USING MATHEMATICS TO TEACH ACCOUNTING PRINCIPLES
USING MATHEMATICS TO TEACH
ACCOUNTING PRINCIPLES
Sony
Warsono, Arif Darmawan, dan Muhammad Arsyadi Ridha
ABSTRACT: As widely acknowledged,
Luca Pacioli discussed accounting in his mathematics book Summa de Arithmetica, Geometria, Proportioni et Proportionalita.
Using the
perspective of mathematics, this paper shows that the majority of available
accounting principles literature employs accounting equations
positioning
the elements
of both assets and expenses in opposite accounting equations, rather than
placing the two elements in the same side of the accounting equation. More than
just offering consistent rationality, the use of mathematics rationality will
make it much simpler to explain why the elements of assets and expenses should
receive the same treatment in relation to debits and credits. Furthermore, this
paper shows that the rules of debits and credits are entirely based on
mathematical logics. Finally, this paper proposes the need for learning
accounting from the perspective of mathematics, in addition to those of GAAP
and engineering skills.
Keywords: Accounting education
methods; definition of equity; expanded accounting equations; mathematics
rationality; rules of debits and credits; mathematics-oriented study of
accounting principles
INTRODUCTION
A large
number of accounting software generate information which is reliable and
relevant, comply with generally accepted accounting principles (GAAP), and fulfill the various needs for
corporate financial information. The development of the accounting software must have involved many
professionals other than accountants, including programmers who are used to
mathematical thinking. These
programmers are
able to understand the workings of accounting even without having to study it in
detail. Accordingly, there must be some
methods which can be employed to teach accounting principles at college level
to supplement the current teaching because the latter has been questioned in
terms of its effectiveness due to some inappropriateness in the methods of
learning accounting principles. This paper employs the perspective of
mathematics in solving crucial issues which typically come to the surface in
class discussions about accounting principles.
Discussions about accounting
principles teaching methods are always appealing. The traditional teaching of
accounting has been criticized in many countries (Duff and McKinstry 2007)
because they are considered either too narrow procedural (Patten & Williams
1990; Nelson 1995) or unable to catch up with current development in business
to the extent that students can hardly receive any perfect picture of the real
business world (Adler 1999).
A number of experts have questioned the importance of
teaching of debits
and credits in classes of accounting principles because it is considered too
mechanical, unintuitive, and forcing the student to rely on memory only (Ingram
1998), and susceptible of providing an incorrect picture about accounting to
students who do not major in accounting (Pincus 1997; Diller-Hass 2004). Furthermore,
the double-entry system in accounting has experienced a significant decrease in
relevance with the advent of software which is capable of providing a variety of information
without having to set up general ledgers (Elam 1995 in Pincus 1997). On the
other hand, a number of other experts have tried to maintain the teaching of
debits and credits in accounting principles because debits and credits are believed to be “part of
the vocabulary in our language” (Wallace 1997, 230), and because debits and credits are an indispensable part in
the learning process of accounting (Vangermeersch 1997).
A great number of experts have been discussing the need of changes in the teaching methods of
accounting (Rankin et al. 2003; Hartnett et al. 2004). Albrecht and Sack (2000)
stated that the study of accounting needs to be modified to catch up with
changes in technology and globalization. Saudagaran (1996) and Springer and
Borthick (2004) noted that the traditional curriculum of accounting, which
emphasizes memorizing skills, may actually hinder the student’s effort to
develop the requisite competencies in accounting, such as critical thinking.
The AECC has suggested the needs for restructuring accounting principles
through learning by using a user model instead of a preparer model (in Lee and Bisman 2006). The user model was
perceived to be able to provide the student with a better understanding of the
concept of accounting (Baldwin and Ingram
1991; Bernardi and Bean
1999). Other researchers suggested the use of information technology to improve
the effectiveness of accounting study (Elliot, 1992; Pincus 1997; Mohamed and
Lashine 2003; David et al. 2003; Goldwater and Fogarty 2007).
