Arithmetic Refresher

Arithmetic Refresher

by A. A. Klaf

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Product Details

ISBN-13: 9780486212418
Publisher: Dover Publications
Publication date: 06/01/1964
Series: Dover Books on Mathematics
Pages: 464
Sales rank: 416,883
Product dimensions: 5.50(w) x 8.50(h) x (d)

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Arithmetic Refresher


By A. Albert Klaf

Dover Publications, Inc.

Copyright © 1964 Mollie G. Klaf
All rights reserved.
ISBN: 978-0-486-14193-0


Contents

Dover Books on Mathematics,
BOOKS BY A. ALBERT KLAF,
Title Page,
Copyright Page,
FOREWORD,
INTRODUCTION,
CHAPTER I - ADDITION,
CHAPTER II - SUBTRACTION,
CHAPTER III - MULTIPLICATION,
CHAPTER IV - DIVISION,
CHAPTER V - FACTORS—MULTIPLES—CANCELLATION,
CHAPTER VI - COMMON FRACTIONS,
CHAPTER VII - DECIMAL FRACTIONS,
CHAPTER VIII - PERCENTAGE,
CHAPTER IX - INTEREST,
CHAPTER X - RATIO—PROPORTION—VARIATION,
CHAPTER XI - AVERAGES,
CHAPTER XII - DENOMINATE NUMBERS,
CHAPTER XIII - POWER—ROOTS—RADICALS,
CHAPTER XIV - LOGARITHMS,
CHAPTER XV - POSITIVE AND NEGATIVE NUMBERS,
CHAPTER XVI - PROGRESSIONS—SERIES,
CHAPTER XVII - GRAPHS—CHARTS,
CHAPTER XVIII - BUSINESS—FINANCE,
CHAPTER XIX - VARIOUS TOPICS,
CHAPTER XX - INTRODUCTION TO ALGEBRA,
APPENDIX A - ANSWERS TO PROBLEMS,
APPENDIX B: TABLES,
INDEX,


CHAPTER 1

ADDITION


34. Why is addition merely a short way of counting?

If we have four apples in one group and five in another, we may count from the first object in one group to the last object in the other and obtain the result, nine. But seeing that 4 + 5 = 9 under all conditions, we make use of this fact without stopping to count each time we meet this problem.

The addition of two numbers is thus seen to be a process of regrouping. We do not increase anything, we merely regroup the numbers.


35. What is our standard group or bundle?

Our number system is based on groups or bundles of ten.

EXAMPLE: 9 + 8 = 17. Two groups of 9 and 8 are regrouped into our standard arrangement of 17, or one bundle of 10 and 7 units. While we say "seventeen" we must think "ten and seven" or "1 ten and 7 units."


36. What is thus meant by addition?

It is the process of finding the number that is equal to two or more numbers grouped together.


37. What is meant by sum?

It is the result obtained by adding numbers.


38. Of the total number of 45 additions of two digits at a time for all the nine digits, which give single numbers as a sum and which give double numbers?

(a) The following 20 pairs result in one-number sums:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) The following 25 pairs give double numbers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


39. What is the rule for addition?

Write the numbers so that units stand under units, tens under tens, hundreds under hundreds, etc. Begin at the right and add the units column. Put down the units digit of the sum and carry the "tens" bundles to the next column representing the "tens" bundles. Do the same with this column. Put down the digit representing the number of tens and carry any "hundreds" bundles to the hundreds column. Continue in the same manner with other columns.


40. What is the proper way of adding?

Add without naming numbers, merely sums.

EXAMPLE:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


41. What is the simplest but slowest way of adding?

Column by column and one digit at a time. Add from the top down or from the bottom up; each way is a check on the other.

EXAMPLE:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


42. What is a variation of the above?

Add each column separately. Write one sum under the other, but set each successive sum one space to the left. A subsequent addition gives the total or sum.

EXAMPLE: (as above)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


43. How can grouping of numbers help you in addition?

Add two or more numbers at a time to two or more others in the columns.

EXAMPLE:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


44. How is addition accomplished by multiplication of the average of a group?

When you have a group of numbers whose middle figure is the average of the group, then:

sum = average number times number of figures in the group


EXAMPLES:

(a) Of 4, 5, and 6 number 5 = average of the three

[therefore] Sum = 5 × 3 = 15 = (4 + 5 + 6)

(b) Of 8, 9, and 10 9 = average

[therefore] Sum = × 3 = 27 = (8 + 9 + 10)

(c) Of 12, 13, and 14 13 = average

[therefore] Sum = 13 × 3 = 39 = (12 + 13 + 14)

(d) Of 6, 7, 8, 9, and 10 8 = average

[therefore] Sum = 8 × 5 = 40 = (6 + 7 + 8 + 9 + 10)

(e) Of 11, 12, 13, 14, and 15 13 = average

[therefore] Sum = 13 × 5 = 65


Note that whenever an odd number of equally spaced figures appears, you can immediately spot the center one or average and promptly get the sum of all by multiplying the average by the number of figures in the group.


