9.5: Revenue problems - Mathematics

9.5: Revenue problems - Mathematics

5th Grade Math Curriculum Topics, Word Problems, Worksheets with Answers, Practice Tests

In 5th Grade Math Problems, we have explained all types of topics with solutions. Keeping in mind students’ mental level we took all efforts and introduced new concepts in a simple language. Mathematical Concepts are explained in the simplest possible way so that you can retain the concepts for a long time. Each Topic is provided with various examples and you can understand the applications easily.

In Grade 5 Math you will find the concepts on Roman Numerals, Integers, Decimals, Rounding Numbers, Percentage, Profit and Loss, Simple Interest, Quadrilaterals, Data Handling, etc. Grade 5 Worksheets provided helps teachers and students track the child’s learning progress. Develop efficient ways to tackle the Fifth Grade Math Topics and clear the exams with better grades.

Interpreted in hexadecimal: A-5=5

move the center bar of the 9 to the lower left position of that digit to make a 0

look at it from the other side of the table (i.e. rotate it 180 degrees), so it appears to say $5=5-0$

Move the upper-left stick of the 9 to cross the equals sign so we get 3 - 5 ≠ 5

This is the first thing that came to mind (when the question was originally posed as "Move one matchstick to make an equation"):

moving the top match from the first 5

I'm not familiar with the process by which y'all are rendering these (some type of MatchstickML based on TeX and MathJax?) But as there is no specified alignment and you assume they may only be vertical or horizontal, if you move the U+002D matchstick to the initial 9 , one could consider that that a crossed or dotted zero, (as all zeros should be), leaving something akin to

(Something akin to padding a simple 8-bit decimal in UTF-16 with a BOM, for ridiculous encodings like that.


Now that you have seen how Cole’s incorrect problems were analyzed, it is time for you to practice analyzing Student 2’s mathematics errors. Click here to begin.

For this activity, click on each problem you wish to mark incorrect. For each incorrect response, analyze the correct and incorrect digits and try to determine the student’s error pattern(s). Next, in the description box below the probe, fill in the type(s) of errors Student 2 is making. After doing this, check your answer by clicking on the feedback link.

9.5: Revenue problems - Mathematics

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Understanding Mathematics

I wrote this page for students at the University of Utah. You may find it useful whoever you are, and you are welcome to use it, but I'm going to assume that you are such a student (probably an undergraduate), and I'll sometimes pretend I'm talking to you while you are taking a class from me.

Let's start by me asking you some questions. If you are interested in some suggestions, comments, and elaborations, click on the Comments. Do so in particular if you answered "Yes!". (In making the comments I assume you did say "Yes!", so don't be offended if you didn't and are just curious.)

  • Do you feel
    • That being lost in mathematics is the natural state of things? Comments.
    • That lectures and textbooks are incomprehensible? Comments.
    • That the amount of material in any math course is so overwhelming that you (or anybody else) could not possibly absorb it? Comments.
    • That mathematics is just a collection of formulas and theorems that one somehow has to cram into one's head? Comments.
    • That the solution of problems requires a collection of tricks whose conception was based on a generous allowance of magic? Comments.
    • That math courses are just hurdles one has to cross as an undergraduate student? Comments.
    • That mathematics is irrelevant? Comments.

    If your answer to all of these questions is a resounding "No!" then you should read no further and return to the study of mathematics. I'd also like to meet you, please drop me an e-mail! Otherwise I'm hoping that this page has something to offer you (and of course you may send me an e-mail anyway). I believe that many students struggle with mathematics only because they don't know what it means to understand Mathematics and how to acquire that understanding.

    The purpose of this page is to help you learn how to approach mathematics in a more effective way.

    Understanding Mathematics

    • Explain mathematical concepts and facts in terms of simpler concepts and facts.
    • Easily make logical connections between different facts and concepts.
    • Recognize the connection when you encounter something new (inside or outside of mathematics) that's close to the mathematics you understand.
    • Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

    By contrast, understanding mathematics does not mean to memorize Recipes, Formulas, Definitions, or Theorems.

    Clearly there must be some starting point for explaining concepts in terms of simpler concepts. That observation leads to deep and arcane mathematical and philosophical questions and some people make it their life's work to think about these matters. For our purposes it suffices to think of elementary school math as the starting point. It is sufficiently rich and intuitive.

    All of this is neatly summarized in a letter that Isaac Newton wrote to Nathaniel Hawes on 25 May 1694.

    People wrote differently in those days, obviously the " vulgar mechanick" may be a man and "he that is able to reason nimbly and judiciously" may be a woman, (and either or both may be children).

    for complicated mathematics building on simpler mathematics.

    The following examples illustrate the difference between the two approaches to understanding mathematics described above.

    You won't be able to learn how to understand mathematics from abstract principles and a few examples. Instead you need to study the substance of mathematics. I'm hoping that the answers to the following

    will illustrate how mathematics is meant to make sense and is built on a logical procession rather than a bunch of arbitrarily conceived rules.

    One of the main things that turned me on to mathematics were certain concepts and arguments that I found particularly beautiful and intriguing. I'm listing some of these below even though they may not be frequently asked about. But I'll hope you'll enjoy them, and perhaps get more interested in mathematics for its own sake.

    Solving Mathematical Problems

    The most important thing to realize when solving difficult mathematical problems is that one never solves such a problem on the first attempt. Rather one needs to build a sequence of problems that lead up to the problem of interest, and solve each of them. At each step experience is gained that's necessary or useful for the solution of the next problem. Other only loosely related problems may have to be solved, to generate experience and insight.

    Students (and scholars too) often neglect to check their answers. I suspect a major reason is that traditional and widely used teaching methods require the solution of many similar problems, each of which becomes a chore to be gotten over with rather than an exciting learning opportunity. In my opinion, each problem should be different and add a new insight and experience. However, it is amazing just how easy it is to make mistakes. So it is imperative that all answers be checked for plausibility. Just how to do that depends of course on the problem.

    There is a famous book: G. Polya, "How to Solve It ", 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6. It was first published in 1945. This is a serious attempt by a master at transferring problem solving techniques. Click here to see an html version of Polya's summary.