Even though the business has experienced dynamic changes,
the study of accounting remains essentially the same (Albrecht and Sack 2000; Sangster, et al. 2007), passive
(Bonner 1999; Boyce et al. 2001), procedural (Dempsey and Stegmann 2001), inadequate
in equipping the student with the necessary competencies (Mohamed and Lashine
2003), and relying merely on a one-way
direction of knowledge distribution (Williams 1993; Saunders and Christopher, 2003). This traditional learning
of accounting makes accounting books look similar to one another (Sullivan and Benke 1997) which in turn make accounting less
than appealing to the student. For the next ten years there will be a shortage
of faculty members with Ph.D’s (AACSB 2003). The shortage of American lecturers
and doctorate students in accounting “already exists and may grow” (Plumlee et
al. 2006, 113). Fogarty and Markarian (2007) indicate that there has been a
decrease in the number of accounting lecturers – one that may escalate to a
serious problem of sustainability for the discipline of accounting.
Furthermore, there is an increasing number of students who decide to major in
accounting after enrolling in a university (Nelson et al. 2008).
The Teaching and Curriculum Section of the American Accounting
Association states that research into the history of accounting may provide a
precious lesson to comprehend the discipline of accounting (in Sangster et al.
2007). In line with this idea, this paper employs the perspective of
mathematics to discuss important topics in accounting principles. As widely
acknowledged, Luca Pacioli is a university professor of mathematics who
discusses accounting in his book Summa de
Arithmetica, Geometria, Proportioni et Proportionalita (Sangster et al.
2007). Using the
perspective of mathematics, this paper presents solutions which would make the
goal of learning accounting attainable, that is, by introducing the student to
the fact that accounting is in reality a much appealing knowledge that would
encourage the student to try to find out more about it.
Actually, the mathematical perspective has been employed
in a number of accounting books, but it is seldom mentioned explicitly, and
sometimes it is used inconsistently. On
the basis of some observations on the majority of literature on accounting
principles, the present paper concludes that only a few books state with enough
emphasis that accounting is one of the sciences that based on mathematics. This
paper also concludes that the majority of available literature employs accounting equations positioning the elements of both assets and expenses in opposite accounting
equations, rather than placing the two elements in the same side of the
accounting equation, even though both equations are mathematically correct.
Furthermore, this paper shows that the rules of debits and credits are entirely
based on mathematical logics. Finally, this paper discusses the need for
learning accounting from the perspective of mathematics, in addition to those
of GAAP and engineering skills.
THE DEFINITION OF EQUITY
Accounting
is based on the basic equation that assets equal to liabilities plus equity.
Equity is a residual interest, namely
the arithmetic difference between assets and liabilities (Alfredson et al. 2007).
This definition of equity is intended to maintain a balance between the left
side and the right side of the accounting equation. However, most books of
accounting principles simplify the definition of equity as “owner’s equity,”
which reflects the owner’s claim over the firm. The use of the term “owner’s equity”
narrows the real meaning of equity. In general, primarily at the formation of a
firm, the element of equity is likely to be the owner’s investment. Under
certain circumstances, however, equity may come from grants, donations, or aids
from the government or other outside parties which may not be categorized as
owners. In other words, the use of
the term “owner’s equity” is very likely to raise a dilemma: that the balance
in the accounting equation cannot always be attained due to the reception of grants,
donations, or government aids which do not meet the criteria either liabilities
or owner’s equity. Of course accounting teachers have their own ready answers
to this dilemma, but should they give any advanced answers to simple questions
posed by accounting novices?
A number of textbooks provide an additional description of
owner’s equity as a residual value so that assets always equal the total amount
of liabilities and owner’s equity (Horngren et al. 2002; Williams et al. 2005; Anthony
et al. 2007; Weygandt et al. 2008). FASB and IASB define equity or net assets as
“the residual interest in the assets of an entity that remains after deducting
all its liabilities” (FASB 1985, par. 49; IASB Technical Summary 2008). Thus, both
standards emphasize that equity is merely a mathematical rule intended to
maintain a balance in the accounting equation. Therefore, it is appropriate to
use the terms equity, net assets, or residual interest of assets in the study
of accounting principles, instead of the term “owner’s equity” commonly used in
accounting principles textbooks.
THE RATIONALITY OF ACCOUNTING EQUATIONS
Assets are
resources under the firm’s
control, whose funds come from liabilities and equity (sources of funds).
