Commercial arithmetic

8th Grade Maths

Chapter 1 – Unit 4

How to deal with percentage problems

How to change percentages, into fractions and decimals

A percentage is a special fraction.

The % says, “per hundred.”

For example, 50%, is simply another

way of writing the fraction 50/100

You can almost see the 100 in the percentage sign

Percentage Fraction

17% 17/100

20% 20/100

75% 75/100

Etc.

The 1The

noughts

24% = 24/100 12/50 6/25

So to write a percentage as a fraction, you simply put it over

100 like this:

24% = 24/100

Yes, but remember to try to write the fraction in its

simplest terms

Copy and complete this chart

Percentage Fraction Percentage Fraction

1. 23% 23/100 7. 80%

2. 24% 8. 84%

3. 55% 9. 12%

4. 20% 1/5 10. 48%

5. 17% 11. 71%

6. 78% 12. 36%

To change a fraction into a decimal, you divide the bottom number (the denominator) into the top number

(numerator). For example, to change ¾ into a decimal, you would divide the 3 by the 4.

¾ 3 ÷ 4 = 0.75

Remember that a percentage is a fraction always has 100 as its

denominator

So, to change a percentage into a decimal, you always divide by

100

For example, 35% as a fraction is 35/100 so to change this into a decimal, you divide 35 by

100

35% 0.35÷ 100

To change decimal into a percentage

Decimal

PercentageTo change a decimal into a percentage, you X by 100

0.23 23%X 100

You Should Know

A a percentage is a special fraction.

23% = 23/100

Percentage to a decimal

÷ 100

Decimal to a

percentage

X 100

Copy and complete this chart

Percentage Decimal Decimal Percentage

1. 27% 7. 0.87

2. 2% 8. 0.08

3. 55% 9. 0.42

4. 5% 10. 0.04

5. 17% 11. 0.71

6. 98% 12. 0.07

Finding one quantity as a percentage of another quantity

Megha obtained a score of 23 out of 32 in a science test. She wants to know what this will be as a percentage?

To do this, you

• Write the result as a fraction 23/32

• 4. 72%

• Change the fraction into a decimal (23 ÷ 32 = 0.71875)

• Change the decimal into a % (0.71875 X 100 = 71.9%)

There are lots of similar problems and all of

these can be tackled in the same way 1. Write the quantities as a fraction

23/32

2. Change the fraction into a decimal (23 ÷ 32 = 0.71875). It’s probably a good idea to round off to 2 decimal places.

3. Change the decimal into a % (0.72 X 100 =72%)

4. So, 23 is 72% of 32

The total quantityThe quantity that

you want as a %

Car Par Survey

Colour Frequency

Green 2

Silver 4

Black 5What percentage of the cars are green?

Total is 11 Want 2/11 as a percentage

Change 2/11 into a decimal = 0.1818

(round off to 2 d,p. Change 0.18 into a %

(0.18 X 100 = 18%) Answer 18%

Try these:

1 Work out your percentage if you got 12 out of 25

2  Work out your percentage if you got 15 out of 25

3 Work out your percentage if you got 17 out of 20

4 Work out your percentage if you got 19 out of 24

5 Work out your percentage if you got 17 out of 31

6 A player scores 3 times out of 6 shots.What is their score percentage

7 A player scores 4 times out of 10 shots. What is their score percentage

8 A player scores 44 times out of 69 shots. What is their score percentage

9 Work out 12.5 out of 26 as a percentage

10 Calculate the score percentage if out of 15 shots 12 are successful

The ‘percentage of’ problem

What is 12% 0f

34?

What is 13% of 56?

What is 78% of

57?

To answer these questions

First: Change the percentage into a decimal

Second: Change the word ‘of’ into a X sign

Third: Workout

Example: What is 23% of 4523% = 0.23 and ‘of’ X

0.23 X 45 = 10.35

Finding Percentages of Different Numbers

 

Problem:

In a survey of 120 pupils it was found that 20% had personal stereos. How many pupils had personal stereos?

 

Change the percentage to a decimal, then multiply by the number of children.

20% = 0.20

0.20 times 120 = 24

 

1. 25% of 32

2. 75% of 36

3. 80% of 35

4. 10% of 142

5. 20% of 60

6. 35% of 70

7. 6% of 220

8. 15% of 150

9. 12% of 20

10.  27% of 30

Try these

Profit & Loss Statement

Profit & Loss Statement

A basic profit and loss statement reports the following for a specified period of time:

• Sales

• Expenses • Profits/losses

Theory and Concepts

• Manufacturer or Producer

• Whole-saler or Dealer• Retailer or sales person• Customer A Customer can get things

in above sequences.

Cost Price (CP)

• Definition:- The money paid by Shopkeeper to the

manufacturer or Whole-saler to buy the goods is called the cost- price(CP) of the goods purchased by the shopkeeper.

Selling Price (SP)

• Definition:- The price at which the Shopkeeper sells the goods

is called the selling price(SP) of the goods sold by shopkeeper.

Profit(gain)

• Definition:- If the selling price of an article is more than its cost price, then the dealer (or shopkeeper) makes a profit(or gain).

– Profit = SP-CP; SP>CP

LOSS

• Definition:-If the selling price of an article is less than its cost price, then the dealer suffers a loss.– Loss=CP-SP; CP>SP

Important Formulae1. Profit=SP-CP 2. Loss=CP-SP3. Loss or Profit is always based on CP.4. Profit% = (Profit*100)/CP.5. Loss% = (Loss*100)/CP6. Selling Price(SP)=[(100+Profit%)/100]*CP7. Selling Price(SP)=[(100-Loss%)/100]*CP8. Cost Price(CP)=[100/(100+Profit%)]*SP9. Cost Price(CP)=[100/(100-Loss%)]*SP

Marked Price

• Definition:- Basically to avoid loss due to bargaining by the customer and to get the profit over the cost price trader increases the cost price by certain value, this increase in value over cost price is known as markup and the increased price(CP+markup) is called the marked price or printed price.

