col112 unit 1 mathematical thinking and modeling in business(1)

COL112 Mathematical Modeling For Business Unit 1 Section 1.1 Page 1

COL112

Mathematical Modeling for Business

Unit 1

Mathematical Thinking and Modeling in Business

Section 1

Percentages in Business Department of Mathematics and Statistics University College Zayed University


Unit 1 Section 1 Percentages in Business


1.1 Why Use Percentages? Consider this problem, which students are faced with many times in their studies:

Fatima took two tests and got a score of 62 out of 75 on the first test and 97 out of 120 on the second test. Which was the better grade?

Solution

Because the total number of points on each test is different, it is hard to tell.

But, changing the grades to percent grades means that both grades will appear to be out of the same total of 100.

So, 62 out of 75 is the same as saying that Fatima got 62

75 of the total available marks.

Notice that you can think of this as a fraction.

To write this as a percent, divide the 62 by 75 (which gives us a decimal) and then multiply by 100.

So, 62

75= 0.8267

Which, as a percent = 82.67

Therefore, 62 out of 75 is the same as 82.67%

And in the same way, 97 out of 120 is the same as 80.83%

Which means that Fatima did better on the first test.

This is meant to show that, by writing fractions as decimals and percentages, it is easier to compare quantities.

Self-Assessment Question [S.A.Q.] Ahmed took three tests and got the scores: 78 out of 90; 36 out of 40 and 26 out of 30. Which was his best grade? What do you notice about the other two grades?


Things You Must Know.

Percentages are just another way of writing fractions and decimals.

You can convert between all three forms using these rules: To Convert A Percentage To A Common Fraction:

Replace the % symbol with division by 100 and simplify the fraction.

24% = 24 6 4 6

100 25 4 25

To Convert A Percentage To A Decimal:

Remove the % symbol and divide by 100 (the result is that the decimal point appears to move two places to the left.)

24% = 0.24 To Convert A Decimal To A Percentage: Multiply the decimal by 100 and add the % sign. (The result is that the decimal point appears to

move two places to the right) 0.24 0.24 x 100 24% To Convert A Common Fraction To A Percentage: First convert the common fraction to a decimal fraction (by dividing denominator into numerator)

and then convert the decimal fraction to a percentage.

1

0.25 254 %

0.25

4 1.00


Percent Problems Explained There are six types of basic percent problems:

A. % of problems

30% of 450 is what number?

30% of what number is 450?

What percent of 450 is 30?

And

B. % change problems (Increase or decrease)

What is the result of a 30% increase on the number 450?

What is the result of a 30% decrease on the number 450?

What is the percentage increase from 30 to 450?

What is the percentage decrease from 450 to 30?

They all have different answers and all need a different method.

The secret of success in percentage problems is to learn all of them and know when to apply them.

This section will be part skills and part applications.

It is very important to know how to work out percentage problems in their real life situations and it

is also very important to be able to do them quickly and efficiently.

This unit will give you chance to do all of these.

There are more difficult types, but we will deal with those later.


A. % of Problems: Skills

Examples

1. 30% of 450 is what number?

Let the unknown number be N

Then, N = 30% of 450

= 0.30 x 450

= 135

Therefore, 30% of 450 is 135

2. 30% of what number is 450?


Then, 30% of N = 450

i.e. 0.30 x N = 450

i.e. N = 450

0.3

N = 1500

Therefore, 450 is 30% of 1,500

Or, 30% of 1,500 is 450

3. What percent of 450 is 30?


Then, N = 30

450x 100 (%)

= 6.6667 %

Therefore, 30 is 6.6667% of 450

Or, 6.6667% of 450 is 30


SAQ 1. 75% of 1250 is what number?

Therefore,


i.e.

i.e.

Therefore,

Or,



Then,

Therefore,

Or,


1. 45% of 258 is what number?

Therefore, 45% of 258 is


Therefore,


Therefore,

S.A.Q.* Use the method on the previous page to calculate the following problems.


B. % Change Problems (Type 1): Skills

1. What is the result of a 30% increase on the number 450?

Let the unknown number be N .

Then, N = 450 + 30% of 450

i.e. N = 450 + 0.3 x 450

I.e. N = 450 + 135

= 585

Therefore, the result of a 30% increase on 450 is 585

Some algebra can be used in this example.