Accounting conveys the elements of revenues, expenses, and dividends in the
accounting equation (called expanded accounting equation) because the firm conducts business and distributes
dividends. The three elements are part of equity; revenues increase equity, while expenses and
dividends decrease equity. That is the rationality of the accounting equation
employed in most accounting textbooks. The rationality is primarily based on
the balance-sheet approach so that other accounting variables (including
revenues and expenses) are “considered secondary and derivative” (Dichev 2008, 454).
The emphasis
on
the balance-sheet approach is “unclear” (Dichev 2008) and “requires revaluations that often are not trustworthy” (AAA’s Financial
Accounting Standards Committee 2007).
The basic accounting equation can be expressed
as equation 1 (see Figure 1), and the expanded accounting equation can
be expressed as equation 2 (see Figure 2).
Insert
Figure 1 and Figure 2 here
Many textbooks employ the basic
accounting equation (equation 1) to analyze transactions which result in changes
in the element of revenues, expenses and dividends (Ainsworth et al. 2000; King
et al. 2001; Porter and Norton 2001; Warren et al. 2002; Libby et al. 2004;
Williams et al. 2005; Anthony et al. 2007). Several of these textbooks write
down the expanded accounting equation (equation 2) in their books (Horngren et
al. 2002; Weygandt et al. 2008).
Is there any rationality inappropriateness in the
existing textbooks in their explanations of the accounting equation? Although
the expanded accounting equation (equation 2) is mathematically correct, the
rationality which applies to equation 1 is applied in an inconsistent manner to
equation 2 because the elements of expenses and dividends are not sources of
funds. In other words, the rationality employed to explain basic accounting
equation is different from that employed to explain expanded accounting
equation. Learning which employs different rationalities to explain two things
which in essence are closely related is liable to confuse students.
Ingram (1998) employs equation 3 (see Figure 3) to
simplify the understanding of the logics of debits and credits. However, it is
hard to find textbooks which express accounting equation as expressed in
equation 3 although mathematically equation 2 and equation 3 are both correct. Therefore,
it is interesting to find the rationality of the accounting equation expressed
in equation 3 because mathematically it is more proper to place elements with
the same signs (positive or negative) on the same side. For the sake of
simplicity, this paper calls the rationality of equation 3 as mathematical
rationality, while this paper calls the rationality of equation 2 as conventional
rationality.
Insert
Figure 3 here
Subramanyam
and Wild (2009) and
Anthony et al. (2007) state that the basic accounting equation can be perceived
as sources and uses of fund. Therefore, by using the mathematical rationality,
the left side of equation 3 reflects the uses of fund, while the right side
reflects the sources of fund. The company uses the funds to acquire assets, pay
expenses and/or distribute dividends with funds taken from
the sources of liabilities, equity, and/or revenues. This mathematical rationality
can consistently explain both the basic
accounting equation (equation 1) and the expanded accounting equation (equation 3).
Vangermeersch (1997) noted that revenues and expenses are separate
elements, and not subdivisions of
equity. Therefore, the placement of revenues, expenses, and dividends on the same side (equation 2) is a compulsion that
runs the risk of confusing the student. Besides raising the problem of inconsistency in the
rationality of accounting equation, two additional reasons make the use of
equation 2 unacceptable. First, by definition equity is limited to a residual
interest or net assets to the effect that there is no appropriate justification
for an explanation as to why the element of revenues, expenses, and dividends
should belong to equity. Secondly, the attachment of one element to the other
may result in their being less than optimal. Analogizing the approaches of data
management in the computer system, the database approach provides information which is more up-to-date,
standardized, and easier to access than the application-oriented approach because the
database approach separate data from their application software (Romney and Steinhart
2009).
More than just offering consistent rationality, the use
of equation 3 will make it much simpler to explain why the elements of assets
and expenses should receive the same treatment in relation to debits and credits
even though by definition assets and expenses markedly differ from one another;
assets represent sources which provide future benefit, while expenses represent a sacrifice of
assets (FASB 1985). Moreover, the use of equation 1 or 2 forces the student to
think twice when identifying changes in the accounts due to expense
transactions and dividend transactions; the recognition of expenses
(dividends) make expense (dividend)
accounts increase, but must be recorded as a decrease because expenses
(dividends) decrease equity. This is an unnecessary step and is liable to raise
confusion to the student especially when he or she should identify accounts which must be debited or
credited). Unlike equation 1 or 2, the
use of equation 3 dispenses with this unnecessary step and minimizes confusion in the student’s mind when identifying debit or credit accounts.