45. What is the procedure for adding two columns at a time?

37 Start at bottom. Add 96 to 80 of above, then the 2 getting 24178. Add 178 to the 20 above, then the 4 getting 202. Add 82202 to the 30 above, then the 7 getting 239 = sum. 96/239

A variation would be to add the units of the line above it first and then the tens, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


46. How are three columns added at one time?

Start at bottom. Add hundreds, then tens, then units as you continue up.

EXAMPLES:

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


47. What is a convenient way of adding two small quantities by making a decimal of one of them?

Make a decimal of one by adding or subtracting and reverse the treatment for the other.

EXAMPLE: 96 + 78.

Add 4 to 96 getting 100 = decimal number. Subtract 4 from 78 getting 74.

[therefore] Sum = 174 at once.


48. How may decimalized addition be carried out to a fuller development?

Reduce each number to a decimal. Add the decimals. Add or subtract the increments.

EXAMPLE:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


49. How may sight reading be used in addition?

By use of instinct you get an immediate result.

EXAMPLES:

(a) Add 26 to 53. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) Add 67 to 86. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


Fix eyes between the two columns where the dots are and at once see a 7 and a 9 or a 13 and a 14 to make 153. Actually 70 is added to 9 and 140 to 13 but each is done instinctively.


50. What simple method is used to check the correctness of addition of a column of numbers?

First begin at the bottom and add up. Then begin at the top and add down. When the columns are long it is often better to write down the sums rather than to carry the "bundles" from column to column. Place sums in proper columns.

EXAMPLE:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


51. What is meant by a check figure in addition?

One which, when eliminated from each number to be added and from the sum, will give a key number that may indicate the correctness of the addition. The check numbers 9 and 11 are generally used.


52. What are the interesting facts on the use of the check number 9?

(1) The fact that the remainder left after dividing any number by 9 is the same as the remainder of the sum of the digits of that number divided by 9.

Ex. (a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Ex. (b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


(2) Also note that the sum of the digits alone will give the same number as a remainder as the division of the number by 9. Thus in (a) 6 + 5 + 4 = 15 and 1 + 5 = (6). In (b) 2 + 6 + 7 + 7 = 22 and 2 + 2 = (4).

(3) Also the fact that 9's can be discarded when adding the digits. Thus in (a) 6 + 5 + 4, discard 4 + 5 right away and the remainder is again (6). In (b) 2 + 6 + 7 + 7, discard 2 + 7 but add 6 + 7 = 13 and 1 + 3 = (4).


53. What is the procedure in checking addition by the use of the check figure 9, often called "casting out nines"?

(a) Add the digits in each number horizontally and get each remainder.

(b) Add the digits of these remainders and get the key figure.

(c) Add the digits horizontally of the answer and get the same key figure if the answer is correct.

EXAMPLE:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


In practice it is sufficient to add the numbers mentally to get the remainders.

Note that all 9's and digits that add up to 9 are discarded right away. Each digit so discarded is shown with a dot at the upper right corner.


54. Why is "casting out nines" not a perfect test of accuracy in addition?

It is possible to omit or add nines or zeros without detection. Also figures may be transposed; 27 is quite different in value from 72 although the sum of the digits is the same.

This method is not generally recommended as a practical test in addition work but has its greatest value in multiplication and division work. However, it is sometimes useful as a quick check of addition.


55. What are the interesting facts on the use of the check number 11?

(1) The remainder left after dividing any number by 11 is the same as the remainder left after subtracting the sum of the digits in the even places from the sum of the digits in the odd places. If the subtraction cannot be made add 11 or a multiple of it to the odd-places sum.

EXAMPLES:

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


(2) The same remainder is also obtained by starting with the extreme left digit in the number and subtracting it from the digit to its right. When necessary add 11 to make the subtraction possible. Subtract the remainder from the next digit. Again add 11 if necessary. Repeat the process of subtraction until all the digits of the number have been used.


56. Why is the checking of addition work by the use of the check figure 11 (often called "casting out elevens") superior to that of "casting out nines"?

"Casting out elevens" can indicate an error due to transposition of digits which is not possible with the "nines" method.

EXAMPLE: Suppose our number is 8,706

8 from (11 + 7) = 10 10 from (11 + 0) = 1 I from 6 = (6) = Remainder = Check number


Now suppose the transposed number is 8,076

8 from (11 + 0) = 3 II3 from 7 = 4 4 from 6 = (2) = Remainder = Check number


The check numbers are seen to be different and we have uncovered a transposition of digits.


57. What is the procedure in checking addition by the use of the check figure 11?

(a) Cast out elevens from each row and get each remainder.

(b) Add the remainders and cast out elevens from this sum, getting the key figure.

(c) Cast out elevens from the answer and get key figure. Compare.