Since N = 450 + 0.3 x 450

we can factorize , giving N = 450(1 + 0.3)

= 450(1.3)

= 585

2. What is the result of a 30% decrease on the number 450?

Let the unknown number be N .

Then, N = 450 30% of 450

i.e. N = 450 0.3 x 450

I.e. N = 450 135

= 315

Therefore, the result of a 30% decrease on 450 is 315

Applying the same logic as above: N = 450(1 0.3)

= 450(0.7)

= 585


SAQ

1. What is the result of a 65% increase on the number 240?

Therefore

Algebra.

Since

we can factorize , giving

2. What is the result of a 65% decrease on the number 240?

Therefore,

Applying the same logic as above:


B. % Change Problems: Applications

Percentages are often used as comparisons, or changes, between two amounts. In business, the comparison is often between old and new prices or between other sets of data.

Example 1

A car cost AED 120,000 in 2010. If the price of the car increased by 6.5% in one year, what would be the cost of the same model in 2011?

Let the new cost be N.

Then, N = 120000 +6.5% of 120000

i.e. N = 120000 + 0.065 x 120000

= 120000 ( 1 + 0.065)

= 120000 (1.065)

= 127800

Therefore, the cost of the car in 2011 would be AED 127,800 S.A.Q.* Last year a boat cost AED 8,900 and the price of the boat increased by 17.5% this year. What is the cost now?


Example 2 A computer originally sells for AED 9,870, but the retailer marks the price down by 20% in a sale. How much would the computer cost at the sale price?

The new price is now going to be less than the old price, because the change is a decrease.

That is, the change is negative.

Let the new cost be N.

Then, N = 9870 20% of 9870

i.e. N = 9870 + 0.2 x 9870

= 9870 ( 1 - 0.2)

= 9870 (0.8)

= 7896

The new price of the computer is AED 7,896

S.A.Q.* A Home Theatre cost AED 18,900 in 2010 and its price decreased by 19.5% over two years. What is the cost now?


Of course, once you become comfortable with these kinds of problems, you can also do them in the following way:

Example 1 Revisited

The cost of the car in 2010 is the old price. The increase in price of the car because of the 6.5% is the change in price The cost of the car in 2011 is the new price. To make calculations easier, call the old price 100% (of AED 120,000).

The increase of 6.5% means that the new price is 106.5% of the old price.

Therefore, the new price = 106.5% of old price = 1.065 x 120000 = 12780

The increase of means that the new price is % of the old price.

Therefore, the new price = of old price = = ______________

Example 2 Revisited

The new price is now going to be less than the old price, because the change is a decrease. That is, the change is negative. Let the old price be 100% (of AED 9,870) The decrease in price is 20% of the old price.

So, the decrease in price = 20% of 9870 = 0.20 x 9870 = 1,974

The decrease of 20% means that the new price is 80% of the old price.

Therefore, the new price = 80% of old price = 0.8 x 9870 = 7896

S.A.Q.* A smart phone cost AED 3,950 last year and its price decreased by 12.5% during this year. Using the method above, what is the cost now?

S.A.Q.* A washing machine cost AED 2,900 in 2010 and its price increased by 22.5% over two years. Using the method above, what is the cost now?


% Changes (Type 2): Skills 3. What is the percentage increase from 300 to 450?

The actual change = New quantity Old quantity

The fraction change = Actual change Old quantity

In this case;

Actual change = 450 300 = 150 (Absolute Change)

Fraction change = 450 300 150 0.5300 300

(Relative Change)

Percent change = 450 300 100300

= 50%

This is positive, which means an increase.

Therefore, the percentage increase is 50 %

4 What is the percentage decrease from 450 to 300? Actual change = New quantity Old quantity Fraction change = Actual change Old quantity

In this case; Actual change = 300 450 = 150 (Absolute Change)

Fraction change = 300 450 150 0.3450 450

(Relative Change)

Percent change = 300 450 100 33.3450

This is negative, which means an decrease

Therefore, the percentage decrease is 33.3333 %

The percent change = New quantity Old quantity x 100 Old quantity

The percent change = New quantity Old quantity x 100 Old quantity


% Change Problems: Applications

Sales at the minimarket one month were AED 789,500 and the next month the sales they were AED 981,300. What was the percentage change?

METHOD The actual change = New quantity Old quantity

The fraction change = Actual change Old quantity

The percent change = New quantity Old quantity x 100

Old quantity In this case; Actual change = ___________________ = ___________________ (Absolute Change)

Fraction change = (Relative Change) Percent change = x 100 = _____________

This is positive, which means an increase in sales.

