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Examples for the Slate.
58. ILLUSTRATIVE EXAMPLE I. If 147 trees are taken from a nursery of 489 trees, how many trees will be left ? WRITTEN WORK. Explanation. To find how many will be left, we take 147 of the number 489 away. In subtracting a large number like this we take away the units, the tens, and the hundreds separately; hence, for convenience, we write the minuend and the subtrahend as in the margin, so that units of the same order shall be expressed in the same column. Drawing a line beneath, and beginning with the units, we subtract thus: 7 units taken from 9 units leave 2 units, which we write under the line in the units' place; 4 tens taken from 8 tens leave 4 tens, which we write under the line in the tens' place; 1 hundred taken from 4 hundreds leaves 3 hundreds, which we write under the line in the hundreds' place; and we have for the whole remainder 3 hundreds 4 tens and 2 units, or 342. Answer, 342 trees.
1. If a man having 375 oranges in a box should sell 234 of them, how many would be left?
2. I had a farm of 493 acres, and sold a part containing 172 acres. How many acres had I left?
(6) (15) (8) (12) Min. 7 5 9 2 Sub. 3 6 7 4 Rem. 391 8
59. ILLUSTRATIVE EXAMPLE II. If a minuend is 7592 and the subtrahend 3674, what is the remainder?
Explanation. We write these numbers and subtract as before. As we have but 2 units in the minuend, we cannot now take the 4 units away, so we change one of the 9 tens (leaving 8 tens) to units. This 1 ten equals 10 units. We add the 10 units to the 2 units, making 12 units. Subtracting 4 units from the 12 units, we find 8 units left, which we write as part of the remainder. Subtracting 7 tens from the 8 tens we now have, we find 1 ten left, which we write. As we have but 5 hundreds in the minuend, we cannot now take 6 hundreds away, so we change one of the 7 thousands (leaving 6 thousands) to hundreds, and add the 10 hundreds thus obtained to the 5 hundreds, making 15
hundreds. Subtracting 6 hundreds from 15 hundreds, we find 9 hundreds left, which we write. Subtracting 3 thousands from the 6 thousands we now have, we find 3 thousands left, which we write; and we have for the whole remainder, 3 thousands 9 hundreds 1 ten and 8 units, or 3918.
This explanation may be given briefly thus: 4 from 12 leaves 8, 7 from 8 leaves 1, 6 from 15 leaves 9, 3 from 6 leaves 3; remainder, 3918. In actual work, however, all explanation should be omitted. Do not stop to say "4 from 12 leaves 8,” etc., but do the work, naming only results as you write them, thus: “8, 1, 9, 3; remainder, 3918." In this way you will learn to work rapidly.
What are the remainders in the following examples?
(1) (9) (9) (10) $2000
965 Ans. $1 0 3 5
7. If I had $685 in a bank and withdrew $328, how many dollars remained?
8. How old was a person in 1876 who was born in 1798?
60. ILLUSTRATIVE EXAMPLE III. If a farm is bought for $965 and sold for $2000, how much is gained?
Explanation. To find how much is gained, we take away a part of $ 2000 equal to $965.
As we have no units, no tens, and no hundreds in the minuend, we change one of the thousands (leaving 1 thousand) to 10 hundreds; then change one of the 10 hundreds (leaving 9 hundreds) to 10 tens; and one of the 10 tens (leaving 9 tens) to 10 units. 2000 is thus changed to 1 thousand 9 hundreds 9 tens and 10 units, from which taking 9 hundreds 6 tens and 5 units, we have for the remainder 1035. Ans. $1035.
9. From 2000 years take 1028 years.
10. From 3000 oxen take 229 oxen.
11. How many more birds are there in a flock of 960 birds than in one of 487 birds?
Subtraction of Decimals.
61. ILLUSTRATIVE EXAMPLE IV. What is the difference between 20.69 and 8.745?
Explanation. Writing these numbers so that units of the same order shall be expressed in the same column, and beginning with the units of the lowest order (in this case thousandths) to subtract, we have for the remainder 11.945.
12. Take 20.5 from 199.
13. From $27.68 take $15.96.
14. Find the difference between one thousand and one thousandth.
62. From the examples above explained we may derive the following
Rule for Subtraction.
1. Write the minuend and underneath write the subtrahend, so that units of the same order may be expressed in the same column. Draw a line beneath.
2. Begin with the units of the lowest order to subtract, and proceed to the highest, writing each remainder under the line in its proper place.
3. If any term of the minuend is less than the corresponding term of the subtrahend, add ten to it and then subtract; but consider that the next term of the minuend has been diminished by one.
Add the remainder to the subtrahend: the sum ought to equal the minuend.
63. Examples in Subtraction.
Examples with Decimals.
a. From 7282 subtract 4815. b. Take 3084 from 6231.
How many are 64037 less 5908?
d. Subtract 807605 from 1740932.
e. What number taken from 71287 will leave 40089 ? Ans. 31198. f. How many more than 94736 is 104083? Ans. 9347. g. Find the difference between 86045 and 708406.
Ans. 622361. Ans. 1920466.
h. 2684753 - 764287 = how many ? i. From four hundred twenty thousand six hundred eighty-three, take two hundred fifty-nine thousand seventyfive. Ans. 161608.
j. Take eight hundred ten thousand twenty-three from one million sixty thousand forty-one. Ans. 250018. Ans. 91802.
k. 1001001 minus 909199 equals what?
1. Subtract the sum of the numbers in example c from the sum of the numbers in example d. Ans. 2478592.
m. Find the difference between the amount of the numbers in example a and the amount of the numbers in example b. Ans. 2782.
n. From $17.60 take $5.25. o. From 426.17 take 11.723.
Ans. $12.35. Ans. 414.447.
p. Subtract three hundred sixty-four thousandths from
q. What must be added to 0.0476 to make 1?
NOTE. The examples on this page embrace the chief varieties in form of examples in Subtraction. After performing these, and before taking the Applications on page 26, pupils will usually need additional practice in similar work. Examples for such practice will be found on pages 59–63.
15. A farmer who raised 948 bushels of corn sold all but 198 bushels. How much did he sell?
16. The year's earnings of a family were $1172. If their expenses were $875, what was saved?
17. A and B together own 5740 acres of land. If B owns 2964 acres, how much does A own?
18. Mount Washington is 6234 feet high, which is 2286 feet higher than Vesuvius. How high is Vesuvius?
19. The several items of an account amount to $9867.62; of this amount $ 7985.75 has been paid. Find the balance.
20. Franklin was born in 1706, and died in 1790. What was his age at the time of his death?
21. The difference between A's and B's estates is $1463; B's, which is the greater, is worth $7638. What is A's worth? 22. In one week a grain elevator received 984560 bushels of grain; of this 769386 bushels were delivered. How much remained in the elevator?
23. The sailing distance from New York to Queenstown is 2890 miles. If a Cunard steamer has run 1368 miles on her course from New York, how far has she still to run?
The population of the city of New York was 60489 in the year 1800; 96373 in 1810; 123706 in 1820; 202589 in 1830; 312710 in 1840; 515547 in 1850; 813669 in 1860; and 942292 in 1870. What was the increase in population From 1840 to 1850?
24. From 1800 to 1810?
From 1850 to 1860?
From 1860 to 1870?
25. From 1810 to 1820? 26. From 1820 to 1830? 27. From 1830 to 1840? 32. The population of London in 1871 was 3266987. How many times may you subtract from this a population equal to that of New York in 1870?
33. The equatorial diameter of the earth is 41847194 feet, and the polar diameter 41707308 feet. What is the difference?