    The main thing that keeps mathematics alive and interesting of course are unsolved problems . Many open problems that are "important" in the contemporary view are hard just to understand. But here are

    for which you can form your own conjectures. The word " simple" in this context means that the problem is easy to state and the question is easy to understand. It does not mean that the problem is easy to solve. In fact all of these open problems are difficult. (That's why they are unsolved, it's not that nobody tried!)

    Acquiring Mathematical Understanding

    Since this is directed to undergraduate students a more specific question is how does one acquire mathematical understanding by taking classes? But that does not mean that classes are the only way to learn something. In fact, they often are a bad way! You learn by doing. For example, it's questionable that we should have programming classes at all, most people learn programming much more quickly and enjoyably by picking a programming problem they are interested in and care about, and solving it. In particular, when you are no longer a student you will have acquired the skills necessary to learn anything you like by reading and communicating with peers and experts. That's a much more exciting way to learn than taking classes!


    A student was asked to find the arithmetic mean of the numbers 3, 11, 7, 9, 5, 3, 8, 19, 17, 21, 14 and x. He found the mean to be 12. What should be the number in place of x ?    

    Given :  Average of  3, 11, 7, 9, 5, 3, 8, 19, 17, 21, 14 and x is 12. 

    Sum of all the given numbers / number of numbers  =  12

    (3 + 11 + 7 + 9 + 5 + 3 + 8 + 19 + 17 + 21 + 14 + x) / 12  =  12

    Multiply both sides by 12. 

    Subtract 117 from both sides. 

    So, the number should be in the place of 'x' is 27. 

    The average age of 30 kids is 9 years. If the teacher's age is included, the average age becomes 10 years. Find the teacher's age.   

    Given :  The average age of 30 kids is 9 years.

    Total age of 30 kids / 30  =  9

    Multiply both sides by 30. 

    Total age of 30 kids  =  9  ⋅ 30

    Total age of 30 kids  =  27 0

    Given :  If the teacher's age is included, the average age becomes 10 years

    (Total age of 30 kids  + Age of the teacher) / 31  =  10

    (270  + Age of the teacher) / 31  =  10

    Multiply both sides by 31. 

    270  + Age of the teacher  =  10  ⋅  31

    270  + Age of the teacher  =  310

    Subtract 270 from both sides.

    Age of the teacher  =  40 years

    The average of 6 numbers is 8. What is the 7 th number , so that the average becomes 10 ?   

    Given :  The average 6 numbers is 8.

    Given :  If the 7 th number is included, the average becomes 10. 

    (Sum of 6 numbers  +  7 th  number) / 7  =  10

    Subtract 48 from both sides.

    David's average score in the last 9 tests is 80. What should be his score in his next test, so that his average score will be 82 ?   

    Given :  The average score of 9 tests is 80.

    Sum of scores in 9 tests / 9  =  80

    Sum of scores in 9 tests  =  80  ⋅ 9

    Sum of scores in 9 tests  =  720

    Let "x" be his score in his next test. 

    Given : Average score of 10 tests is 82.

    (Sum of scores in 9 tests  + x) / 10  =  82

    Multiply both sides by 10. 

    Subtract 720 from both sides.

    So, David score in the next test should be 100. 

    In Kevin's opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Kevin and he thinks that Kevin's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight is less than 68 kg. If all of them are correct in their estimation, what is the average of different probable weights of Kevin ?   

    Let Kevin's weight be "x" kg.

    According to Kevin, we have

    According to Kevin's brother, we have

    According to Kevin's mother, we have

    The values of 'x' which satisfy all the above inequalities are 66 and 67. 

    So, the different probable weights of Kevin are 66 kg and 67 kg. 

    Average of 66 and 67  =  (66 + 67) / 2

    Average of 66 and 67  =  133 / 2

    Average of 66 and 67  =  66.5

    So, the average of different probable weights of Kevin is 66.5 kg.

    Apart from the stuff given in this section, if  you need any other stuff in math, please use our google custom search here.

    If you have any feedback about our math content, please mail us : 

    We always appreciate your feedback. 

    You can also visit the following web pages on different stuff in math. 

    Section 9. Methods of Proof

    (1) This transmits revised IRM 9.5.9, Methods of Proof.

    Material Changes

    (1) IRM 9.5.9, Methods of Proof has been revised throughout to delete Personally Identifiable Information (PII) and/or Federal Tax Information (FTI). Specifically, Exhibits 9.5.9-1 through Exhibits 9.5.9-6 and all references to them have been deleted.

    Effect on Other Documents


    Effective Date

    James C. Lee FOR
    Don Fort
    Chief, Criminal Investigation


    This section will explain the various methods of proof available to the special agent in determining a subject ’s correct taxable income, and how to properly document each method of proof. The following methods of proof will be discussed in this section:


    "Proof" is the establishment by evidence of a requisite degree of belief concerning a fact in the mind of the trier of fact or the court. Proof is the logically sufficient reason for assenting to the truth of a proposition advanced. In its judicial sense, it is a term of wide import and encompasses everything that may be adduced at a trial, within the legal rules, for the purpose of producing a conviction in the mind of judge or jury.

    "Evidence" is a much narrower term. It includes only such proof as is admissible at trial by the act of the parties or through such concrete facts as witnesses, records, or other documents. Proof is the end result or effect of evidence, while evidence is the medium or means by which a fact is proved or disproved.

    Direct evidence proves a fact, without an inference or presumption, and conclusively establishes that fact without reference to any supporting evidence. Direct evidence is evidence of the precise fact in issue and is distinguished from circumstantial i.e., "indirect, " evidence.

    Tax crimes are often acts of individual greed and, therefore, very little "direct evidence" is usually available. Depending on the facts and circumstances of each investigation, the subject’s correct taxable income may be established by " direct" or several "indirect" methods of proof, usually using circumstantial or "indirect" evidence.

    Direct (Specific Item) Method

    Among the various methods of proving unreported or underreported taxable income, the specific item method is the most preferred. Most subjects report their income and expenses by the specific item method using books and/or records in which their financial transactions are contemporaneously recorded. Their transactions are usually summarized and shown on the tax return.

