Number Sense Tricks - Bryant Heath

IntroductionAs most who are reading this book already know, the UIL Number Sense exam is an intense 10 minutetest composed of 80 mental math problems which assesses a student’s knowledge of topics ranging fromsimple multiplication, geometry, algebraic manipulation, to calculus. Although the exam is grueling (with7.5 seconds per problem, it is hard to imagine it being easy!), there are various tricks to alleviate some ofthe heavy computations associated with the test. The purpose of writing this book is to explore a variety ofthese “shortcuts” as well as their applications in order to better prepare students taking the Number Sensetest. In addition, this book is a source of practice material for many different types of problems so thatbetter proficiency of the more straight-forward questions can be reached, leaving more time for harder andunique test questions.This book is divided into three sections: Numerical Tricks, Necessary Memorizations (ranging from conversions to formulas), and Miscellaneous Topics. The difficulty of tricks discussed range from some of themost basic (11’s trick, Subtracting Reverses, etc.) to the more advanced that are on the last column of themost recent exams. Most of the material is geared towards High School participants, however, after lookingthrough some recent Middle School exams, a lot of the tricks outlined in this manual are appropriate for thatcontest as well (albeit, more simplified computations are used). Although this book will provide, hopefully,adequate understanding of a wide variety of commonly used shortcuts, it is not a replacement for practicingand discovering methods that you feel most comfortable with. In order to solidify everything exhibited in thisbook, regular group and individual practice sessions are recommended as well as participation in multiplecompetitions. For further material, you can find free practice tests for both Middle and High School levelson my website at the following URLS:Middle School Exams: practice-tests/High School Exams: /The best way to approach this book is to read through all the instructional material first (and, if you are aMiddle School student, skip certain sections – such as the Calculus stuff – that are not applicable to yourexam) then go back and do the practice problems in each section. The reason why this is needed is becausemany sections deal with combinations of problems which are discussed later in the book. Also, all problemsin bold reflect questions taken from the state competition exams. Similarly, to maintain consistent nomenclature, all (*) problems are approximation problems where 5% accuracy is needed.It should be noted that the tricks exhibited here could very easily not be the fastest method for doing theproblems. I wrote down tricks and procedures that I follow, and because I am only human, there could veryeasily be faster, more to-the-point tricks that I haven’t noticed. In fact, as I’ve been gleaning past teststo find sample problems, I’ve noticed faster methods on how to do problems and I’ve updated the bookaccordingly. One of the reasons why Number Sense is so great is that there is usually a variety of methodswhich can be used to get to the solution! This is apparent mostly in the practice problems. I tried to chooseproblems which reflects the procedures outlined in each section but sometimes you can employ differentmethods and come up with an equally fast (or possibly faster!) way of solving the problems.Finally, I just want to say that although Number Sense might seem like a niche competition with limited value,there are a variety of real-world applications where being able to calculate quickly or estimate accuratelycan benefit you immensely both now and in your future career. One of the most immediate benefit you’llsee is that your standardized math test scores will probably improve (if you can do the rote calculationsquickly, it leaves more time to really think about the more difficult problems). Even fifteen years past mylast competition, the ability to make good back-of-the-envelope calculations in my head quickly has givenme an edge when it comes to on-the-fly interpretations of data I see regularly in my career. Although you’llbe competing in Number Sense for just a few years in Middle and High School, the skills you acquire willlast a lifetime.-Bryant Heath5

1Numerical Tricks1.1Introduction: FOILing/LIOFing When MultiplyingMultiplication is at the heart of every Number Sense test. Slow multiplication hampers how far you are ableto go on the test as well as making you prone to making more errors. To help beginners learn how to speedup multiplying, the concept of FOILing, learned in beginning algebra classes, is introduced as well as someexercises to help in speeding up multiplication. What is nice about the basic multiplication exercises is thatanyone can make up problems, so practice is unbounded.When multiplying two two-digit numbers ab and cd swiftly, a method of FOILing – or more accurately namedLIOFing (Last-Inner Outer-First) – is used. To understand this concept better, lets take a look at what wedo when we multiply ab cd:ab 10a b and cd 10c d(10a b) (10c d) 100(ac) 10(ad bc) bdA couple of things can be seen by this:1. The one’s digit of the answer is simply bd or the Last digits (by Last I mean the least significant digit)of the two numbers multiplied.2. The ten’s digit of the answer is (ad bc) which is the sum of the Inner digits multiplied together plusthe Outer digits multiplied.3. The hundred’s digit is ac which are the First digits (again, by First I mean the most significant digit)multiplied with each other.4. If in each step you get more than a single digit, you carry the extra (most significant digit) to the nextcalculation. For example:74 23 3 4 123 7 2 4 1 302 7 3 171702Units:Tens:Hundreds:Answer:Where the bold represents the answer and the italicized represents the carry.Similarly, you can extend this concept of LIOFing to multiply any n-digit number by m-digit number in aprocedure I call “moving down the line.” Let’s look at an example of a 3-digit multiplied by a 2-digit:493 23 3 3 93 9 2 3 333 4 2 9 3 332 4 3 1111339Ones:Tens:Hundreds:Thousands:Answers:As one can see, you just continue multiplying the two-digit number “down the line” of the three-digit number,writing down what you get for each digit then moving on (always remembering to carry when necessary).The following are exercises to familiarize you with this process of multiplication:6