THE RULES OF DEBITS AND CREDITS
The
rules of debits and
credits have been much debated by
experts. In their consideration, the mechanism of debits and credits does not make sense (“debits and credits are
nothing more than pluses and minuses”, Ingram 1998, 411), demands the student simply to memorize
(Pincus 1997), is too narrowly procedural (Patten and Williams 1990; Nelson 1995), and is liable to
convey a mistaken picture about accounting to the student (Pincus 1997;
Diller-Haas 2004). As far as we know, all accounting textbooks discuss these
rules of debits and credits. A number of textbooks state that the rules of
debits and credits are arbitrary (Anthony et al. 2007), a rule of thumb
(Williams et al. 2007), or customs “like the custom of driving on the
right-hand side…” (Weygant et al. 2008, 49). Other textbooks briefly describe
these rules by providing mathematical illustrations expected to facilitate the
student’s understanding (Ainsworth et al. 2000). Nevertheless, the description ends
up with an appeal that the student simply memorizes the rules (Walther 2009).
From the
mathematical perspective, this debit and credit mechanism actually has an argument which is very clear and easily
understandable to the student. In essence, this debit and credit mechanism
represents a consequence of the accounting equation whose recording is
reflected in the double-entry system.
Why should the asset accounts be debited when they
increase and credited when they decrease? We can get an answer to that question
by taking a close look at the following figures and its two illustrative cases.
Insert
Figure 4 here
Figure 4
indicates the position of each element in the accounting equation:
assets, expenses, and dividends on the left (debit) side of the equation, while
liabilities, equity, and revenues on the right (credit) side.
Insert
Figure 5 here
Case A (see Figure 5): Suppose Company
A purchases supplies on account. This
transaction engenders changes in the Supplies
account
and the Account payable account; both
accounts increase. The Supplies
account is recorded on the debit side, while the Account payable account is recorded on the
credit side. This is in line with the position of each account in the
accounting equation.
Insert
Figure 6 here
Case B
(see Figure 6):
Suppose Company
A purchases supplies
in
cash. This transaction engenders changes in the Supplies account and the Cash
account; the Supplies
account increases, while the Cash account decreases. In this case, both accounts are assets. To
maintain internal consistency mathematically, the Supplies account must be debited because the supplies account is an element of assets; assets have positive values, and are on the
left (debit) side of
the accounting equation. Next,
following mathematical rules, the Cash account must be credited because of a decrease in cash as
a result of the transaction.
Insert
Figure 7 here
Case C (see Figure 7): Suppose Company A converts
its notes payable into bonds. This
transaction engenders changes in the Notes
payable
account and the Bonds payable
account; the Bonds payable
account increases, while the Notes
payable
account decreases. In this case, both
accounts are liabilities. To maintain internal consistency
mathematically, the Bonds payable
account must be credited
because the bonds payable
account is an element of liabilities; liabilities have positive
values, and are on the right
(credit) side of the accounting equation. Next, following mathematical rules, the Notes payable account must be debited
because of a decrease in notes
payable
as a result of the transaction.
Analogyzing the
transactions in Case B and C, we establish the rules of debits and credits for
other elements of the expanded accounting equation. This
rule has been in force in accounting now. Therefore the debit and credit rule
is based on a mechanism which entirely follows the mathematical logics. In our
experience, students can understand and accept this debit and credit rule more
easily than if they have to memorize it. In other words, the use of the
mathematical perspective has made irrelevant the assumption that the debit and
credit rule is something that should be memorized. With a good reasoning, students
may find it easier to apply the debit and credit rule to all kinds of algebraic
equations, not just in relation to accounting equation.
It is true that the debit and credit rule is essentially
mechanical. Is it relevant, then, that this debit and credit rule be taught in
classes of accounting principles? It is
still relevant. First, the debit
and credit rule conveys a picture to the student that accounting is based on established
knowledge, especially mathematics. Second, as computer science with its binary digits (0 and 1) and the science of
electricity with its “on” and “off,”
accounting is endowed with debits and credits as a unique knowledge, which is used only in accounting. Third,
debits and credits can be used to enhance the concreteness of knowledge of
accounting; the study of debits and credits tangibilizes the workings of accounting. Tangibilizing accounting mechanizm is important to help the
student understand accounting topics related to keeping journals, posting,
which are indeed in the heart of accounting as an academic discipline. Fourth, as accounting
students are expected to compile or construct information, not just to use
information, they must have acquired basic knowledge of data processing into
some useful information (Vangermeersch, 1997). Fifth, knowledge of debits and
credits encourages the student to think systematically and logically, and to
develop the knowledge about accounting dynamics as a fast growing science through the
implementation of mathematical knowledge.
