chap 1-5-business mathematics-updated july 7

8/4/2019 CHAP 1-5-Business Mathematics-Updated July 7

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Chapter 1

Numeration


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Chapter 1: Numeration

1.1 Place Value for Numbers

Arithmetic

Arithmetic is a basic tool in the study of Business Mathematics. The extent ofpractical applications whether social or business, make use of the arithmeticaloperations such as addition, subtraction, multiplication and division.

Suggested Steps in Solving Problems

1. Read very carefully the problems until the conditions are clear.

2. Determine the given and set what is to be found.

3. Form the relationship between the given and the required

4. Decide what process or processes to use. If a problem involves a series ofsteps, decide what process applied first.

5. Use formulas connecting the known quantities with the unknown quantities.

6. Also, if possible, try to give an estimate of the possible answer.


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7. Solve the problem in accordance with the process involved. The most importantis to be careful in every aspect of computations. A simple error will make allcomputations collapsed.

8. Check the solutions with the condition of the problem.

Number and Numeral

A number is one or more units or things. A number that denotes one or more whole

units is a whole number or an integer. A number that denotes a part of portion of aunit is called a fraction. A mixed number is a combination of a whole number and afraction. A number proceeded by a decimal point and whose value is less than oneunit is called a decimal. Combination of a whole number and a decimal is a mixeddecimal. A decimal fraction or decimal is another way or representing a fractionwhose denominator is in the powers of 10. For example the fraction 7/10 represents

0.7 in decimal. A complex decimal is a number consisting of a decimal or mixeddecimal and a fraction. Examples are 0.33 1/3 and 0.66 2/3.

A numeral is a symbol that stands for a number while a number is the idea. We cansee a numeral but we cant see a number.


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Numeration System

A system of reading and writing numbers is a numeration system. This consists ofsymbols and rules or principles on how to use these symbols. Our system of

reading and writing numbers is the decimal system or the hindu-arabic system. Tendigits are used -0,1,2,3,4,5,6,7,8 and 9. this system is based on groups of tens. Ituses the place value concept.

Simple and Relative Values

The value represented by a figure depends on its position in relation to other

figures. The simple value of a figure is the value it has when it stand alone. Forexample, 3 when it stands alone has a value that is one greater than 2, or one lessthan 4. if 2 is placed to the right of 3 making it 32, figure 2 has a new value. It is tentimes 3 or 3 tens. The new value that is given to it by placing another figure to theright of it is called its relative value.

Place value

The place value of digit determines its value and each place has a value of tentimes as that of the place to the right. Let us take a look at the place value chart.The digit 5 on the chart has a value that changes with its position or place. Itsfirst value is 5 units or ones. Its second value is 5 hundred thousands and its

third value is five billions.


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Place value chart for whole numbers

Periods Quadrillion Trillions Billions Millions Thousands Units

Place Names

Digits 2 9, 4 2 5, 0 7 1, 5 6 8, 3 1 5


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Rounding off Numbers

Rounding off numbers is sometimes convenient to adopt for simple forms of thenumbers rather than taking the exact values. The rounding off numbers is frequently

used in business to facilitate computations. For instance, a store sale of P4,992.75in a day may give a rounded figure of P5,000 for a better understanding and fastlook of the sale.

Guidelines to be followed in rounding off numbers

1. When portion to be dropped begins with 0,1,2,3,4 or a digit less than 5, the lastdigit to be retained is unchanged.

Example:

34,214.4184 34,214 rounded off to the nearest ones

34,210 rounded off to the nearest tens34,200 rounded off to the nearest hundreds

2. The digits dropped in the whole number are replaced by zero or zeros. Using theexamples above.


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Examples:

34,214.018 34,210 rounded off to the nearest tens

34,200 rounded off to the nearest hundreds

34,000 rounded off to the nearest thousands

3. When portion to be dropped begins with 6,7,8,9, the last digit to be retained isincreased by 1.

Example:

8,579.251 8,579 rounded off to the nearest ones

8,580 rounded off to the nearest tens

8,600 rounded off to the nearest hundreds

4. When the portion to be dropped is 5, and the preceding digit is: a. even, retainthe preceding digit; b. odd, increase the preceding digit by 1.

Examples:

8,575 8,580 rounded off to the nearest tens

8,000 rounded off to the nearest thousands.


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Chapter 2

Fundamental Operations on

Whole Numbers & Decimals


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Short Cuts in Multiplication

1. When a number is multiplied by 10 or a multiple of 10, move the decimal point ofthe given number to as many places to the right as there are many zeros in themultiples of 10.