Problem Set 1.1:95 30 90 78 51 11 83 51 64 53 65 81 92 76 25 46 94 92 27 64 34 27 11 77 44 87 86 63 54 92 83 68 72 65 81 96 57 89 25 98 34 32 88 76 22 11 36 69 35 52 15 88 62 48 56 40 62 78 57 67 28 44 80 71 51 61 81 15 64 14 47 37 79 97 99 87 49 54 29 67 38 98 75 47 77 34 49 94 71 29 85 66 13 65 64 11 62 15 43 65 74 72 49 41 23 70 72 75 53 59 82 91 14 17 67 27 85 25 25 99 137 32 428 74 996 47 654 45 443 39 739 50 247 87 732 66 554 77 324 11 111 54 885 78 34 655 52 532 33 334 45 301 543 543 606 212 657 322 543 230 111 121 422 943 342 542 789 359 239 795 123 543 683 429 222 796 7

1.21.2.1Multiplying: The BasicsMultiplying by 11 TrickThe simplest multiplication trick is the 11’s trick. It is a mundane version of “moving down the line,” whereyou add consecutive digits and record the answer. Here is an example:523 11 1 3 31 2 1 3 51 5 1 2 71 5 55753Ones:Tens:Hundreds:Thousands:Answer:As one can see, the result can be obtained by subsequently adding the digits along the number you’remultiplying. Be sure to keep track of the carries as well:6798 11 Ones:Tens:Hundreds:Thousands:Ten Thousands:Answer:89 8 177 9 1 176 7 1 146 1 774778The trick can also be extended to 111 or 1111 (and so on). Where as in the 11’s trick you are adding pairsof digits “down the line,” for 111 you will be adding triples:6543 111 Ones:Tens:Hundreds:Thousands:Ten Thousands:Hun. Thousands:Answer:34 3 75 4 3 126 5 4 1 166 5 1 126 1 7726273Another common form of the 11’s trick is used in reverse. For example:1353 11 or11 x 1353Ones Digit of x is equal to the Ones Digit of 1353:Tens Digit of x is equal to:Hundreds Digit of x is equal to:Answer:5 3 xtens3 2 xhund321123Similarly you can perform the same procedure with 111, and so on. Let’s look at an example:46731 111 or111 x 46731Ones Digit of x is equal to the Ones Digit of 46731:Tens Digit of x is equal to:Hundreds Digit of x is equal to:Answer:3 1 xtens7 2 1 xhund124421The hardest part of the procedure is knowing when to stop. The easiest way I’ve found is to think abouthow many digits the answer should have. For example, with the above expression, we are dividing a 5-digitnumber by a roughly 100, leaving an answer which should be 3-digits, so after the third-digit you know you8

are done.The following are some more practice problems to familiarize you with the process:Problem Set 1.2.1.:1. 11 54 18. 87 111 2. 11 72 19. 286 11 3. 11 38 20. 111 53 4. 462 11 21. 297 11 5. 11 74 22. 2233 11 6. 66 11 23. 198 11 7. 1.1 2.3 24. 297 11 8. 52 11 25. 111 41 9. 246 11 26. 111 35 10. 111 456 27. 111 345 11. 198 11 28. 2003 111 12. 357 11 29. 3 5 7 11 13. 275 11 30. 121 121 14. 321 111 31. 33 1111 15. 1.1 .25 32. 22 32 16. 111 44 33. 36963 111 17. 374 11 34. 20.07 1.1 9