EXAMPLE:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


PROBLEMS

1. Count from 3 to 99 by 3's.

2. Count from 4 to 100 by 4's.

3. Count from 6 to 96 by 6's.

4. Count from 9 to 99 by 9's.

5. Start with 3 and count by 2's, 4's, 6's, 8's to just below 100.

6. Start with 2 and count by 3's, 5's, 7's, 9's to just below 100.

7. Start with 9 and count by 4's, 7's, 9's, 2's to just below 100.

8. Start with 14 and count by 6's, 2's, 4's, 8's to just below 100.

9. Add 269, 745, and 983.

10. Add, using "carry overs."

11. Add $5.25, $17.60, $0.85, $175, $4.565.

12. Find the sum of:

13. What is the sum of 10, 20, 30 by the average method?

14. What is the sum of 14, 15, 16 by the average method?

15. What is the sum of 17, 18, 19, 20, 21 by the average method?

16. What is the sum of 3, 4, 5, 6, 7, 8, 9 by the average method?

17. What is the sum of 5, 7, 9 by the average method?

18. What is the sum of 13, 15, 17 by the average method?

19. What is the sum of 14, 16, 18, 20, 22 by the average method?

20. What is the sum of 9, 12, 15 by the average method?

21. Add two columns at a time.

22. Add three columns at a time.

23. Add the following by the decimalizing method:

(a) 94 + 75

(b) 86 + 69

(c) 92 + 48

(d) 89 + 52

(e) 468 + 982 + 429

(f) 346 + 899 + 212

(g) 589 + 913 + 165

(h) 862 + 791 + 386


24. Add by sight reading:

(a) 27 + 56

(b) 21 + 43

(c) 32 + 65

(d) 49 + 57

(e) 68 + 87

(f) 76 + 82


25. A gasoline station owner had 275 gallons left after selling 632 gallons. How many gallons did he have originally?

26. One pipe from a tank discharges 76 gallons per second while another pipe from the same tank discharges 16 gallons per minute more than the first. How many gallons will both pipes discharge in a minute?

27. An automobile travels 386 miles on the first day and 416 miles the second day, at which time it is 237 miles from its point of destination. What is the distance from its starting point to its destination?

28. A suburban house was built with the following expenses: masonry, $3,565; lumber, $4,850; millwork, $1,485; carpentry, $3,800; plumbing, $2,758; painting, $679; hardware, $1,508; heating, $1,250; and electricity, $687. What did the house cost when completed?

29. If a family of two persons spends $135 for rent, $205 for food, $85 for clothing, $35 for fuel, $7 for light, $22 for insurance, $6 for carfare, $12 for charity, and saves $18, what is the income after taxes and other payroll deductions?

30. The twenty-second of February is how many days after New Year's? How many days from New Year's to the fourth of July?

31. Check the following by first adding up and then by adding down. Place check marks as proof.

32. Prove the following by use of the check figure 9.

33. Prove the following by use of the check figure 11.

34. Add horizontally and vertically.

CHAPTER 2

SUBTRACTION


58. What is subtraction?

It is the reverse of addition. Since we know that five apples + three apples = eight apples, it follows reversely that taking five apples away from eight apples leaves three apples.

Or taking three apples away from eight apples leaves five apples.

8 - 5 = 3 || 8 - 3 = 5

As with addition, subtraction is thus seen to be merely a regrouping:

group (a) + group (b) = group (c) = 8. group (c) - group (a) = 3. || group (c) - group (b) = 5.


59. Why may subtraction be said to be a form of addition?

Ex. (a) 9 - 4 = 5.

May be thought of as "4 and what make 9?" 4 and 5 make 9.

Ex. (b) 16 - 9 = 7.

9 and what make 16? 9 and 7 make 16.


60. What three questions will lead to the process of subtraction?

(a) How much remains?

(b) How much more is required?

(c) By how much do they differ?

In (a) if Bert has $10 and pays out $6, how many dollars remain? Here the $6 was originally a part of the $10.

In (b) Bert has $65 and would like to buy a 35-mm. camera that costs $89. How much more does he require?

In (c) if Bert has $10 and Charles has $6, by how much do they differ? Here the $10 and the $6 are distinct numbers.


61. What are the terms of a subtraction?

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


If the subtrahend was originally a part of the minuend then the answer is called the "remainder." If the minuend and subtrahend are distinct numbers the answer is called the "difference."


62. Why is it said that we can always add but we cannot always subtract?

Subtraction is not always possible. It is not, when the number of things which we wish to subtract is greater than the number of things we have.


(Continues...)

Excerpted from Arithmetic Refresher by A. Albert Klaf. Copyright © 1964 Mollie G. Klaf. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents

INTRODUCTION
I ADDITION
II SUBTRACTION
III MULTIPLICATION
IV DIVISION
V FACTORS-MULTIPLES-CANCELLATION
VI COMMON FRACTIONS
VII DECIMAL FRACTIONS
VIII PERCENTAGE
IX INTEREST
X RATIO-PROPORTION-VARIATION
XI AVERAGES
XII DENOMINATE NUMBERS
XIII POWER-ROOTS-RADICALS
XIV LOGARITHMS
XV POSITIVE AND NEGATIVE NUMBERS
XVI PROGRESSIONS-SERIES
XVII GRAPHS-CHARTS
XVIII BUSINESS-FINANCE
XIX VARIOUS TOPICS
XX INTRODUCTION TO ALGEBRA
APPENDIX A-ANSWERS TO PROBLEMS
APPENDIX B-TABLES
INDEX

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