Examples: 42.25 x 1,000 = 43,250

2. A number ending in 5 when multiplied by itself will have 25 as the two extremeright digits in the product. The remaining digit/s will be multiplied by one greaterthan itself.

Examples: 25 x 25 = (2 x 3) then 25 = 62575 x 75 = (7 x 8) then 25 = 5625

3. Two numbers whose ending digits make a sum of 10 and whose remaining digitsare the same, the ending digits will be multiplied and the remaining digits willalso be multiplied by one greater than itself.

Examples: 64 x 66 = (6 x 7) then (4 x 6) = 4224

71 x 79 = (7 x 8) then (1 x 9) = 5609


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4. In multiplying a number by 9,99,999, move the decimal point as many nines in

the multiplier, then subtract the multiplicand from the result.

Examples: 23.25 x 999=23250 23.25 = 23226.75

865 x 99= 86500 865= 85635

5. In multiplying a number by 11, 101, 1001, , multiplying the number by 10, 100,

1000, , then, add the multiplicand to the result.

Examples: 753 x 101= 75300 + 753= 76053

7.563 x 101=753 + 7.53= 760.53

6. In multiplying a number by 5, multiply the number 10 and divide it by 2 since 5 is of 10.

Examples: 329 x 5=3290 / 2= 1645

3.9 x 5=39 / 2=19.5

7. In multiplying a number by 25, multiply the number by 100 and divide it by 4 since25 is 1/4 of 100.

Examples: 618 x 25= 61800 / 4= 15450

9.7 x 25=970 / 4= 242.5


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8. In multiplying a number by 50, multiply the number by 100 and divide it by 2 since50 is 1/2 of 100.

Examples: 7761 x 50=776100 / 2=388050

4.26 x 50=426 / 2=213

9. In multiplying a number by 0.1, 0.01, 0.001, , move the decimal point in the

multiplicand to the left as many places as there are decimal places in the multiplier.

Examples: 3742 x 0.1=374.2

35.9 x 0.01= 0.359

10. When either the multiplicand or the multiplier ends in zero, bring down the zeros

into the product and continue to multiply by the next digit to the left.

Examples: 3370

x 420674

1348

1415400


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Basic Operations in Business

This section aims to:

1. analyze and solve problems involving addition; and

2. appreciate the role of Mathematics as a tool for solving problems.

Simple Average

To get the simple average, we get the sum of all the given values or itemsand divide the sum by the number of values.

Weighted Average

To get the weighted average, we multiply the quantities by the measuresinvolved. Then, we divide the sum of the products by the sum of the quantities.

Profit and Loss


1.differentiate average for the simple averages and define profit and loss;

2. solve business problems and problems and bank reconciliation involvingprofit, loss and averages.


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Profit and Loss

Profit is the amount by which the sales are greater than the cost of goodssold and the operating expenses. Margin also means gain or profit.

The formulas in computing profit are as follows;

Net Sales = Gross Sales Refunds and Allowances

Net Profit = Gross Profit Operating Expenses

The net sales and profit may also be given in scheme diagram as follows:

Gross Sales

- Sales Returns and Allowances

Net Sales

- Costs of Goods Sold

Gross Profit

- Operating Expenses

Net Profit


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1. Gross Sales is the actual amount received for selling the goods.

2. Refunds are amounts returned usually if goods are defective.

3. Net sales are obtained when refunds are being deducted from the gross

sales.

4. Costs of goods sold or buying price is the amount paid for articlesbought including the buying expenses.

To compute for the cost of goods sold for a period of time, we have the formula asfollows:

Available Goods = Beginning Inventory + Purchases

Costs of Goods Sold = Available Goods Ending Inventory

The above may be diagrammed as follows:

Beginning Inventory

+ Purchases

Goods For Sale

- Ending Inventory

Costs of Goods Sold


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5. Inventory is an itemized lists of goods on hand.

6. Gross Profit is the difference between the net sales and the cost ofgoods sold.

7. Operating expenses or overhead are selling expenses such as salariesor wages, traveling expenses, rentals, water, electric bills, commissions, taxes.

8. Net Profit is the amount obtained when all the selling expenses or othercost of doing business are deducted from the gross profit.