Therefore, the percentage increase is %

Actual change = (Absolute Change)

Fraction change = (Relative Change)

Percent change

Therefore,

S.A.Q.* The population of a country in 2000 was 12,580,000. In 2010 the population of the same country was 13,459,500. What was the percentage change?

The percent change = New quantity Old quantity x 100

Old quantity


Example 1.9

The number of people attending university basketball game last week was 234. This week the number attending was 159. What was the percentage change?

Actual change = 159 234 = 75

Fraction change = 75 234

Percent change = 75 x 100 = 32.05 234

This is negative, which means an decrease in numbers.

Therefore, the percentage decrease is 32.05%

S.A.Q.* The number of students registering for IT majors last semester was 148. This semester is 125. What is the percentage change?


Example 1.6 A New Method For A Different Type Of Problem In a sale a television costs AED 8,700 after it has been reduced by 25%. What was the original price of the television?

We still think of the old price (before the sale) as the 100%. (100% of the old price)

But, we dont know what this old price is.

The decrease in price of 25% means that the new price (8,700) is 75% of the old price.

To calculate the old price we use a method called the unitary method.

This is a method of carrying out a calculation to find the value of a number of items by first finding the cost of ONE OF THEM.

Method.

We know that: 75% of the old price is 8,700

So, 1% of the old price is 8700

75

Therefore, 100% of the old price is 8700

10075

= 11,600

The original price of the television was AED11,600

We dont work this out yet

S.A.Q.* A smart phone cost AED 4,250 after it had been reduced by 15%. Using the method above, what was the original cost?


Example 1.7

An apartment is advertised for sale at AED 1,830,000. This price is 20% more than it was the year before. What was the original price of the apartment.

Let the original (old) price be the 100%. (100% of the old price)

The increase in price means that the new price (1,830,000) is 120% of the original price.

The Method We know that: 120% of the original price is 1,830,000

So, 1% of the original price is 1830000

120

Therefore, 100% of the original price is 1830000

100120

= 1525000

The original price of the apartment was AED 1,525,000

We dont work this out yet

S.A.Q.* A watch costs AED 24,250. This price is 12.5% more than it was last year. Using the method above, what was cost of the watch last year?


Compound Percent Problems Examples in which one percentage change is accompanied by another are called compound percent problems. Example 1.11 A bank employee is earning AED 27,000 a month is given a 10% raise one month followed by another 10% raise the next month.

(a) What is the salary after the second month?

(b) What is the total salary increase over the two months?

Solution

(a) Old salary is AED 27000. The new salary is 110% of 27000. That is, New salary = 1.10 x 27000

Now, the (1.10 x 27000) becomes the old salary.

So, the new salary is 110% of this.

New salary = 1.10 x (1.10 x 27000)

= (1.10 x 1.10) x 27000

= 1.21 x 27000

= 32670

Therefore, the salary after the second month is AED 32,670

(b) The new salary is 1.21 times the original salary

The new salary is 121% of the original salary (1.10 x 1.10 = 1.21)

In other words, the total salary has increased by 21%

Notice that the salary has NOT increased by 10% +10%.

Dont work this

out

.. bring it down to

here.


S.A.Q.*

The number of students registering for Business majors in Spring 2011 was 215. In Fall 2011 this increased by 5%. This semester it increased by 7%. (a) What is the number of students registering for Business this semester? (b) What is the total percentage increase over the two semesters? [Check your answer to part (b)]


Class Exercise

A restaurant advertises a special offer It offers diners 10% discount. But, the restaurant also adds a tax of 15% to the bill. Which would you rather do:

Get the discount first and then pay the tax, or pay the tax first and then get the discount?

Solving the problem

First: try some an easy numerical example and compare costs.

For example, suppose your restaurant bill comes to costs AED1000. Discount First Method Calculate the amount you would pay after the discount is taken off the bill, then calculate the final amount after the tax is added to the bill. Tax First Method Calculate the amount you would pay after the tax is added to the bill., then calculate the final amount after the discount is taken off the bill. What happened?

Next: try another example.

Suppose your bill comes to AED1500 and compare costs. Repeat the calculations. What happened?

That is, you are taking special examples.

What you are doing here is SPECIALIZING.

The next step is to GENERALIZE. .


Generalizing To generalize we let letters stand for numbers.