35. 11% of 22 is:48. 55 33 % (dec.)36. 13 121 49. (*) 32 64 16 48 37. 27972 111 50. 2002 11 38. 2006 11 51. 77 88 39. 114 52. (*) 44.4 33.3 22.2 40. 33 44 53. 11 11 11 11 41. 2 3 11 13 54. 25553 1111 42. 121 22 55. 11 13 42 43. 44 55 56. 1111 123 44. 2 3 5 7 11 57. 11 7 5 3 2 45. 2553 111 46. 114 121 58. 121 124 47. 44 25 11 59. (*) 33 44 55 1.2.2Multiplying by 101 TrickIn the same spirit as the multiplying by 11’s trick, multiplying by 101 involves adding gap connected digits.Let’s look at an example:438 101 Ones:Tens:Hundreds:Thousands:Tens Thousands:Answer:1 81 31 4 1 81 3 11 444238831244So you see, immediately you can write down the ones/tens digits (they are the same as what you aremultiplying 101 with). Then you sum gap digits and move down the line. Let’s look at another example:8234 101 Ones/Tens:Hundreds:Thousands:Tens Thousands:Hundred Thousands:Answer:10342 48 32 188316343461138

Problem Set 1.2.21. 1234 101 6. 202 123 2. 10.1 234 7. If 6 balls cost 6.06, then 15 balls cost: 3. 369 101 8. 404 1111 4. 34845 101 9. (*) (48 53) 151 5. 22422 101 1.2.310. (*) 8888 62.5% 511 Multiplying by 25 Trick100The trick to multiplying by 25 is to think of it as. So the strategy is to take what ever you are multiplying4with, divide it by 4 then move the decimal over to the right two places. Here are a couple of examples:84 100 21 100 21004166 100 41.5 100 4150166 25 484 25 In a similar manner, you can use the same principle to divide numbers by 25 easily. The difference is youmultiply by 4 and then move the decimal over to the left two places415415415 41660 16.6100251001004Problem Set 1.2.31. 240 25 7. 25 147 2. 25 432 8. 418 25 3. 2.6 2.5 9. 616 25 4. 148 25 10. 2.5 40.4 5. 25 33 11. 1.1 2.5 6. 64 25 12. 3232 25 11

13. (*) 97531 246 14. Which is smaller:23. 2006 25 6or .25?2524. 25 307 15. 209 25 125. 32 is 2 % of:216. (18 16)(9 16) 26. (*) 47985 246 17. (*) 334455 251 27. 25 2003 18. 21.4 is% of 25.28. 15 25 11 19. 404 25 29. 11 24 25 20. 303 25 30. 11 18 25 21. (*) 97531 246 22. Which is larger:1.2.47or .25?2531. (*) 248 250 252 Multiplying by 75 trick3In a similar fashion, you can multiply by 75 by treating it as · 100. So when you multiply by 75, first divide4the number you’re multiplying by 4 then multiply by 3 then move the decimal over two places to the right.76 · 3· 100 19 3 100 5700442 · 342 75 · 100 10.5 3 100 3150476 75 Again, you can use the same principle to divide by 75 as well, only you multiply by(or move the decimal place over two digits to the left).8127 · 48181 · 4 1.08 3 · 100753 · 1001004Problem Set 1.2.41. 48 75 4. 84 75 2. 64 75 5. (*) 443322 751 3. 66 75 124then divide by 1003

6. 28 75 11. 96 75 7. 75 24 12. 75 11 24 8. (*) 7532 146813. 4800 75 9. 48 75 14. 75 48 15 10. (*) 566472 748 1.2.515. 8.8 7.5 1.1 Multiplying by Any Fraction of 100, 1000, etc.You can take what we learned from the 25’s and 75’s trick (converting them to fractions of 100) with a1variety of potential fractions. ’s are chosen often because:835137.5 · 1006.25 · 10125 · 10008881111’s, ’s, ’s, and sometimes even’s for approximation problems (because they do63912not go evenly into 100, 1000, etc., they have to be approximated usually).In addition, you see223 2· 100098333.3 51· 10000 · 100000612327 1· 10003For approximations you will rarely ever see them equate to almost exactly to the correct fraction. For2example you might use · 1000 for any value from 654 678. Usually you can tell for the approximation3problems what fraction the test writer is really going for.Problem Set 1.2.51. 125 320 8. (*) 774447 111 2. (*) 8333 24 9. (*) 62.5 3248 3. 138 125 10. 12.5 480 4. (*) 57381 128 11. (*) 17304 118 5. (*) 245632 111 12. (*) 87% of 5590 6. (*) 16667 8333 555 13. (*) 457689 111 7. 625 320 14. (*) 625 648 13