In case the sales are less than the cost of goods sold, there is a loss. The formulas in

computing losses are as follows:

Gross Loss = Costs of Goods Sold Net Sales

Net Loss = Gross Loss + Operating Expenses

The relationship of the terms of loss may be diagrammed schematically as follows:

Cost of Goods Sold

- Net Sales

Gross Loss

+ Operating Expenses

Net Loss


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Bank Reconciliation

Bank Reconciliationis the process of bringing the bank's monthly report foreach depositor showing deposits made, check written, cancelled checks, and servicecharges. Checkbook contains checks and check stubs. The checks are filled out by

the depositors made, and of charges made by the bank. Reconciliation statementputting an agreement the bank statement balance and the checkbook balance.

The differences in the balances may be due to:

1. Outstanding checks. These are checks issued by the depositor but havenot yet been presented to the bank for payment.

2. Deposits in transit. These are deposits made but late to be included in themonthly bank statement.

3. Service charges.

4. Errors in the check stub entries.

5. Cancelled checks. These are checks that have been paid by the bank.


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Depreciation Schedule

this section aims to:

1. relate the total and average years in depreciation in the preparation ofdepreciation schedule; and

2. construct the depreciation schedule.

Depreciation is the lost in value of physical assets through its use. Theyearly deposits into the depreciation fund are called depreciation charges. Thedepreciation fund is the portions of a given amount at the end of its useful life or thedifference between the original cost of the asset and the sum in the depreciation fundis called the book value of the asset. At the end of the year.


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Fractions

Chapter 3


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Chapter 3: FRACTIONS

FRACTION is one or more of the equal parts into w/c a whole is divided.

Terms: numerator & denominator.

Numerator- number abovethe line, showing how many of the equal parts

are expressed or taken.

Denominator- number belowthe line, showing into how many equal partsthe whole is divided.

vinculum- divided by- line between numerator & denominator

Ex: 3 (numerator) 4 (denominator)


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Kinds of Fractions

Proper Fractions- numerator is less than the denominator.

- Value is less than 1.

Ex: 2/7, 6/9

Improper Fractions-numerator is equal to or greater than thedenominator

Ex: 3/3, 7/2

Mixed Number-whole number & fraction

Ex: 5 6/7, 10 7/7

Similar Fractions-same denominators

Ex: 2/9, 5/9, 12/9


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Dissimilar/ Unlike Fractions-different denominators

Ex: , 5/3, 5/7

Decimal Fractions-decimal pt. is used to indicate that the denominator is apower of 10.

Ex: 0.3= 3/10 ; 12.25= 12 25/100

Other terms

LCD- two numbers is the smallest number w/c is exactly divisible by thedenominators of the dissimilar fractions.

Ex: LCD of 2/5 and 1/2 LCD of 2 & 5 (10)

GCF- betwn. 2 numbers is the highest or biggest number w/c can be used as acommon divisor of the numerator & denominator of a fraction.

Ex: GCF of 12 & 24 in the fraction 12/24. GCF of 180 and 168 = 12


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Lowest-term of a fraction-numerator & denominator have no commonfactors except 1.

Ex: 2/5, 11/13, 15/7

Reciprocal of a Fraction-quotient of 1 divided by the given number.

Ex: Reciprocal of 2/5 = 5/2, reciprocal of 4= 1/4

Laws of Fractions

1. The value of a fraction does not change if its terms aremultipliedbythe same number except 0.

2. The value of a fraction does not change if its terms aredividedbythe same number except 0.


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Conversion of a Fraction

Improper Fraction to a whole ormixed number

Mixed number to an improperfraction

Lower terms fraction to a higherterms fraction

Higher terms fraction to a lowestterms fraction

Dissimilar fractions to similarfractions

Ex: 4/3 = 1 1/3, 12/4 =3

Ex: 6 2/3= 20/3

Ex: raise 3/5 to twenty-fifths= 15/253/7= 12/28

Ex: reduce 12/16 to lowest term

12/16 divided by 4/4 = 3/4

Ex: change 3/4 &5/6 to similar fractionLCD= 12

= 9/12 5/6 = 10/12


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Addition of Fractions

Similar fractions

Dissimilar Fractions

Mixed Numbers

Subtraction of Fractions

Similar fractions

Dissimilar Fractions

Mixed Numbers

3/4 + = 4/4 5/8+5/8= 10/8

+ 5/6 = 9/12

6 1/5+ 2 2/5 = 8 3/5 3 + 2 2/7 = 5 15/28

Examples

4/7- 2/7 = 2/7

6/7 = 12/14 7/14 = 5/14

6 8/9 + 2 1/9 =

= 5+ (3/3 + 1/3)- 2 2/3

= 5 4/3 2 2/3 = 3 2/3


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Multiplication & Division of Fractions

A Fraction & Whole No.

A Fraction & Mixed No.

A Whole No. & Mixed No.

Mixed No.& Mixed No.