    There are three broad categories of schemes which are suited to the specific item method of proof:

    overstatement of expenses

    fraudulent claims for credits or exemptions

    A false tax return may include any or all of these schemes. Unreported Income can be proved using the basic or aggregate approaches discussed in the following subsections:

    see IRM, How to Use the Specific Item Method of Proof

    see IRM, Basic Approach

    see IRM, Aggregate Approach

    Small amounts of expenses claimed on the false return sometimes have to be allowed or accepted because the special agent is unable to properly trace or document the actual amounts, or he/she lacks the time to do so.

    Indirect Methods

    Indirect methods require the special agent to gather and present evidence to support the allegation. The special agent will use evidence to determine what income should have been reported on the subject ’s return and compare that to the amount shown on the return, if a return was actually filed.

    Sources of income may not be identifiable, as in a specific item method of proof. Therefore, taxable income often has to be computed indirectly based upon the taxpayer’s application or use of funds.

    The courts have upheld the use of the net worth, expenditures, bank deposits and cash methods of proving income, on the theory that proof of unexpended funds or assets may establish a prima facie understatement of income which requires a defendant to overcome the logical inference drawn therefrom.

    With respect to the establishment of a prima facie investigation by such evidence, courts have been careful to point out that findings of fraud have been sustained if, but only if, the defendant has offered no adequate explanation of the discrepancies between (on the one hand) expenditures, bank deposits, and increases in net worth and (on the other hand) the amount of income reported by the defendant.

    Another indirect method of proof is the percentage markup method of proof. Pending the establishment of judicial precedents, the percentage markup method of proof should only be used as a primary method of proof, on a limited basis, and not used to corroborate other methods. ( See IRM, Percentage Markup Method of Proving Income.)

    The unit and volume method of proof can be utilized when the number of units handled and the price or profit charged per unit is known.

    Distinguishing Between Accounting Systems, Accounting Methods, and Methods of Proving Income

    For many years, there has been much confusion regarding the synonymous use of the terms "accounting system, " "accounting methods," and "methods of proving or determining income." It is not unusual to hear reference made to the net worth and expenditures method as a method of accounting when, in fact, it is a method of proving income.

    There are two basic accounting systems, i.e., the single entry system and the double entry system, but there are various methods of accounting, e.g., accrual, hybrid, installment, and long-term or completed contract methods. The most frequently used methods of proving or determining income are the specific item, net worth, expenditures, bank deposits, cash and percentage markup methods of proof.

    For the purpose of criminal prosecution, taxable income must be computed by way of the accounting method regularly used by the subject to compute his/her income. In Morrison v. United States, 270 F. 2d 1(4th Cir. 1959), it was necessary to establish not only that the tax liabilities at issue were understated, but that the understatement was attributable, at least in part, to the fact that the subject’s returns were not honestly prepared. Proof of the latter fact could only be accomplished by adopting and consistently applying the subject’s method of accounting.

    If no method of accounting has been regularly used by the subject, or if the method used does not clearly reflect income, special agents may use whichever method they believe clearly reflects the subject’s income. Whatever method is used, it must be used for all prosecution years.

    Specific Item Method of Proving Income

    Where the government is using the specific item method of proof (in an investigation of alleged tax evasion), the government attempts to document specific transactions that were not completely or accurately reflected on the subject’s income tax return. Additionally, the government must show that the specific omissions of income were made willfully for the purpose of understanding the subject’s income tax liability.

    The specific item method offers the most direct method of proving unreported income. The specific item method is the preferred method of proving income because it is the easiest to understand, include in a prosecution report, and present at trial. Additionally, the specific item of proof method is the hardest for the subject to rebut.

    Types of Evasion

    Omitted income, fictitious deductions, false exemptions, or false tax credits are means whereby taxes may be evaded.

    Omitted income results from a subject ’s failure to report any of the numerous items of taxable income set forth in the Internal Revenue Code (IRC).

    During investigations of proprietors and/or their businesses, sales or gross receipts are the most frequently encountered item of omitted income.

    During investigations of individuals, omitted income is frequently encountered in the form of salaries, interest, dividends, commissions, gains from the sale of property, and fees.

    Overstatement of expenses results from the subject ’s attempt to reduce taxable income by claiming false or inflated expenses. During investigations of proprietors and/or their businesses, overstated expenses can be hidden in any expense reported on the tax return. During investigations of individuals, the overstated expenses are most frequently encountered on Schedules A, C, D, and F. Additionally, subjects attempt to evade income taxes by claiming false deductions and exemptions. In all of the investigations described above, the specific item method of proof is ideally suited to proving the violation.

    Unreported Income from Certificates of Deposit

    There are two types of certificates of deposit, i.e., a standard certificate of deposit and an original issue discount certificate.

    Standard Certificate of Deposit

    A standard certificate of deposit pays interest at specific intervals over the term of the note.

    Although the interest may be withdrawn without penalty, the principal normally may not be withdrawn without incurring a substantial penalty.

    Financial institutions issue Forms 1099 INT to the owner reflecting the interest earned.

    An Original Issue Discount Certificate

    An original issue discount certificate pays interest only upon the note’s maturity.

    Title 26 USC §1272 requires holders of this type of certificate of deposit to report the interest earned on the basis of a constant interest rate.

    Title 26 USC §6049 contains specifics as to when Form 1099–Original Issue Discount Certificates (OID) will be issued to holders of such certificates.

    Department of Justice, Tax Division’s Position on Original Issue Discount Certificates

    The position of the Department of Justice (DOJ), Tax Division is that, except for unusual circumstances, it will not recommend the prosecution of criminal investigations which are based upon the subject ’s failure to report interest from original issue discount certificates before maturity, except under unusual circumstances.

    In these investigations, a willfulness issue usually arises from the subject’s lack of actual possession, use, and enjoyment of the interest during the holding period.

    Similar problems should not arise in investigations involving standard certificates of deposit when the interest is made available to the subject.

    When preparing a prosecution report proving the omission of income from a certificate of deposit, the special agent must properly identify the type of certificates of deposit in question. Copies of Forms 1099, as well as copies of the actual certificates of deposit, must be exhibited in the prosecution report.

    In addition, the prosecution report should address the issue of willfulness by discussing whether the principal and interest on a matured certificate was rolled over into a new certificate of deposit, whether premature withdrawal of the principal is subject to penalties, and whether there was a premature redemption.