Discount

• Definition:- Discount means reduction of marked price to sell at a lower rate or literally discount means concession. Basically it is calculated on the basis of marked price.

• Discount is always given on the marked price of the article.

Discount Formulae:-

• Discount = marked price – selling price

• Marked price is sometimes called list price.

• Discount = rate of discount times the marked price.

• Net price = marked price - discount.

Commission

• Commission is percentage of sales price of an item.

• Also, Transaction that are mediated by person other than buyer and owner.

• The mediator who helps in buying and selling is Commission agent or Broker.

• The money that the broker or agent receives in deal is brokerage or commission.

Commission Contd…

• Commission in per hundred rupees is called Commission rate.

• For e.g. If a salesperson receives a 10% commission on their sales and sells Rs1500 worth of merchandise, they would earn Rs150 in commission.

Simple Interest

Formula

I = PRT

What is Interest

INTEREST IS

• If you DEPOSIT amount in a bank account then bank will pay you a percentage as INTEREST.

I = PRT• I = interest earned (amount of money the bank pays

you)

• P = Principle amount invested or borrowed.• R = Interest Rate usually given as a percent

(must changed to decimal before plugging it into formula)

• T = Time (must be measured in years) or converted to years by dividing by 12 months

Converting• Change % to decimal

1) 12%2) 5%3) 2 ½ %4) 8.5%

• Change from decimal to %5) .0986) .455

• Answers

1) .122) .053) .0254) .085

5) 9.8%6) 45.5%

Move 2 places to left & drop % sign

Move 2 places to right & add % sign

I = PRTSolve for one of variables:

• Solving for I• Write value for P, R,

& T.• Then multiply• The result is required

Interest.

• Solving for other variables

• Multiply the numbers that are on same side then divide by that answer.

1. A savings account is set up so that the simple interest earned on the investment is moved into a separate account at the end of each year. If an investment of Rs5,000 is invested at 4.5%, what is the total simple interest accumulated in the checking account after 2 years.

• I = PRT• I=• I=Rs450

• Interest paid by bank is unknown

• Principle (invested)• Rate changed to decimal• Time is 2 years• Multiply

(5,000) (.045)(2)

2. A savings account is set up so that the simple interest earned on the investment is moved into a separate account at the end of each year. If an investment of Rs7,000 is invested at 7.5%, what is the total simple interest accumulated in the checking account after 3 years.

• I = PRT• I=• I=Rs1575

• Interest paid by bank is unknown

• Principle (invested)• Rate changed to decimal• Time is 3 years• Multiply

(7,000)(.075) (3)

3. When invested at an annual interest rate of 6% an account earned Rs180.00 of simple interest in one year. How much money was

originally invested in account?

• I = PRT• 180=• 180 = .06P

.06 .06

3,000 = P

• Interest paid by bank • Principle (invested) is

unknown• Rate changed to

decimal• Time is 1 year• Multiply• Divide

P(.06) (1)

4. When invested at an annual interest rate of 7% an account earned Rs581.00 of simple interest in one year. How much money was

originally invested in account?

• I = PRT• 581=• 581 = .07P

.07 .07

Rs8,300 =P

• Interest paid by bank • Principle (invested) is

unknown• Rate changed to

decimal• Time is 1 year• Multiply• Divide

P(.07) (1)

5. A savings account is set up so that the simple interest earned on the investment is moved into a separate account at the end of each year. If an investment of Rs7,000

accumulate Rs910 of interest in the account after 2 years, what was the annual simple interest rate on the savings account?

• I = PRT• 910=• 910 = (7,000)(2)R• 910 = 14,000 R

14,000 14,000

0.065 = R

6.5% = R

• Interest paid by bank• Principle (invested)• Rate is unknown• Time is 2 years• Regroup & Multiply• Divide• Change to %

(7,000)(R)(2)

6. A savings account is set up so that the simple interest earned on the investment is moved into a separate account at the end of each year. If an investment of Rs2,000

accumulate Rs360 of interest in the account after 4 years, what was the annual simple interest rate on the savings account?

• I = PRT• 360=• 360 = (2,000)(4)R• 360 = 8,000 R

8,000 8,000

0.045 = R

4.5% = R

• Interest paid by bank• Principle (invested)• Rate is unknown• Time is 4 years• Regroup & Multiply• Divide• Change to %

(2,000)(R)(4)

7. Mohan M. bought a 6-month Rs1900 certificate of deposit. At the end of 6 months, she received a Rs209 simple interest. What rate of interest did the

certificate pay?

• I=PRT209=209=(1900)(6/12)R209=950R950 9500.22 = R 22% = R

• Interest paid by bank• Principle (invested)• Rate is unknown• Time is 6 months (divide by 12)• Regroup & Multiply• Divide• Change to %

1900(R) (6/12)

8. An investment earns 4.5% simple interest in one year. If the money is withdrawn before the year is up, the interest is prorated so that a proportional amount of the interest is paid out. If Rs2400 is invested, what is the total amount that can be withdrawn when the account is closed out after 2 months?

•I=PRT

•I=

• Interest paid by bank - Unknown

• Principle (invested)• Rate is .045

• Time is 2 months (divide by 12)

• Multiply• Now, since the money

is being withdrawn, add the interest to the

principal.

(2400)(.045) (2/12)

I=Rs18

Rs18 + Rs2400 = Rs2418

Rs2418 will be withdrawn