8 x 2/3 = 16/3 3/5 x 10 = 6

5/9 x 1 2/5= 7/9 2 1/3 x 2/4 = 1 1/6

12 2/3 x 4 = 50 2/3

2 x 3 5/6 = 10 13/24

Division of Fractions

In dividing fractions, we invert the divisor & multiply

1/8 / 8 = 1/64

4 1/5 / 1/5 = 21

2 x 2 = 25/4

2 7/8 / 2 = 23/16


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Ratio & Proportion

Ratio- relation betwn. 2 like nos. or quantities expressed as a quotient or

fraction.

Ratio of one no. a to another no. b ( a:b )

The fraction a/b provided b not = to 0. a :b = a/b

Proportion- two ratio are equal. Equality of 2 ratios or fractions

Ex: a:b = c:d or a/b= c/d

as a is to b as c is to d.

The are 4 terms in a proportion

Ex: 6: 24 = 1: 4

where 6 & 4 (extremes)

24 & 1 (means)


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Chapter 4

Percentage in

Business


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Chapter 4: Percentage in Business

Percent, denoted by %, is expression which indicates the number of parts takenfrom a hundred. It also means hundredths. Thus 5% is the same as the fraction5/1000, or as the decimal 0.05.

Conversion techniques1. To reduce a decimal to a common fraction, we write the given decimal numberregarding the decimal point as the numerator of a common fraction with adenominator of the power of 10 of the given decimal.

Examples:

0.7 = 7/10 there is 1 decimal place so the denominator is 10.

0.16 = 16/100 there are 2 decimal places so the denominator is 100.

2.075 = 2 75/1000 for 3 decimal places, 1000 is the denominator.

2. To reduce a common fraction to a decimal, we divide the numerator by thedenominator.Examples: solutions:

= 0.5 2) 1.00.5


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3. To change a percent to decimal, we move the decimal point two places to theleft and drop the percent sign. If the percent is in fractional units, we change firstthe fraction to decimal before moving the decimal point.

Examples:

40% = 0.40

0.05% = 0.0005

4. To change a decimal to a percent, we move the decimal point two places to the

right and add a percent sign.

Examples:

1 = 100%

0.23 = 23%

1/8 = 0.125 = 12.5%

90%


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6. To change a fraction to a percent, we divide the numerator by the denominator,then we move the decimal point of the quotient two places to the right and add thepercent sign. For a mixed number, we change it first to an improper fraction beforeperforming the indicated division.

Examples:

27% = 27 / 100

1.25% = 0.0125 = 125 / 10,000

5. To change a percent to a fraction, we drop the percent sign and replace it by100 as denominator. If the percent is in decimal, we move the decimal point twoplace to the left after dropping the percent sign. Then we convert the decimal to itsfractional equivalent. If the percent is in fraction, divide it by 100 and drop the

percent sign.

Example:

3/5 = 0.60 = 60% 5) 3.00

0.60


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Aliquot Parts

Any number that is contained in another number in an exact number of times iscalled an aliquot part of that number. For instance, P0.20(1/5), P0.25(1/4),P0.50(1/2) are all aliquot parts of P1, since

P0.20 is contained 5 times in P1



Thus aliquot parts are certain fractional parts of a larger number.Examples:

2 is an aliquot part of 5 (5/2 is contained 2 times in 5).

12 is an aliquot part of 100 (12 is contained 8 times in 100).

Sometimes numbers which are not aliquot parts are found to be multiples of

aliquot parts.Examples:

150 is a multiple of 50 taken thrice.

66 2/3 is a multiple of 33 1/3 taken twice.


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Illustration for exact number of Aliquot parts

An aliquot part of 100% is any number that is contained in 100%an exact number of times. The common aliquot parts of 100% are as

follows:

Examples:

of 100% = 50%

of 100% = 25%

1/3 of 100% = 33 1/3%

Illustration for Multiples of Aliquot parts

Some numbers that are not aliquot parts of 100% are found to bemultiple of aliquot parts. Below are common multiples of aliquot parts:

Examples:3/4 of 100% = 75%

2/3 of 100% = 66 2/3%

3/8 of 100% = 37 %


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Multiplying an Aliquot part of 100%

To multiply any number by an aliquot part of 100% we multiply the givennumber by the fractional equivalent and by the corresponding 100%.Examples:

800 x 0.37 = 800 x 3/8 = 300

Note that 0.371/2 is equivalent to 3/8 of the larger number 1, i.e,

0.375 = 375 /125 = 3/8

1000/125

Dividing by an Aliquot part of 100%

In dividing a number by an aliquot part of 100% we divide the given number by100% and divide again by the fractional equivalent.