    When to Use the Specific Item Method of Proof

    Specific omissions of gross income are most easily shown when the subject has a small number of significant sources of income. During the investigation, the special agent determines the specific amount of income reported from each source and then compares those figures with the total amount of income documented in the subject’s books and records, and reported on his/her tax return. The following examples illustrate the appropriate use of the specific item method of proof:

    While investigating a physician, the special agent found all receipts from patients had been reported but that amounts paid by insurance companies on behalf of patients were omitted. By contacting the various insurance companies, specific omissions of income were determined.

    While investigating a self-employed subject who re-upholstered furniture, the special agent noted an inconsistency between reported income and living expenses. An analysis of reported gross receipts showed the subject reported small amounts of income received from individual customers. An analysis of the subject’s bank records showed checks deposited from a large department store. Further investigation revealed that the subject failed to report substantial income earned on a contract basis with the department store.

    When the subject of an investigation generates small amounts of income from numerous customers or clients, as would be the investigation with subjects owning bars, restaurants, and grocery stores, it is difficult, if not impossible, to match reported amounts of income with specific sources of income. In these situations the specific item method may not be the best method to use indirect methods may be more appropriate.

    How to Use the Specific Item Method of Proof

    There are two approaches to the specific item method of proof, i.e., the basic approach and the aggregate approach. Depending upon the facts and circumstances of the investigation, the special agent will use one of the two approaches to prove unreported income.

    The basic approach to the specific item method of proof requires the special agent to trace the reported items of income through the subject’s books and records to the tax return. Upon doing so, the special agent can specifically identify the unreported income items.

    The aggregate approach to the specific item method of proof simply requires that the special agent identify the total amount of income the subject should have reported in any given year. The special agent then compares the total amount of income with the aggregate amount of income reported on the return, and arrives at an understatement of income.

    Basic Approach

    The basic approach to the specific item method of proof involves a two step process:

    I. First, the special agent identifies the sources and amounts of reported income and expenses shown on the tax return by reconciling them to the subject ’s records.
    II. Second, he/she determines the correct amounts of income, expenses, and credits using the subject’s records, bank records, investment account records, and/or contacts with third parties and compares the correct amounts to those reported on the tax return. The comparison will yield specific items of unreported income and false or inflated expenses or credits.

    The reconciliation of reported gross receipts to the subject’s records. In examining the return of a self-employed geologist, the special agent reconciles reported gross receipts with the subject’s records as follows:

    Source Date Amount Total
    Actor Company 01/27/1998 $3,000.00
    Actor Company 08/14/1998 $5,000.00
    Barber Company 01/05/1998 $6,000.00
    Barber Company 03/20/1998 $4,000.00
    Barber Company 06/14/1998 $2,000.00
    Chef Company 05/01/1998 $2,500.00
    Chef Company 07/22/1998 $1,500.00
    Total Reported Schedule C Gross Receipts $24,000.00

    Determine specific items of omitted income. By contacting each of the subject’s three reported clients in the above example, the special agent was able to determine the correct income from those clients. An analysis of bank records disclosed a fourth customer, which the special agent also contacted.

    Source Date Check No. Amount Reported Amount Omitted
    Actor Company 01/27/1998 4517 $3,000.00
    Actor Company 06/09/1998 6248 $17,000.00
    Actor Company 08/14/1998 9704 $5,000.00
    Barber Company 01/05/1998 204 $6,000.00
    Barber Company 03/20/1998 413 $4,000.00
    Barber Company 06/14/1998 785 $2,000.00
    Barber Company 10/14/1998 1032 $19,000.00
    Chef Company 05/01/1998 817 $2,500.00
    Chef Company 07/22/1998 1042 $1,500.00
    Chef Company 11/14/1998 1324 $2,000.00
    Driver Company 12/02/1998 205 $13,000.00
    Total Reported Schedule C Gross Receipts $24,000.00
    Total Omitted Schedule C Gross Receipts $51,000.00
    Aggregate Approach

    When it is not possible to specifically identify the items of income which were not reported on a subject’s tax return, due to a lack of accurate books and records, the special agent may use the aggregate approach to the specific item method of proof in calculating the subject’s correct taxable income. This approach requires that the special agent specifically identify all of the subject ’s items of income and then compare that amount to the subject ’s total reported taxable income. For example, if the subject ’s return shows gross receipts of $150,000, the special agent may develop a specific item investigation by showing through third-party documentation that the subject has actually received $200,000 in gross receipts during the same period. The special agent does not have to identify the specific items of income that were not reported ($50,000) as he/she has specifically identified the individual items that make up the gross receipts and determined that amount exceeds the aggregate amount of gross receipts reported on the subject ’s income tax return. The following example will illustrate the aggregate approach to the specific item method of proof:

    Gross Receipts Reported $150,000
    Receipts Documented by Third Party Contacts:
    Witness A $ 50,000
    Witness B 100,000
    Witness C 50,000
    Corrected Gross Receipts $200,000
    Less: Reported Receipts (150,000)
    Equals: Unreported Gross Receipts $ 50,000

    As shown by the above example, the special agent can use the aggregate approach and prove that gross receipts are understated without examining the subject’s books and records. However, if the subject’s books and records are available, the special agent must attempt to reconcile them to the tax return.

    The basic approach to the specific item method of proving income should be used whenever possible. The aggregate approach to the specific item method of proof should only be used when specific sources and amounts of income reported on a tax return cannot be identified.

    Net Worth Method of Proof

    An investigation utilizing the net worth method of proof differs from a specific item method in that direct comparisons of income, expenses, and credits can not be made. The net worth method of proof utilizes evidence of income applications such as asset accumulation, liability reduction, expenditures, and other financial data to indirectly establish correct taxable income.

    An accounting is made showing how funds generated from income were applied by identifying increases to net assets and various expenditures.

    After making adjustments for exemptions, itemized deductions, nontaxable income, and nondeductible losses, the courts permit the IRS to infer, indirectly, that the remainder is taxable income.

    By comparing this to taxable income reported on the subject’s return, if a return was actually filed an understatement of taxable income can be determined.

