Ex. Find the time for payment of the balance of an account, if the debit and credit sides, when equated, stand as follows: Balance of account = $900 - $825 = $75. If the account were settled at the later date, May 30, the $900 on the Dr. side would have been on interest 1 day, and this is equivalent to having the balance, $75, on interest 2 of 1 day = 12 days. Hence, the balance should begin to draw interest 12 days before May 30; that is, May 18. Ex. Find the time for payment of the balance of an account if the debit and credit sides, when equated, stand as follows: DR. CR. Due Jan. 20, 1881, $850 Due Feb. 18, 1881, $950 Difference in equated time, 29 days. Balance of account, $950-$850 = $100. If the account were settled at the later date, Feb. 18, the $850 would have been on interest 29 days, which is equivalent to having the balance, $100, on interest 58 of 29 days == 246 days. Hence, to increase the Dr. side by an equal amount of interest, the balance should remain unpaid 247 days; that is, the balance is due Oct. 23, 1881. 375. From the two preceding examples is derived the following RULE FOR EQUATING ACCOUNTS. Find the equated time for each side of the account. Multiply the side of the account that falls due first by the number of days between the dates of the equated times of the two sides, and divide the product by the balance of the ac count. The quotient will give the number of days to the maturity of the balance, to be counted forward from the later date when the smaller side falls due first, and backward when the larger side falls due first. Find the time for paying the balance in the following June 23, 1881 . $920 $480 26. May 30, 1881.. $1000 27. July 6, 1881... $500 || Apr. 14, 1881 . . NOTE. In finding the equated time of accounts it is customary to neglect cents if less than 50, and if 50 or more to consider them as $1. A fraction of a day in the result is rejected if less than 1; if or more it is called a day. 376. When an account is settled by cash at any other date than that at which the balance becomes due, the interest is found on the balance for the interval between the day of settlement and the day the balance is due, and is added to, or deducted from, the balance, according as the settlement is made after or before the balance is due. 377. Another method is by computing the interest on each item, from its date to the day of settlement. (The time is reckoned in days.) = $1260 $30.35 Hence, cash balance = $210 + $8.54 $218.54. Ans. When the balance of account and the balance of interest fall on opposite sides, the cash balance is their difference. Find (by either method) the cash balance in the following bills, reckoning interest at 6%: NOTE. Since the Dr. items have the same term of credit, find the equated time of these items, and count forward from that date 3 mos. for the term of credit. 30. 1881. DR. 1881. CR. Jan. 3. To mdse. 30 dys. $100.00 || Feb. 25. By cash, $100.00 CHAPTER XXI. POWERS AND ROOTS. 378. The square of a number is the product of two factors, each equal to this number. are Thus, the squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. 379. The square root of a number is one of the two equal factors of the number. are Thus, the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 380. The square root of a number is indicated by the radical sign, or by the fractional exponent written above and to the right of the number. 381. Since (24)2 = (16)2=256=28; and (24)* = (16)* = 4 22, it is evident that, A power of a power of a number, or a root of a power of a number, is that number with an exponent equal to the product of the given exponents. 382. Since 35=30+5, the square of 35 may be obtained as follows: 383. Hence, since every number consisting of two or more figures may be regarded as composed of tens and units, The square of a number will contain the square of the tens twice the tens the units the square of the units. SQUARE ROOT. 384. The first step in extracting the square root of a number is to mark off the figures in periods. Since 112, 100 = 102, 10,000 = 1002, and so on, it is evident that the square root of any number between 1 and 100 lies between 1 and 10; of any number between 100 and 10,000 lies between 10 and 100. In other words, the square root of any number expressed by one or two figures is a number of one figure; of any number expressed by three or four figures is a number of two figures; and so on. If, therefore, an integral number be divided into periods of two figures each, from the right to the left, the number of figures in the root will be equal to the number of periods. The last period at the left may consist of only one figure. Ex. Find the square root of 1225. 12/25 (35 9 65)3 25 3 25 Since 1225 consists of two periods, the square root will consist of two figures. The first period, 12, contains the square of the tens' number of the root. The greatest square in 12 is 9, and the square root of 9 is 3. Hence, 3 is the tens' figure of the root. The square of the tens is subtracted, and the remainder, 325, is twice the tens the units + the square of the units. Twice the 3 tens is 6 tens, and 6 tens is contained in the 32 tens of the remainder 5 times. Hence, 5 is the units' figure of the root. Since twice the tens the units + the square of the units is equal to (twice the tens + the units) × the units, the 5 units are annexed to the 6 tens, and the result, 65, is multiplied by 5. 385. The same method will apply to numbers of more than two periods, by considering the part of the root already found as so many tens with respect to the next figure of the root, |