Let the original bill be represented by the letter C.

Let the discount (as a decimal) be represented by the letter d. (That is: 10% = 0.10)

Let the tax (as a decimal) be represented by the letter t . (That is: 15% = 0.15 (i) What do you pay if you take the discount off first?

Let the cost of the room after discount is taken off be D.

Then, D = C Cd

So, D = C(1 d) (1)

Let the cost of the room after tax is added to the new cost be T.

Then, T = D + Dt

So, T = D(1 + t) (2)

Now substituting for D in (2) we get:

T = C(1 d)(1 + t) (3) Complete the next part yourself. (ii) What do you pay if you add the tax first?

Let the cost of the room after tax is added be T.

Then, ..

So, .. (5)

Let the cost of the room after discount is taken off the new cost be D.

Then, ..

So, .. (6)

Now substituting for T in (6) we get:

. (7) What conclusion can you draw from this?


In the example on the previous page, the restaurant offered a discount to its customers, but added a tax to the bill. Suppose the restaurant offers its diners a discount of 10%, but adds a tax of 12.5% to the bill. We have seen that it does not matter whether the discount is calculated first , or the tax is calculated first; the result is the same.

Discount subtracted first

If the bill comes to AED 1580, then this is the original amount.

With the discount subtracted, the new amount will be 90% of 1580.

90% of 1580 = 0.90 x 1580 = 1422

Now, the AED 1422 becomes the old amount.

With the tax added, the new amount will be 112.5% of 1422.

112.5% of 1422 = 1.125 x 1422 = 1,599.75

The bill will come to AED 1,599.75

Tax added first

If the bill comes to AED 1580, then this is the original amount.

With the tax added, the new amount will be 112.5% of 1580.

112.5% of 1580 = 1.125 x 1580 = 1,777.50

Now, the AED 1777.50 becomes the old amount.

With the discount subtracted, the new amount will be 90% of 1777.50

90% of 1777.50 = 0.90 x 1777.50 = 1,599.75

The bill will come to AED 1,599.75

What we have is:

Final bill = (1.125) x (0.90 x 1580)

i.e. Final bill = 1.0125 times the original bill

i.e. Final bill = 101.25% of the original bill

i.e. The total bill has increased by 1.25%

Discount subtracted

Tax added


Compound Percent Problems : Skills

1. Calculate the effect on a number of an increase of 6% followed by an increase of 4%.

Let the original number be N .

An increase of 6% will make N into the number (1.06) N.

An increase of 4% will make this new number into the number (1.04) (1.06) N

So, the final number will be:

(1.04)(1.06)N = 1.1024 N

Therefore the result will be an increase of 1.1024 of the original number.

i.e. the number will be 1.1024 times the original number.

i.e. the number will be 110.24% of the original number.

i.e. the number will increase by 10.24%

2. Calculate the effect on a number of an increase of 6% followed by an decrease of 4%.


An increase of 6% will make N into the number (1.06) N.

An decrease of 4% will make this new number into the number (0.96) (1.06) N


(0.96)(1.06)N = 1.0176 N

Therefore the result will be an increase of 1.0176 of the original number.



i.e. the number will increase by 1.76%


3. Calculate the effect of an decrease of a decrease of 6% followed by a decrease of 4%.


An decrease of 6% will make N into the number (0.94) N.

An decrease of 4% will make this new number into the number (0.96) (0.94) N


(0.94)(0.96)N = 0.9024 N

Therefore the result will be an decrease of 0.9024 of the original number.



i.e. the number will decrease by 9.76%. (100% - 90.24%)


Commission

In many areas of work, such as insurance, retail, car sales, employees are paid a basic wage (sometimes not) per month and then can earn extra money from the quantity of sales they make in that month.

This is usually a percentage of the sales the employee makes.

This payment is called commission.

Example 1.12 Ahmed receives 9% commission on his net sales. If he sold AED 25,000 in goods during the last month. What is the amount of his commission?

We need to find 9% of AED 25,000

9% of 25,000 = 0.09 x 25000

= 2250

Therefore, Ahmed receives AED 2,250 in commission

S.A.Q.* Bushra receives 7.5% commission on her net sales. If she sold AED 32,500 in goods during the last month. What is the amount of her commission?


Example 1.13 Fatima is paid a basic salary of AED17,000 per month and can make 8% commission on her sales.

In one month, her sales come to AED 15,600. How much will her total salary for that month be?