THE USE OF THE WORKSHEET
Learning about the worksheet is
one of the important topics discussed in textbooks about accounting principles
because it can give a clear picture of the process of compiling financial statements. The worksheet format
can be designed in a variety of ways as long as it helps in the compilation of
financial statements. The 10-column worksheet format is one of the numerous
worksheet formats that for a long time had been
in common use in accounting textbooks (Porter
and Norton 2001; Williams
et al. 2005; Weygandt et al. 2008). Walther (2009)
discusses the
use of the 12-column worksheet (one with 10 columns plus 2 additional columns
for statements of retained earnings).
The use of the 10-column format as well as the 12-column one reflects the application of mathematics in
accounting. Nevertheless, there are “tricks” in the recording of net income in
the Net income
column and the Balance sheet column in the worksheet. When the firm gets profits, the amount of the monetary profits
is recorded on the
left side of the Net Income column (to maintain balance between the debit and
credit sides of the Net Income column), but then must be recorded on the credit side of the Balance sheet column,
or the other way round when
the company undergoes losses.
This rule indicates inconsistency in the mathematical application, which is
liable to confuse the student (see Table 1).
Insert
Table 1 here
One way among many that can be employed to dispense with
this inconsistency is the use of
a 12-column worksheet, consisting of a
10-column worksheet plus 2 columns for closing entries. In addition to its
usefulness to dispense with the inconsistency of the 10-column worksheet, the
12-column worksheet is also useful for the study of accounting. Firstly, the provision of the 2 columns for the closing
entries indicates that closing entries are among the important activities in
accounting (without which the balance of nominal accounts would be carried to
the next period). Secondly, the 12-column worksheet engenders the Income
Summary account, which comes out due to the existence of the closing entries. This Income Summary account is important to show the firm’s profit or loss. Thirdly, the 12-column worksheet conveys the up-to-date
balance of the retained earnings account so that the Balance sheet column in the
worksheet helps a lot in compiling the financial statements (see Table 2).
Insert
Table 2 here
MATHEMATICS-ORIENTED
STUDY OF ACCOUNTING PRINCIPLES
There are still many important topics about the
accounting principles that can be explained mathematically. Our experience
indicates that when told that the primary working of accounting is based on
mathematics, the student can understand accounting topics more easily, including adjusting entries
–whose debits and credits have often become an object of complaint on the part
of the students (Pincus 1997) – and the crucial problem of closing entries.
Accounting
is a tool to attain a particular aim (Ingram 1998). In other words, accounting
should be treated like technology. As a technology, accounting can be made analogous
to aircrafts, computers, or any other technological products. Those
technologies are developed systematically, logically, and on the basis of
sciences whose validity has been so well established that they are capable of
growing even further and giving a vast contribution to the humankind.
We argue that the development of accounting is
affected by three interrelated pillars:
a.
Mathematics; this pillar should
be firmly founded upon which accounting may grow.
b.
Generally accepted accounting
principles (GAAP); this pillar serves to ensure that the development of
accounting could be well understood and accepted by the users.
c.
Engineering skills; this pillar
provides a space for the user for developing the kind of accounting that is
most suitable to his wants and needs.
The development of accounting
should be done through the development of the three pillars mentioned above.
The tremendous growth of the business world has likewise
increased the complexities of accounting and financial reporting. Up to now the development of accounting (GAAP)
regulations has been intensively done with the hope that such a development may
provide the necessary solutions to existing problems. Nevertheless, “we cannot
expect regulation to completely protect investors” (Scott 2009, 15). Therefore, it is expected
that a development that gives preeminence to the mathematical pillar would
enable accounting to give a significant contribution to mankind.
The addition of the revenues and expenses elements would make accounting study dynamic
(Vangermeersch 1997). By using the mathematical perspective, it is expected
that accounting study would be more dynamic and capable of inviting the student
to develop accounting knowledge, rather than to be content with understanding
accounting simply as a rule of play established by the business game. The use
of the mathematical perspective can also be an initial step toward the
development of new models of determining monetary values in financial
statements, which up to now have been considered within the competence of other
fields.