    The net worth method is a very effective way of proving taxable income in criminal income tax investigations. The formula for calculating the subject’s correct taxable income can be broken down into four steps:

    The special agent must first calculate the change in a subject’s net worth (assets less liabilities). This is done by determining the subject’s net worth at the beginning and end of a period of time (a taxable year or years) and then subtracting the beginning period’s net worth figure from the ending period’s net worth figure. This computation will yield a change in net worth (either an increase or decrease in net worth).

    The amount of this change in net worth is then adjusted for personal living expenses, nondeductible losses, and nontaxable items to arrive at a corrected adjusted gross income figure.

    The corrected adjusted gross income figure is then adjusted for itemized deductions or the standard deduction amount, and then for exemptions, to arrive at a corrected taxable income figure.

    Finally, by comparing the corrected taxable income figure with the taxable income reported on the tax return, the special agent can determine whether the subject failed to report any taxable income.

    Authority for Net Worth Method

    There is no statutory provision defining the net worth method and specifically authorizing its use by the Commissioner. However, every judicial circuit has endorsed the net worth method of proof and the Supreme Court has approved its use in a number of investigations. The following is a listing of some of the more prominent of those investigations:

    Holland v. United States, 348 US 121 (1954)

    Friedberg v. United States, 348 US 142 (1954)

    Smith v. United States, 348 US 147 (1954)

    United States v. Calderon, 348 US 160 (1954)

    Massei v. United States, 355 US 595 (1958)

    United States v. Johnson, 319 US 503 (1943)

    These investigations outline the broad principles governing the prosecution and review of investigations based on the net worth method of proving income.

    Legal Requirements to Establish a Prima Facie Net Worth Investigation

    The Supreme Court, while firmly approving the net worth method of proof, cautioned, in Holland v. United States, 348 US 121, 125 (1954), that " it is so fraught with danger for the innocent that the courts must closely scrutinize its use."

    The Supreme Court set forth three requirements that the government must satisfy prior to using the net worth method of proof:

    establish an opening net worth with reasonable certainty

    negate reasonable explanations by the subject inconsistent with guilt

    establish that the net worth increase is attributable to currently taxable income - Id. at 132 - 137.

    Net worth increases are determined by establishing a net worth at the beginning of a given year and then comparing this beginning net worth with the net worth at the end of the year. The opening net worth is the point from which net worth increases are measured. While every effort should be made to identify all of the assets and liabilities of the subject at the starting point, the government does not have to establish the opening net worth with mathematical certainty.

    Without a doubt, determining how much cash an individual has "on hand" at the beginning or end of a year is an extremely difficult task. To require mathematical certainty would eliminate the possibility of using the net worth method of proof.

    The thoroughness of the investigation is crucial in determining whether the government has established the subject ’s opening net worth with reasonable certainty. When the government chooses to proceed against a subject using the net worth method of proof, " the government assumes special responsibility of thoroughness and particularity in its investigation and presentation." United States v. Hall, 650 F. 2d 994, 999 (9th Cir. 1981).

    Success in overcoming attacks on the legal sufficiency of the evidence supporting an opening net worth is directly related to the extent and thoroughness of the investigation. Although not a model, the Mastropieri investigation does furnish an excellent example of a number of steps that must be taken to establish an opening net worth. US v Mastropieri, 685 F. 2d 776, 779 (1982). For example, in Mastropieri:

    The special agent canvassed 47 banks, 71 brokerage firms, and 13 lending institutions. In addition, the special agent searched the local property records of various counties for the years during the investigation and prior to 1967.

    The special agent checked records of the IRS and the county clerk and interviewed unnamed friends and relatives of the subject.

    When to Use the Net Worth Method

    The net worth method of proof is most often used when one or more of the following conditions exist:

    the subject maintains no books and records

    books and records are not available

    books and records are inadequate

    subject withholds books and records

    The fact that the subject’s books and records accurately reflect the figures on the return does not prevent the use of the net worth method of proof. The government can look beyond the self-serving declarations in the subject’s books and records and use any evidence available to refute the accuracy thereof.

    In addition to being used as a primary method of proving taxable income in civil and criminal income tax investigations, the net worth method can be used:

    to corroborate other methods of proving income

    to verify accuracy of reported taxable income

    Method of Accounting

    The net worth method of proof is not limited by the subject’s method of accounting. The net worth statement may reflect the subject’s corrected taxable income by whichever method of accounting (cash, accrual, etc.) is appropriate. Reflecting a certain accounting method in the net worth computation is accomplished by including certain accounts in the net worth statement and omitting others. For instance, to compute the income of a physician on the cash basis, patient accounts receivable and business accounts payable at the beginning and end of each year would be omitted. If the physician used the accrual method of accounting, these accounts would be included in the net worth computation.

    In preparing a net worth statement or summary for use in a criminal investigation, special agents should ensure that:

    The subject’s method of accounting is used.

    The cost of assets and actual amounts of liabilities are used and that values other than cost, i.e., market value or reproduction value, are not considered in the net worth computation.

    Estimated nondeductible expenditures are eliminated from the net worth computation, unless the subject agrees to the estimated amount or it is proper to include some minimum estimated personal living expense figures.

    Generally accepted accounting principles are followed.

    Technical adjustments that increase income are eliminated, e.g., unintentional errors or omissions relating to capitalized expenses, depreciation, revaluation of the basis of property, and changing inventory basis, or doubtful items such as unidentifiable commingled funds.

    Overview of the Net Worth Method of Proof Formula

    The net worth formula expanded:

    Personal living expenses (including payments that may later be allowed as itemized deductions or adjustments to arrive at adjusted gross income)

    In determining the value of assets, all assets in the computation are entered at cost or other tax basis. Fluctuations in fair market value are of no consequence in determining taxable income. Paper gains or losses resulting from changes in fair market value of assets are not taxable or deductible until said gain or loss is realized.

    Establishing the Starting Point

    The key to a successful net worth investigation is establishing a reliable beginning net worth (opening net worth) which includes all of the assets and liabilities on hand. It is this starting point from which all future increases or decreases will be calculated. This starting point is normally referred to as the base year. In a net worth computation, it is extremely important to firmly establish a beginning net worth (starting point or base year) with the best evidence available.