We need to find

8% of 15,600 = 0.08 x 15600

= 1248

So, Fatima will receive AED 1,248 as commission.

Fatimas total salary for that month = AED 17,000 + AED 1,248 = AED 18,248.

Therefore, Fatimas total salary will be AED 18,248

S.A.Q.* Hamdan receives a basic salary of AED 43,858 per month and can make 12% commission on his sales. If he sold AED 25,560 in goods during the last month. What will his total salary be for that month?


In the previous two examples, Ahmed and Fatima received what is called , fixed commission, where all of the sales earn the same percent commission. In some cases, the commission can vary according to the level of the employees sales.

This payment is called variable commission.

Example 1.14 Sara is paid commission according to the following scale:

4% on the first AED 10,000 in sales each month

6% on the next AED 12,000 in sales each month

9% on sales greater than AED 22,000 each month.

If she sold AED 26,548 in merchandise in one month, what was her commission?

Solution.

For the first AED 10,000 Sara will receive 4% commission.

For the next AED 12,000 she will receive 6% commission.

The remainder will be paid at 9%.

So, the portion of her sales after the first AED 22,000 is:

26548 22000 = 4548

Which means that AED 4,548 of her sales will be paid at 9%

Therefore, Saras commission will be found from the following computation.

0.04 x 10000 = 400.00

0.06 x 12000 = 720.00

0.09 x 4548 = 409.32

Total 1529.32

Therefore, Saras commission is AED 1,529.32


S.A.Q. Mohamed is paid commission according to the following scale:

3% on the first AED 7,000 in sales each month

5% on the next AED 10,000 in sales each month

7.5% on sales greater than AED 17,000 each month.

If he sold AED 18,5385 in merchandise in one month, what was his commission and total salary for that month?


Percentages involving Of And More than

These words can cause confusion and will lead to errors if they are not properly understood.

Example

Suppose your monthly spending triples from 4000 dhs per month to 12000 dhs per month.

You can describe this in two ways using percentages:

a) Using more than

The percentage change = new value old value x 100 old value

= 12000 4000

1004000

= 200 %

So, you can say that your new spending is 200% more than your old spending. b) Using of

A direct fractional comparison of the two values shows that:

New value 12000 = = 3 old value 4000 so, the percentage comparison is

New value 12000 x 100 = x 100 = 300 % old value 4000

So, you can say that your new spending is 300% of your old spending.

Also, your new spending is three times your old spending.

And the actual increase is 12000 4000 = 8000 dhs.


SAQ

Suppose the value of a painting increases from $80,000 to $100,000. Calculate the percentage increase using more than and then using of.


Example

A computer drops in price from AED 6000 to AED4500.

a) Write down the actual decrease

b) Calculate the percentage change in terms of less than.

c) Calculate the percentage change in terms of of.

This means:

a) The actual decrease = 6000 4500 = 1500 (AED) b) The percentage change = new price old price x 100 old price

= 4500 6000

1006000

= 25 %

So, you can say that the new price of the computer is 25% less than the old price.

c) A direct fractional comparison of the two values shows that:

New price = 4500 = 0.75

old price 6000

so, the percentage comparison is

New price 4500 x 100 = x 100 = 75% old price 6000

So, you can say that new price of the computer is 75% of the new price.

Also, the new price is three quarters of the old price.


SAQ A Business College registered 450 students in 2010 and 375 in 2011.

a) Write down the actual decrease in the number of students.

b) Calculate the percentage change in terms of less than.

c) Calculate the percentage change in terms of of.


Relationship between of and more than Notice that there is a relationship between the of value and the more than value.

Examples

25% more than something means 125% of something.

25% less than of something means 75% of something.

100% more than of something means 200% of something

[Also, 100% more than of something means 2 times something] SAQ 45% more than something is

59% less than something is

150% more than something is

75% of something is

300% of something is

Relationship Between Of and More Than

If the compared value is P% more than the reference value,

then it is (100 + P) % of the reference value.

If the compared value is P% less than the reference value,

then it is (100 P) % of the reference value.


A word of warning! DONT CONFUSE OF WITH OFF OF means a part of a whole OFF means a part taken off a whole. OFF is a discount. Example

30% OF 540 dhs is 540 x 0.3 = 162 dhs

30% OFF 540 dhs is (540 30% OF 540 dhs) = 540 162 = 378 dhs

This can also be written as an of statement SAQ A car is advertised for AED 145,000 with 25% off. What does the car cost to buy? SAQ Fatima bought a car which was advertised at AED 145000, but she only paid 75% of that price. How much did she buy the car for?


