Ex. 15. 1 Observing the directions given in the preceding article, find the sum of each of the following columns: 14. Principles of Addition and Subtraction. A certain man's property is in two investments. At the beginning of the year he estimates the income on each investment at a certain amount. By what process can he find his probable total income? The income from each investment forms what element of the operation? The total income? 1. Suppose that the income from one investment is $500 greater than the expected income. How will the total income compare with the expected total income? Increasing an addend has, then, what effect on the amount? 2. Suppose that the income from one investment is $500 less than the expected income. How will the total income compare with the expected total income? Diminishing an addend has, then, what effect on the amount? 3. Suppose that the income from one investment is $500 greater, and from the other $500 less, than the expected in come. How will the total income compare with the expected total income? Increasing one addend by a certain quantity and diminishing another by the same quantity has, then, what effect on the amount? A man at the end of six months estimates his annual income at a certain amount, and his expenditures at a certain amount. By what process can he find his probable net income? The total income forms what element of the operation? The expenditures? The net income? 1. Suppose that the income is $500 greater than the expected income. How will the net income compare with the expected net income? Increasing the minuend has, then, what effect on the remainder? 2. Suppose that the income is $500 less than the expected income. How will the net income compare with the expected net income? Diminishing the minuend has, then, what effect on the remainder? 3. Suppose that the expenditures are $500 greater than the expected expenditures. How will the net income compare with the expected net in come? Increasing the subtrahend has, then, what effect on the remainder? 4. Suppose that the expenditures are $500 less than the expected expenditures. How will the net income compare with the expected net income? Diminishing the subtrahend has, then, what effect on the remainder? 5. Suppose that the income is $500 greater than the expected income, and that the expenditures are $500 greater than the expected expenditures. How will the net income compare with the expected net income? Suppose that the income is $500 less than the expected income, and that the expenditures are $500 less than the expected expenditures. How will the net income compare with the expected net income? Increasing or diminishing both subtrahend and minuend by the same number has, then, what effect on the remainder? For convenience of reference we here express the preceding principles. PRINCIPLES OF ADDITION. 1. Increasing an addend increases the amount. 2. Diminishing an addend diminishes the amount. 3. Increasing one addend and diminishing another by the same number does not change the amount. PRINCIPLES OF SUBTRACTION. 1. Increasing a minuend increases the remainder. 2. Diminishing a minuend diminishes the remainder. 3. Increasing a subtrahend diminishes the remainder. 4. Diminishing a subtrahend increases the remainder. 5. Increasing or diminishing both minuend and subtrahend by the same number does not change the remainder. 15. To Prove Subtraction. What relation has a minuend to a subtrahend and a remainder? Give, then, a method of proving a subtraction. 16. To Add when One, or More, of the Addends is Greater than 9. We are to find the sum of 876, 934, 568, 827, 913, 475, 822, and 936. What is the sum of the units' column? 934 41 units is equal to how many units and how many tens? 568 827 What do we do with the 1 unit? What with the four tens? 913 475 822 936 6351 What is the sum of the tens' column? 35 tens is equal to how many tens and how many hun dreds? What do we do with the 5 tens ? What with the 3 hundreds? What is the sum of the hundreds' column? What, then, is the total sum? Find the sum of 143.56, 897.16, 479.92, 723.42, 974.33. EXPLANATION. We first write the numbers, so that units will stand under units, tenths under tenths, etc. We next add the right-hand column. Its sum we find to be .19, or 1 tenth and 9 hundredths. The 9 hundredths we write under the hundredths' column; the 1 tenth we add, or "carry," to the tenths' column. 974.33 3218.39 The sum of the tenths' column, including the 1 tenth brought from the hundredths' column, is 2.3, or 2 units and 3 tenths. The 3 tenths we write under the tenths' column; the 2 units we carry to units' column. Proceeding in the same way with the other columns, we have as the result of the addition, 3218.39 NOTE I. In finding the sums in the preceding exercises the mental processes are as follows: 2. 1. 8, 16, 23, 31, 41; 11, 20, 30, 35; 11, 20, 25, 33, 42, 46, 54, 63. 5, 13, 19; 6, 16, 23; 5, 12, 21, 28; 6, 15, 24, 31; 4, 12, 16, 23, 32. NOTE 2. It will be observed that in the preceding exercises the left-hand figure of the sum of each column is written above the column containing the figures of its order. The object of this is to make clear the process of 'carrying.' In practice, the recording of the left-hand figure is advisable only when the columns to be added are of unusual length. NOTE 3. In adding very long columns, the necessity of burdening the mind with the number of hundreds may be avoided by placing a light mark near the last figure of each hundred. |