    In calculating annual net worth, be aware that an inverse relationship exists between one year and the next. If the subject ’s opening net worth is understated, there is a resulting overstatement of the increase in net worth for the following year. Conversely, if the subject’s opening net worth is overstated, there would be a resulting understatement of the increase in net worth for the following year.

    The first step to establishing a firm starting point is to determine the date (opening or base year) best suited for the investigation. The interview with the subject will strengthen the starting point. While questioning the subject, the special agent should attempt to develop all information relating to the subject’s assets and liabilities for the years involved. The subject should be questioned about the value of any item which cannot be determined from available books and records, e.g., cash on hand as of a particular date, personal living expenses, assets held in the names of others, gifts, inheritances, loans, and other nontaxable sources of income.

    The establishment of cash on hand is critical. The inability to establish a firm and accurate amount of cash on hand can be fatal to the investigation. Uncertainty about the amount of cash on hand is a common defense in net worth investigations. It will be easier to refute this defense if the special agent has established a firm beginning and an ending cash on hand amount is established. Cash on hand is almost always proved by circumstantial evidence.

    The best source of information in establishing an accurate cash on hand figure may be obtained from the subject during an interview. The special agent may not always have the opportunity to interview the subject in every investigation. However, when the opportunity does exists, the special agent should attempt to establish the beginning and ending cash on hand. In determining a firm cash on hand figure, the following subsections offer insight into possible techniques to employ during a subject interview.

    During the subject interview, the subject should be questioned in detail about cash on hand. The questioning should be preceded with an explanation of what constitutes cash on hand and elicit the subject ’s answer as to cash on hand. Cash on hand is coin and currency (bills, Federal Reserve notes, etc.) in the subject’s possession, i.e., on the subject’s person, in the subject’s residence, or other place, in nominee hands, or in a safe-deposit box. It does not include any money the subject has on deposit in any account with any type of financial institution.

    The special agent should use caution in using terms such as cash because people often refer to money on deposit in banks as cash on hand. The special agent should be specific and explain that he/she is referring to undeposited coin and currency in all locations.

    Most people have difficulty recalling specific dates and amounts, especially when several dates are involved, and they extend back for a number of years. Direct questions, such as "How much cash on hand did you have on December 31, ____" will frequently be answered with "I don’t know" or "I can’t remember that far back" . In such investigations, the special agent should persist in questioning about whether the subject had a depository for coins or currency and/or whether the subject placed any coins or currency in the possession of another person. The special agent should obtain a description of the depository. If the depository is a safe-deposit box or home safe, the special agent should relate the questions to when and where the box was rented or purchased. The special agent should obtain a description of the depository and a description of the funds (their denomination and quantity) to determine whether it was possible to have such a sum of money in that particular depository.

    The special agent may determine the amount of cash on hand by asking questions about the maximum amount of cash that the subject could possibly have had at any particular time. For example, such questions as, "Did you ever have more than $100 in cash on hand? More than $5,000? More than $10,000?," may result in admissions that can establish the total amount of cash on hand at a particular date.

    Discussing the accumulation and purpose of the cash on hand may establish the minimum and maximum amount on a particular date. Determining the ultimate disposition of this cash on hand can provide a lead to a specific amount of cash on hand on a particular date. For example, a statement like "I used all my cash on hand to pay for my house in 1994" indicates how much cash the subject had on the date of payment. It also provides a cut-off date for cash on hand, since the subject evidently had no more cash after using all the cash on hand to pay for the house. The special agent should question the subject further to elicit an admission that the subject did not have any additional cash on hand as of the specified date.

    The special agent’s questioning should be directed toward developing:

    the maximum amount of cash on hand (undeposited currency and coin) claimed at the starting point and at the end of each year under investigation

    the amount of cash on hand at the date of the interview (This data is sometimes useful in computing cash on hand for earlier years.)

    how was the cash on hand accumulated and from what sources

    whether anyone ever counted the cash

    when, where and for what was any cash spent

    whether any record is available with respect to the alleged cash on hand

    the denominations of the cash on hand

    was the cash shown on any net worth or personal financial statements

    ask to see the cash on hand

    In addition to questioning the subject about cash on hand, also:

    question the subject about prior years’ earnings

    obtain prior years’ tax returns to determine if no return was filed or if the returns indicate little or no income in prior years

    determine if the subject had financial difficulties prior to the starting point, e.g., compromises of overdue debts by the subject foreclosure procedures against the subject collection actions against the subject, etc.

    obtain copies of financial and or net worth statements

    question the subject as to the contents of any safe-deposit boxes

    question the subject concerning all taxable and nontaxable sources of income

    determine consistent use of checking and savings accounts

    determine if there are recurring overdrafts on non-sufficient funds (NSF) charges or other bank penalties

    determine the minimum payments on any credit card balances

    determine if there was ever a divorce and division of assets

    In addition to interviewing the subject, the following investigative steps should be taken when establishing a firm starting point in a net worth investigation:

    The special agent should interview the subject ’s spouse, relatives, and close associates to determine if the subject received loans, gifts, or inheritances in prior years. The interview of the subject’s spouse should include cash on hand and sources of taxable and nontaxable income so that the subject cannot claim the increases resulted from funds the spouse received.

    The special agent should canvass banks and stockbrokers to determine whether the subject has or had any accounts that could be a source of funds, or whether he/she submitted any financial statements to the financial institution. When reviewing bank records, the special agent should determine whether the subject has ever had checks returned for insufficient funds.

    The special agent should examine financial statements presented for credit or other purposes at a time prior to or during the periods under investigation. The special agent can obtain these types of documents from banks, loan companies, bonding companies, and the other operating divisions of the IRS (offers in compromise and financial statements).

    The special agent should check the following records for potential assets, liabilities, and sources of funds:

    real estate records to determine if the subject owns or has owned property that could be a source of funds
    bankruptcy, foreclosure, and repossession record (If the subject filed for bankruptcy, this could be used as a starting point for net worth)
    divorce records
    social security records for prior earnings and receipt of any funds from social security
    welfare records
    probation records

    The special agent should determine the subject ’s borrowing habits, especially borrowing at high interest rates.

    The special agent should analyze available Federal and state tax returns. Tax returns can be obtained from the IRS, the state where the subject resided, the subject’s accountant and/or return preparer, or financial institutions where the subject has applied for and/or obtained loans.