Percentages Changes of Percentages A problem often occurs when percentages are compared with percentages.

The results are not obvious. Example cccSuppose you spent 10% of your income on shoes last year and 12% of your income on shoes this year, how much did your spending on shoes increase as a percentage of your income? Actual increase = 2%

Percentage increase = 12 10

100 20%10

Therefore your spending has increased by 20% over last year.

This often comes at surprise, but it must be true. Example Suppose inflation falls from 21.5% to 20% in one year . What is the percentage decrease in inflation? Actual decrease = 1.5%

Percentage increase = 20 21.5

100 6.977%21.5

Therefore inflation decreases by 6.977% over the year

(This would normally be rounded to 7%). Example Suppose you spent 20% of your income on shoes last year and 10% of your income on shoes this year, how much did your spending on shoes decrease as a percentage of your income? Actual decrease = 10%

Percentage decrease = 10 20

100 50%20

Therefore your spending has decreased by 50%.

Remember: %change =

Remember: %change =


SAQ

Suppose you spent 15% of your income on shoes last year and 30% of your income on shoes this year, how much did your spending on shoes increase as a percentage of your income?

SAQ

Suppose you spent 30% of your income on clothes last year and only 15% of your income on clothes this year, how much did your spending on clothes decrease as a percentage of your income?

SAQ

Suppose inflation rises from 14% to 15.5% in one year . What is the percentage increase in inflation?


Unit 1 Section 1.1 Percentages in Business Exercises


Unit 1 Exercises on Percentages: Skills (I) Percentages, Fractions and Decimals 1. Complete the following table:

Fraction (Simplest form)

Decimal

Percentage

5

1

0.9

1%

0.12

85%

5

3

0.25

70%

0.01

8

5

12.5%

4

32

360%

0.125

3

1


Skills (II)

1. What is 20% of 3,000 ? 2. What is 30% of 5,000 ?

3. What is 15% of 400 ? 4 What is 12% of 35 ?

5. What is 75% of 600 ? 6. What is 10% of 45 ?

7. What is 35% of 5,600 ? 8. What is 17.5% of 1,000 ?

9. What is 90% of 2,000 ? 10. What is 25% of 4,000 ? 11. 100 is what percent of 200 ? 12. 250 is what percent of 2,000 ?

13. 75 is what percent of 400 ? 14. 3 is what percent of 200 ?

15. 60 is what percent of 720 ? 16. 1,500 is what percent of 12,000 ?

17. 100 is what percent of 12,00? 18. 5 is what percent of 200?

19. 17 is what percent of 500? 20. 11 is what percent of 3,600? 21. What is 100% of 3,000 ? 22. What is 200% of 3,000 ?


25. What is 175% of 6,000 ? 26. What is 300% of 300 ?

27. What is 1000% of 5 ? 28. What is 500% of 1,000 ?

29. What is 100% of 0.5 ? 30. What is 200% of 0.5 ?





39. 1000 is what percent of 200 ? 40. 75 is what percent of 15 ? Mental Exercises [Try these problems without a calculator, then check with your calculator]




47. What is 25% of 400 ? 48. What is 10% of 4,000 ?

49. What is 1% of 200? 50. What is 2% of 400 ?







Unit 1 Applications (I) Exercises on Tax and Discount 1 Applications 1. A restaurant advertises a special offer of 12% discount on meals. But, the restaurant also adds a

tax of 17.5% to the bill. If your bill comes to AED754, how much will you pay for your meal?

2. A restaurant advertises a special offer of 15% discount on meals. But, the restaurant also adds a tax of 12.5% to the bill. If your bill comes to AED1250, how much will you pay for your meal?

3. Sometimes restaurants add service charge to the bill and then add another tax (sometimes a Tourist tax) to that. Suppose you are dining and your bill comes originally to AED1250, and the service charge is 10%. How much will the bill come to after the service charge is 6 pt

If the tourist tax is 6%, how much will the bill come to after the tourist tax is added.

4. Would it matter whether the tourist tax is added first, then the service charge?

Show how you came to your decision.

5. Suppose you are dining and your bill comes originally to AED855, and the service charge is 16%. How much will the bill come to after the service charge is added?

If the Tourist tax is 8%, how much will the bill come to after the tourist tax is added.