CONCLUSION
Historically,
accounting was based on mathematical knowledge as codified in Luca Pacioli’s
book of mathematics. Using the mathematical perspective, the present paper
reviews several basic issues in
textbooks of accounting principles commonly employed so far, and presents a
rationality based on clear arguments over the rules of debits and credits. By
designing a mathematics-oriented learning, it is expected that the study of
accounting principles would be dynamic and capable of developing the capacities
for inquiry, abstract logical thinking, and critical analysis (AECC 1990).
The development of the mathematical pillar in accounting
would enable it to develop faster rather than remaining just a tool to provide
information as it essentially is right now. As a result, the way is open wide for the addition of the
elements of accounting equations as well as new accounting topics developed
mathematically.
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|
|
|
Figure 4:
Position of Each Elements of Expanded Accounting Equation
|
Figure 5: Analysis of Transaction of Purchasing Supplies
on Account
Figure 6: Analysis
of Transaction of Purchasing Supplies in Cash
Figure 7: Analysis
of Transaction of Debt Conversion
Table
1: The 10-Column
Worksheet
ACCOUNTS
|
|
ADJUSTED TRIAL-BALANCE
|
NET
INCOME
|
BALANCE
SHEET
|
|||
Debits
|
Credits
|
Debits
|
Credits
|
Debits
|
Credits
|
||
Assets
|
|
$200,000.00
|
|
|
|
$200,000.00
|
|
Liabilities
|
|
|
$80,000.00
|
|
|
|
$80,000.00
|
Capital
stock
|
|
|
$100,000.00
|
|
|
|
$100,000.00
|
Dividend
|
|
$7,000,00
|
|
|
|
$7,000.00
|
|
Retained
earnings
|
|
|
$15,000.00
|
|
|
|
$15,000.00
|
Revenues
|
|
|
$50,000.00
|
|
$50,000.00
|
|
|
Expenses
|
|
$38,000.00
|
|
$38,000.00
|
|
|
|
Net
Income*
|
|
|
|
$12,000.00
|
|
|
$12,000.00
|
Total
|
|
$245,000.00
|
$245,000.00
|
$50,000.00
|
$50,000.00
|
$207,000.00
|
$207,000.00
|
*Net
income is not an account.
Table
2: The 12-ColumnWorksheet
(Including Closing entries)
ACCOUNTS
|
|
ADJUSTED TRIAL-BALANCE
|
NET
INCOME
|
CLOSING
ENTRIES
|
BALANCE
SHEET
|
||||
Debits
|
Credits
|
Debits
|
Credits
|
Debits
|
Credits
|
Debits
|
Credits
|
||
Assets
|
|
$200,000.00
|
|
|
|
|
|
$200,000.00
|
|
Liabilities
|
|
|
$80,000.00
|
|
|
|
|
|
$80,000.00
|
Capital
stock
|
|
|
$100,000.00
|
|
|
|
|
|
$100,000.00
|
Dividend
|
|
$7,000,00
|
|
|
|
|
d)$7,000.00
|
|
|
Retained
earnings
|
|
|
$15,000.00
|
|
|
d)$7,000.00
|
c)$12,000.00
|
|
$20,000**
|
Revenues
|
|
|
$50,000.00
|
|
$50,000.00
|
a)
$50,000.00
|
|
|
|
Expenses
|
|
$38,000.00
|
|
$38,000.00
|
|
|
b)$38,000.00
|
|
|
Net
Income*
|
|
|
|
$12,000.00
|
|
|
|
|
|
Income
summary***
|
|
|
|
|
|
b)$38,000.00
c)$12,000.00
|
a)$50,000.00
|
|
|
Total
|
|
$245,000.00
|
$245,000.00
|
$50,000.00
|
$50,000.00
|
|
|
$200,000.00
|
$200,00.00
|
*Net income is not an
account
**Ending balance of
retained earnings is $20,000.00 (beginning balance of retained earnings + Net
income – Dividends). The beginning balance of retained earnings is $15,000.00
as shown in the adjusted trial-balance column.
***Income summary is an
important clearing account used to show the number of profit or loss.
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