    In the event the special agent is unable to firmly establish a starting point through the above-described steps, the special agent may have to rely upon an indirect approach to establishing a starting point. This can be accomplished by using a Source and Application of Funds computation.

    An Indirect Approach for Establishing a Starting Point

    Another method of establishing a starting point for cash on hand is to analyze the subject’s available finances for the years leading up to the starting point. Such a "source and application of funds" approach can also be used to bridge the years to the starting point from some point in time when cash on hand has been firmly established. The following is an example of how a source and application of funds computation can be used to establish a firm starting point in a net worth investigation.

    The subject filed bankruptcy in 1993. Immediately following the bankruptcy, the subject did not have any assets or liabilities. The starting point for the investigation is December 31, 1996, the prosecution years are 1997 and 1998. For the purposes of using the source and application of funds computation in determining a firm starting point (cash on hand figure on December 31, 1996), the years 1993 through 1996 would be treated as one unit.

    First, the special agent must determine the total amount of funds available (taxable and nontaxable) during 1993 through 1996. From this amount, he/she will subtract the subject’s personal expenditures for the period. This will yield the maximum amount of funds available for the subject’s net worth at the beginning of 1997.

    Second, the special agent subtracts the subject ’s beginning net worth figure (the amount the investigation revealed as of December 31, 1996, without the cash on hand figure) from the total funds available for net worth. This will account for non-personal living expenditure payments by reflecting the payments made to increase assets and decrease liabilities.

    Funds used to purchase assets disposed of prior to the starting point can be included as funds applied, if their disposition is traced and the funds from the disposition are accounted for as funds available. The advantage of using this method is that the beginning net worth can be used as funds applied. If the subject has a large beginning net worth, it may be possible to overcome the subject’s reported income for prior years and show that he/she could not have had cash on hand at the starting point. This can also be used to establish a maximum possible cash on hand figure. It is important that the subject be given credit for all sources of funds available (both taxable and nontaxable) in the period for which the source and application method is used.

    When using the one unit source and application of funds method to establish a firm starting point, the beginning net worth must be adjusted for any asset purchased and completely paid for prior to the source and application years. This is necessary because no funds were applied during the source and application period to purchase the asset. This point is illustrated in the following example:

    The subject purchased and paid off a residence 10 years prior to the starting point. The cost of the residence $20,000, is included in the beginning net worth. The source and application of funds only covers a period of six years prior to the starting point. The beginning net worth must be adjusted by subtracting the cost of the residence because the residence was purchased with funds acquired by the subject prior to the years included in the computation. This is illustrated as follows:

    Funds available (1991–1996) $105,000
    Less: Funds applied to personal living expenses -50,000
    Equals: Maximum funds available for an increase in net worth 55,000
    Beginning net worth per investigation $72,000
    Less: Cost of residence purchased prior to 1991 -20,000
    Funds applied by the subject to acquire the adjusted beginning net worth $52,000
    Maximum funds available for an increase in net worth $55,000
    Less: Funds applied by the subject to acquire the adjusted beginning net worth -52,000
    Equals: Maximum possible cash on hand at starting point 12/31/1996 $ 3,000

    This method can be used to establish cash on hand at the starting point if the subject does not cooperate during the investigation, or to corroborate the subject’s admission of cash on hand. A source and application of funds cannot be used in every investigation but, in certain instances, can be a valuable tool in determining possible cash on hand.

    Presenting Cash on Hand Figures

    As mentioned earlier, the cash on hand figure is often the most difficult item to establish. Whenever possible, it is best to establish specific cash on hand figures for each year. However, after exhausting all of the various leads, contradictions may still exist or the special agent may have no specific information at all. In order to work around this issue, approximate figures are often used however, this may not be the best solution. In investigations where no cash on hand information can be found, the special agent can enter beginning cash on hand as zero.

    The Dash Theory

    In situations where the subject had some available currency which was used in previously identified currency transactions, a constant figure of an unknown amount represented by a dash (-) can be used in a net worth calculation to symbolize cash on hand.

    The dash (-) indicates that the " inventory" of undeposited coin and currency cannot be quantified, but that facts and circumstances, i.e., evidence in the investigation, indicate that cash on hand or inventory of undeposited currency either remained constant or increased during the period. United States v. Giacalone, 574 F.2d 328, 333 (6th Cir. 1978) (" The recognition of a cash bankroll treated as a constant, together with proof which would support a finding that no significant cash hoard existed, [is] a sufficient accounting for cash in the opening net worth computation. " ) See also United States v. Sabino, 274 F.3d 1053, 1072 (6th Cir. 2001). The Sixth Circuit makes clear the dash method cannot be used to overcome the defense of a cash hoard, or as a way to avoid determining an opening balance of cash on hand.

    Taxable Source of Income

    In order for income to be taxable, it must come from a taxable source, Commissioner v. Glenshaw Glass Co,75 S. Ct. 473 (1955). In the Holland investigation, the Supreme Court opined that, an "Increase in net worth, standing alone, cannot be assumed to be attributable to currently taxable income. But proof of a likely source, from which the jury could reasonably find that the net worth increases sprang, is sufficient. . ." Holland, supra at 138.

    Following the Holland decision, it appeared that proof of a likely source was necessary in every net worth investigation. This premise was clarified by the Supreme Court in United States v. Massei, 78 S. CT. 495 (1958) when it stated:

    In Holland we held that proof of a likely source was "sufficient" to convict in a net worth investigation where the government did not negate all the possible nontaxable sources of the alleged net worth increases. This was not intended to imply that proof of a likely source was necessary in every investigation. On the contrary, should all possible sources of nontaxable income be negated, there would be no necessity for proof of a likely source.

    In view of these decisions, it appears that the government must either prove a likely source of taxable income or negate all nontaxable sources of income. In investigations where the government resorts to negating all nontaxable sources of income, it is even more critical to establish a firm starting point, particularly with reference to cash on hand.

    Proof of a likely taxable source of income has been found sufficient in a number of criminal investigations by:

    Showing that the subject did not report certain income on the tax returns. United States v. Chapman,168 F. 2d 997 (7th Cir 1948).