6. Would it matter whether the tourist tax is added first, then the service charge? Show how you came to your decision.


If the Tourist tax is 5%, how much will the bill come to after the tourist tax is added?

How much extra will you pay for your meal?


If the Tourist tax is 8%, how much will the bill come to after the tourist tax is added?

How much extra will you pay?

9. Questions 7 and 8 raise a very important point: The extra amount you pay is not just the sum of the two percentage.

That is, the extra amount you pay is not 15% extra nor is it 20% extra.

What single tax would make the bill come to the total amounts in question 7 and 8?

10. If a hotel gives a discount of 20% but has a tax of 16%, how much would you pay for a room which normally costs AED2,500?

11. If there were four people in your group, how much would you have to pay for a seven night stay?


12. A hotel room is advertized as AED 3,700 with a 17% discount. The hotel also adds a tax of 12.5% to the bill. How much would you pay for a five night stay?

13. A store is having a sale. A handbag is advertized as costing AED3,400 with 30% discount.

(i) How much would you pay for the bag?

(ii) How much would you be saving?

14. Generalizing. Write down the formula for the total amount paid C, when the service charge (as a decimal) is s,

and the tourist tax (as a decimal) is t.


Applications (II) Exercises 1. Al Salam Computers had AED 2,534,000 in sales last year and spent AED 4,759 on advertising.

What percentage was spent on advertising?

2. Asma bought a sofa for AED 93000 plus 12% sales tax. How much was the tax?

3. Al Bada Gyms announced that 43 of their staff would lose their jobs in one months time. This is 35% of their staff. What is the size of their total work force?

4. Mobility Mobile Shops are having a 17.5% off sale. What will the markdown on a mobile costing AED 600?

5. Ajman population at over 262,000; Population up by 12,000 and Emiratis account for 16%. Posted on 16/08/2011 [UAE The Official Website News] a) What is the percentage increase of the Ajman population?

b) What is the number of Emiratis in this increase?

7. A shop reduces the price of a laser printer from AED 800 to AED 720. What percent markdown can the shop advertise?

8. All Shammas investments are in two banks. She has AED 125,000 invested in one bank and the other 45% invested the other bank. What is the total amount of her investments?

9. Khaloud earned AED 240,000 last year and she will be given a 5.5% raise this year. What will be her new yearly salary?

10. Ghada invested AED 3,000 in the stock market last year. Today its value is AED 3,750. What was the percent increase?

11. A bookstore offers 20% off all books. If a book normally sells for AED 150, what will the sale price be?

12. The label on a sofa says 15% OFF, and the price tag says AED 9,300. What was the price before the discount?

13. Salama will receive a 5% increase in pay this month and a 6% increase next month. What will be her total percent increase over these two months?

14. A kitchen company receives a discount of 30%, followed by a 20% discount from its suppliers. What is the total percent discount?

15. If a hotel gives a discount of 20% but has a tax of 16%, how much would you pay for a room which normally costs AED2,500?


16. If there were four people in your group, how much would you have to pay for a seven night stay?

17. A hotel room is advertized as AED 3,700 with a 17% discount. The hotel also adds a tax of 12.5% to the bill. How much would you pay for a five night stay?

18. A store is having a sale. A handbag is advertized as costing AED3,400 with 30% discount.

(i) How much would you pay for the bag?

(ii) How much would you be saving?

19. What will be the effect on AED 300 of a 10% increase, followed by a 10% decrease.

20. A supplier normally offers a 30% discount. Due to competition, he wants to increase the discount to 40%. What additional discount is needed?

[Hint: Consider the result of one percent decrease followed by another.]


Skills (III) Find each percent change. 1. a 12% increase followed by a 16% increase.

2. a 25% decrease followed by a 12% decrease.

3. a 15% increase followed by a 12% decrease.

4. a 20% decrease followed by a 17% increase.

5. a 20% increase followed by a 20% decrease.

6. a 50% decrease followed by a 50% decrease.

7. a 10% increase followed by a 10% increase followed by a 10% increase.

8. a 5% decrease followed by a 5% decrease followed by a 5% decrease.

9. the additional percent increase needed to change a 7% increase to a 14% increase.

10. the additional percent decrease to change a 9% decrease to a 15% decrease.


Applications (II) Compounding 1. Hessa will receive a 5% increase in pay this year and a 6% increase next year. What is her

percent increase for the two years combined?