Example:

10/0.33 1/3 = 10/ 1/3 = 10.3 = 30

Note that 0.33 1/3 is equivalent to 1/3 or the larger number 1.


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Percentage Formulas

A percentage is the result obtained by taking a certain percent of a number.Percentage (P) is equal to the base (B) times the rate (R) . The base is thenumber on which the percentage is computed. The rate is the number indicating

how many percent of hundreths are taken.

Percentage Formulas

Percentage P = R x B Rate R = P / B Base B = P / R

The percentage (P) refers to the actual quantity or number of items representedby the rate.

The Base (B) is usually preceded by the preposition of in word problems. Of

indicates multiplication. The word is is symbolized by the equal sign +. Other

words may be used instead of of such as as many as , as great as, asmuch as.The rate (R) is identifiable because it is usually in the form of a percent.However, it can also be in decimal or in fraction.

Examples:

P25 is what part of P130?

S


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Solution:

From P = R x B, we have

R = P/B R = 26 / 130 = 1/5

Percentage Variations


1. relate the percentage formulas on solving problems; and

2. solve problems on percentage increase or decrease.

Note that we dont multiply or divide a number by percent. We always change

the percent to a decimal or a fraction first before multiplying or dividing.

1. To find the percentage of increase or decrease, we multiply the base by therate and add the product to the base if it is an increase but subtract the

product from the base if it is a decrease.B + (B x R) = Percentage of Increase

B (B x R) = Percentage of Decrease


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2. To find the rate of increase or decrease, get the difference between the twogiven related values and divide it by the base or base or original quantity.Change the fraction to percent if it is needed.

Larger Value Smaller Value

Base or original quantity = Rate of Increase or Decrease

3. To determine the base when a number that is a fractional part or percent isgreater than or smaller than that of the unknown value, we divide the givennumber or percentage by the sum (if greater than) or the difference (if smallerthan) between 1 and the given fraction or 100% and the given rate.

P

1 + Fraction = Base of Increase

P

1 + Fraction = Base of Decrease


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P

100% + Given % = Base of Increase

P

100% + Given % = Base of Decrease


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Chapter 5: MARKETING GOODS, BUYING & SELLING

Discounts: trade discounts, cash discounts & retail discounts

Trade discounts - deduction given by manufacturers & wholesalers. May beseries or single discount.

Invoice net price difference between the list price & trade discount

Formulas for single trade discount

Trade Discount = list price x trade discount rate

List Price = Trade discount

trade discount rate

Trade Discount Rate = Trade Discount

List Price


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Invoice Price = List Price Trade Discount

List Price = Invoice Price + Trade DiscountTrade Discount = List Price Invoice Price

Examples: A Dining table listed at P1,285 is sold to a retailer at20% discount. Find the amt of discount & the invoice price.

Trade Discount = List Price x Trade Discount Rate

= P 1,285 x 0.2= P 257

Invoice Price = List Price Trade Discount

= P 1,285 P257

= P1,028


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Trade discount series. When 2 or more trade discounts are offered on asingle purchase. To find the invoice price when given a list price less aseries of discounts, 2 methods may be used.

Method 1

1. Apply the 1st discount to the list price

1st discount = list price x 1st discount rate

1st invoice price = list price 1st discount

2. Apply the 2nd discount to the 1st invoice price

2nd discount = 1st invoice price x 2nd discount rate

2nd net price = 1st invoice price 2nd discount

Method 2

1. Subtract each rate from 100 %

2. Express all the results as decimals & multiply them.

3. Subtract the product from 1 & the difference is the corresponding

single equivalent rate.


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Cash & Retail Discounts

Cash Discountsare special deductions from invoice price given to thebuyers who pay their accounts within a specified period of time.

Terms of Payment: Cash on delivery (COD)

: n/10 , n/20, n/60

: End of the Month

: Discount from prompt payment


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Retail Discounts

Are offered by retailers to consumers. May also be a single or a series rate ofdiscount.

Marked price/ list price- price which the retailers offer to sell an article

Markdown- amount of discount

Net price/ selling price- price to be paid by customer after deducting a

possible discount


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Markdown/ Retail Discount = Marked Price x

Marked Price = Markdown

Retail Discount Rate

Retail Discount Rate = Markdown

Marked Price

Selling Price = Marked Price Markdown

Marked Price = Selling Price + Markdown

Markdown = Marked Price Selling Price


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chap 1-5-business mathematics-updated july 7

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