    Showing that the subject did not report certain income for years prior to indictment period. United States v. Skidmore, 123 F. 2d 604 (7th Cir 1948).

    Comparing the business operations and profits of the subject for the years under investigation with profits or prior operations for a comparable period. In the Holland investigation, the Supreme Court pointed out that the business of the defendant, a hotel, apparently increased during the years in question, whereas the reported profits fell to approximately one quarter of the amount declared by the previous management in a comparable period.

    Effectively contradicting the subject’s assertions as to nontaxable sources.

    Opportunities of the subject to receive graft.

    The nature of the business has the capacity to produce income in amounts determined by the net worth method.

    A likely source of income is established in net worth investigations by showing the source of income identified by the subject had the potential to produce income substantially in excess of that reported.

    Negating nontaxable sources of income may be accomplished by substantiating the subject’s admissions as to the receipt or non-receipt of loans, gifts, and inheritances. If the subject alleges to have received nontaxable sources of income, the special agent should verify the claim by reviewing Federal gift tax returns filed by the alleged donor or probate records of the deceased relatives’ estates. Additionally, the special agent should interview the person who allegedly made the gift to the subject. However, if the subject advances a specific explanation as to the sources of nontaxable funds expended, the government does not have to pursue other possible nontaxable sources of income when the one given is proven false.

    Investigation of Leads

    When a subject offers leads or information during a net worth investigation that, if true, would establish his/her innocence, such leads must be pursued. This also applies if the subject offers leads or information after the completion of an investigation but within sufficient time before trial.

    During the trial, if the government fails to show an investigation into the validity of the leads provided by the subject, the trial judge may consider the defendant’s information as true and the government’s investigation insufficient to go to the jury.

    Most leads refer to cash hoards, gifts, inheritances, and loans. These leads should be checked as routine steps taken during the investigation.

    The courts have held that the government does not have to investigate leads that are not reasonably verifiable. This is a question of judgment and, in the final analysis, is always a matter for the court to determine.

    Summaries and Appendices Prepared by the Special Agent

    In investigations utilizing a detailed computation of net worth, the factual data may be best presented via a summary of the details broken down into at least one main appendix and various sub-appendices.

    An appendix is a document developed to summarize and present, in a concise manner, voluminous information that is contained in the exhibits of an investigation. A sub-appendix supports the main appendix and is generally prepared when there are a number of items of a particular type of asset, liability, or other adjustment. Sub-appendices are also used when there are numerous witnesses or exhibits to support a particular net worth item. Keep the main appendix as simple and brief as possible to aid in its presentation and clarity. While there is no set format for a sub-appendix, it should be organized in a manner which presents the information in a clear and concise manner.

    A copy of each appendix and sub-appendix must accompany each copy of the final prosecution report. The exhibits to the investigation accompany only the original report. If sub-appendices are used, they must refer to the proper witness, the exhibit number, and a description of the evidence used to support the item.

    Sub-appendices are prepared to summarize the pertinent information that is found in the exhibits. The totals from the sub-appendices are forwarded to the main appendix, where the information is summarized. The main appendix is then cited in the body of the final report. Multiple main appendices are common in net worth investigations.

    During a trial of an income tax investigation involving the net worth method of proving taxable income, the special agent may introduce the sub-appendices and main appendix used to support the final report. It is important to remember that the special agent’ s work product (main appendix and supporting sub-appendices) is not evidence. These schedules and appendices should summarize documents and testimony already admitted into evidence during the trial. These schedules and appendices are admitted for the purpose of aiding and assisting the jury in considering the evidence admitted. The admissibility and use of appendices and summaries are discussed in IRM 9.6.4, Trial.

    In addition to appendices, schedules, and summaries, net worth computations have been presented to the jury through the use of graphs and charts.

    Adjustments to Net Worth

    After the special agent has established a firm starting point and identified the amount of cash on hand, the next step is to calculate the subject’s change in net worth for the prosecution years. Once the change (increase or decrease) in the subject ’s net worth has been determined, the special agent makes adjustments to that figure and arrives at the subject’s corrected adjusted gross income. Perhaps the most difficult phase in calculating a subject’s corrected taxable income is identifying, documenting, and correctly applying the adjustments to the subject ’s change in net worth for the nondeductible and nontaxable items. These adjustments are necessary in arriving at the subject ’s corrected adjusted gross income figure from the calculated increase or decrease in net worth. The following paragraphs will identify common adjustments to the calculated increase or decrease in a subject ’s net worth.

    The following are examples of adjustments for personal expenditures and nondeductible items which are added to the subject ’s change (increase or decrease) in net worth:

    nondeductible portion of capital loss

    losses on sale of personal assets

    The following are examples of adjustments for nontaxable items which are subtracted from a subject’s change (increase or decrease) in net worth:

    for capital gain transactions see the appropriate instructions and forms for statutory inclusions and exclusions

    non-taxable portion of social security income

    proceeds from life insurance

    disability income received (USC §104–§106)

    errors in subject’s records (in his favor) which relate to honest mathematical and bookkeeping errors found in the subject ’s books and records, and which tend to account for part of the understated income

    gains on the sale of a personal residence (depending upon the date of the sale, the gain could be entirely non-taxable) pursuant to the applicable law concerning these transactions and to the extent of whatever non-taxable gain the subject may have received

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    Unit Resources

    Fractions, Decimals, and Percents

    Student Reference Book pages 61-62

    Student Reference Book pages 53-54

    Converting "Easy" Fractions to Decimals and Percents

    Student Reference Book pages 38-39

    Using a Calculator to Convert Fractions to Decimals

    Student Reference Book pages 206-207

    Using a Calculator to Rename Fractions as Percents

    Student Reference Book pages 62, 206-207

    Student Reference Book pages 53-54

    Conversions among Fractions, Decimals, and Percents

    Student Reference Book pages 62, 207

    Student Reference Book page 7

    Comparing the Results of a Survey

    Student Reference Book pages 62, 207

    Student Reference Book pages 55-57

    Comparing Population Data

    Student Reference Book pages 38-39

    Multiplication of Decimals

    Student Reference Book pages 55-57

    Everyday Mathematics for Parents: What You Need to Know to Help Your Child Succeed

    The University of Chicago School Mathematics Project

    University of Chicago Press