth in 8 months, and the remainder in 12 months. What is the mean time of payment?

It is evident that it makes no difference what the amount is which A owes B, since it is certain fractional parts which become due at particular times. If we suppose the sum to be $1, then our work will be

mo. mo.

× 6=2

× 8=2

Remainder is 2, and × 12=5

Ans. 9 months.

The least sum which we can take so as to avoid fractions is $12. In this case we have

Hence, 10-9 months for the time.'

4. A merchant has due him $300 to be paid in 2 months; $800 to be paid in 5 months; $400 to be paid in 10 months. What is the equated time for the payment of the whole ? Ans. 51mo., or 5mo. 22da.

5. A merchant owes $1200, payable as follows: $200 in 2 months, $400 in 5 months, and the remainder in 8 months. He wishes to pay the whole at one time. What is the equated time of such payment? Ans. 6 months.

6. A merchant bought goods to the amount of $2400, for one fourth of which he was to pay cash at the time of receiving the goods, one third in 6 months, and the balance in 10 months. What was the equitable time for the payment of the whole?

Hence, 14800mo.÷2400-63 months for the time sought. It is obvious that the time may be estimated in days as well as in months. To illustrate this we will give several examples of this kind.

7. Suppose I owe $100 payable on January 1st, $150 on February 5th, $300 on April 10th. If we count from January 1st, and allow 29 days to February, it being Leap year, on what day ought the whole sum in equity to be paid?

Counting from-January 1st, the $100 will have no time of credit: 100 X Oda.= Oda.

From Jan. 1st to Feb. 5th is

Hence, 35250da.÷550-64, days, or counted from Jan. 1st, gives March 5th for the equated time of the payment of the whole.

NOTE. The table under ART. 76 will be found very convenient for determining the number of days between the different dates.

8. A merchant bought a bill of goods amounting to $1000. He agrees to pay $250 the first day of the next March, $250 on the 3d of the following May, $250 on the

4th of the following July, and the remaining $250 on the 15th of the following September. What would be the equitable time for paying the whole ?

In this example, all the payments being equal, we may take for each one any sum we please. For simplicity we will consider each payment as $1.

Counting from March 1st, we see that the first payment has no credit: 0 days= 0 days.

From March 1st to May 3d

is 63 days:

From March 1st to July 4th

is 125 days:

1 x

1x 63 days= 63 days.

1x 125 days 125 days.

From March 1st to Sept.

15th is 198 days:

1x 198 days 198 days.

$4

386 days.

Hence, 386 days÷4-964 days. Calling this 97 days, and counting from March 1st, we have June 6th for the time sought.

When a debt due at some future period has received several partial payments before the time due, to find how long beyond this time the balance may in equity remain unpaid.

9. Suppose $1000 to be due at the end of 6 months; that 3 months before it is due there was paid $100, and that 1 month before the expiration of the 6 months, there was paid $300. How long after the end of the 6 months may the balance of $600 remain unpaid?

100 x 3mo.=300mo.

300 x 1mo.=300mo.

600)600mo.

Ans. 1 month.

Hence we have this

RULE.

;

Multiply each payment by the time it was paid before due then divide the sum of the products thus obtained by the balance remaining unpaid; the quotient will be the equated time.

EXAMPLES.

10. Suppose $1496-41 to be due at the end of 90 days, that 84 days before it is due there is paid $500; 32 days before the 90 days expire there is paid $502.50. How long after the 90 days before the balance of $493-91 ought to be paid? Ans. 117 days.

11. A lent $200 to B for 8 months; at another time he lent him $300 for 6 months. For how long a time ought B to lend A $800 to balance the favor?

Ans. 44 months.

128. It is customary with many merchants to give a credit, of from 3 to 6 months, on their bills of sale. In such cases, in settling up their accounts, which generally consist of various items of debit and credit at sundry times, it is very desirable to have some simple rule by which the cash balance can be found. We have prepared a rule for this purpose.

Suppose A has the following account with B:

What is the cash balance, July 10, 1848, if interest is estimated at 7 per cent., and a credit of 30 days is allowed on all the different sums?

If interest were not considered, the above account could

Had no credit been given, the debits should be increased by the following items of interest: (See Table, Art. 76, and Rule, ART. 114.)

On $100 for 182 days at 7 per cent.=100 x 182 × 907. 66 =400 x 106 × 107.

66 400" 106

66

In like manner the credits should be increased by in

terest:

66

66

3

3

On $50 for 153 days at 7 per cent. 50 × 153 × 9.07. "375" 78 66 =375 x 78 x 7. But, since 30 days credit is given on all sums, it follows that by the above, we should increase the debits by an excess of interest equal to the interest of the sum of debits, $500, for 30 days=500 × 30×907. In like manner we should increase the credits by an excess of interest equal to the interest of sum of credits, $425, for 30 days 425 × 30 × 907.

Now if, instead of diminishing the debit items of interest by 500 × 30×7, and the credit items of interest by 425 x30x07, we merely diminish the debit items of interest by the interest on merchandise balance, $75, for 30 days, which is 75×30×7, the result will be the same. And since taking any sum from one side of a book account has the same effect as adding the same sum to the other side, it follows, that instead of diminishing the