15. 375 408 36. (*) 123% of 882 16. (*) 359954 1111 37. (*) 95634 278 17. 88 12.5 .11 38. (*) 273849 165 18. (*) 719 875 39. (*) 5714.28 85 19. (*) 428571 22 40. (*) 9.08% of 443322 20. (*) 85714.2 714.285 41. (*) 8333 23 21. 488 375 42. .125 482 22. (*) 6311 1241 43. (*) 714285 .875 23. (*) 884422 666 44. (*) 87% of 789 24. (*) 106.25% of 640 45. (*) 16667 49 25. (*) 6388 3.75 46. (*) 123456 111 26. 240 875 47. (*) 875421 369 27. (*) 12.75 28300 102 48. (*) 71984 1.371 28. 375 24.8 49. (*) 63% of 7191 29. (*) 857142 427 50. (*) 5714.28 83 30. .0625 .32 51. (*) 1428.57 62 31. (*) 16667 369 52. (*) 80520 131 32. (*) 918576 432 53. (*) 142.857 428.571 33. (*) 456789 123 54. (*) 12509 635 34. (*) 106% of 319 55. (*) 1234 567 35. (*) 571428 .875 56. (*) 789123 456 14

57. 625 65 71. (*) 416666 555 76 58. (*) 1428.57 73 72. (*) 375 833 555 59. (*) 7142.85 34.2 173. (*) 438 9 11% 11.1 60. (*) 333 808 444 74. (*) 857142 428571 7777 61. (*) 571428 34 5% 10.8 75. (*) 546 45 1162. (*) 833 612 76. (*) 54.5454 66.6 58 63. (*) 8333 (481 358)77. (*) 456 18.75% 64. (*) 234678 9111 78. (*) 818 44 94 % 12.5 65. (*) 428.571 87.5 79. (*) 62.5 83.3 888 66. (*) 375.1 83.33 1.595 80. (*) 797 87.5% 71067. (*) 8333 6666 4444 81. (*) 888 87.5% 71168. (*) 8333 12 12 % .12 82. (*) 1250 1666 4444 69. (*) 639 375 28 83. (*) 85858 585 70. (*) 6250 8333 8888 84. (*) (51597 147)2 1.2.614Double and Half TrickThis trick involves multiplying by a clever version of 1. Let’s look at an example:2 15 78278 (15 2) 2 30 39 117015 78 So the procedure is you double one of the numbers and half the other one, then multiply. This trick isexceptionally helpful when multiplying by 15 or any two-digit number ending in 5. Another example is:35 42 70 21 1470It is also good whenever you are multiplying an even number in the teens by another number:18 52 9 104 93615

or14 37 7 74 518The purpose of this trick is to save time on calculations. It is a lot easier to multiply a single-digit numberthan a two-digit number.Problem Set 1.2.610. 18 112 1. 1.5 5.2 2. 4.8 15 11. 27 14 3. 64 1.5 12. 21 15 14 4. 15 48 13. 33.75 1.5 5. 14 203 14. 345 12 6. 14 312 15. 1.2 1.25 7. 24 35 8. 312 14 16. 24% of 44 9. A rectangle has a length of 2.4and a width of 1.5. Its area is1.2.717. 14 25 12.5 28 Multiplying Two Numbers Near 100Let’s look at two numbers over 100 first.Express n1 (100 a) and n2 (100 b) then:n1 · n2 (100 a) · (100 b) 10000 100(a b) ab 100(100 a b) ab 100(n1 b) ab 100(n2 a) ab1. The Tens/Ones digits are just the difference the two numbers are above 100 multiplied together (ab)2. The remainder of the answer is just n1 plus the amount n2 is above 100, or n2 plus the amount n1 isabove 100.103 108 8 3103 8 or 108 3Tens/Units:Rest of Answer:Answer:Now let’s look at two numbers below 100.162411111124

n1 (100 a) and n2 (100 b) so:n1 · n2 (100 a) · (100 b) 10000 100(a b) ab 100(100 a b) ab 100(n1 b) ab 100(n2 a) ab1. Again, Tens/Ones digits are just the difference the two numbers are above 100 multiplied together (ab)2. The remainder of the answer is just n1 minus the difference n2 is from 100, or n2 minus the differencen1 is from 100.97 94 Tens/Ones:Rest of Answer:Answer:(100 97) (100 94) 3 697 6 or 94 318919118Now to multiply two numbers, one above and one below is a little bit more tricky.Let n1 (100 a) which is the number above 100 and n2 (100 b) which is the number below 100, then:n1 · n2 (100 a) · (100 b) 10000 100(a b) ab 100(100 a b) ab 100(100 a b 1) (100 ab) 100(n1 b 1) (100 ab)To see what this means, it is best to use an example:103 94 100 3 6103 6 1Tens/Ones:Rest of Answer:

be competing in Number Sense for just a few years in Middle and High School, the skills you acquire will last a lifetime.-Bryant Heath 5. 1 Numerical Tricks 1.1 Introduction: FOILing/LIOFing When Multiplying Multiplication is at the