2. A kitchen company receives a 30% discount from its supplier, followed by a 20% discount. What is the total percent discount?

3. A government plans a 4% increase in education expenditure from 2011 for the next five years.

What will be the total percent increase in expenditure?

If the expenditure in 2011 was AED 140 billion, what was the expenditure in the last year? 4. The Ministry of Transport in a country as to make cuts of 3% in its budget for each year over

the next four years. What will be the total percent reduction be?

If the Ministrys budget was AED 12.5 million at the beginning of the cuts, what was the final budget?

5. A supplier normally offers its clients 25% discount. Due to competition, the supplier wants to

increase the discount to 40%. What additional discount is needed


Applications (III) Commission 1. Fatima earned 3.5% commission on sales of AED 45,780. What was her commission?

2. Ahmed earns 7% commission on the sale price of each villa he sells. If he sells 3 villas and the total amount comes to AED2,345,500, what will his commission be?

3. A salesman monthly salary is made up of a basic salary of AED 21,570 and 4.5% commission on all her sales. If one month her sales come to 15,590, what will her salary for that month be?

4. Nouras owns an estate agent and she pays herself 6.5% commission on the amount she earns in sales each month. In a particular month she sells 1 villa for AED 2,580,000; two studio apartments for AED1,1345,000 each and a 3-bedroom apartment for AED 2,159,500. What will be her total earnings that month?

5. Fahad works for a recruitment agency. His monthly salary is made up of a basic salary of AED 25,678 and a commission for every person recruited to a company of 8.4% of his salary. If he recruits three people in a particular month, what will be his total salary?


Skills (IV)

1. What is 75% more than 2100? 2. What is 75% of 2100?

3. What is 20% less than 650 4. What is 20% of 650?

5. What is 175% of 2100? 6. What is 80% of 650?

7. What is 120% of 650? 8. What is 25% of 650?

9. What is 100% more than 20? 10. What is 100% of 20?

11. What is 0% of 300? 12. What is 0% less than 300?

13. What is 200% of 25? 14. What is 200% more than 25?

15. What is 1% of 100? 16. What is 1% less than 100?

In the following exercises compete the sentences using more than or less than:

1. 45% of something is

2. 135% of something is

3. 500% of something is (how many times more than?)

4. 3 times something is (what percentage more than?)

5. One-quarter more than something is (what percentage of?)

6. One-third less than something is (what percentage of?)

7. One-half of something is (what percentage less than?)

8. One-and-a-half times something is (what percentage of?)


Skills (V)

In questions 1 to 12, what is the total percentage change? (State whether it is a decrease or an increase)

1. 40% to 50% 2. 50% to 40%

3. 75% to 50% 4. 50% to 75%

5. 2% to 3% 6. 3% to 2%

7. 25% to 26% 8. 76% to 75%

9. 120% to 150% 10. 6% to 0%

11. A change from 20% to 22%, followed by a further change to 24%

12. A change from 50% to 40%, followed by a further change to 20%

13. What is the new percentage figure when a figure of 20% is increased by 5%?

14. What is the new percentage figure when a figure of 20% is decreased by 5%?

15. What is the new percentage figure when a figure of 20% is increased by 100%?


COL112


Unit 1


Section 1. 2

Constant Growth and Percentage Growth Department of Mathematics and Statistics University College Zayed University


Unit 1 Section 1.1.2 Constant Growth and Percentage Growth


1.1.2 Constant Growth and Percentage Growth Solving the problem

This is an ideal problem for Excel. Open a new workbook in Excel.

Part (a) is fairly easy to calculate.

In week 1 the company is producing 1200 computers.

In week 2 the company will produce 1200 + 80 = 1280 computers.

In week 3 the company will produce 1200 + 80 + 80 = 1200 + 80 x 2 = 1360 computers

And so on (Keep the pattern)

In week 10 the company will produce 1200 + 80 x 9 = 1920 computers

Therefore, in week 10 the company will be producing 1,920 computers.

To write this out a bit more mathematically:

Let A be the number of computers made (production value) in week 10.

Then, A = 1200 + 80 x 9

Part (b) is also not too bad. Use Excel to complete this table.

Week 1 2 3 4 5 6 7 8 9 10 Total

Computers 1200 1280 1360

Part (c) There is a mathematical way, but we will use Excel. Extend the table by increasing the week numbers until production exceeds 5,000 per week.

Production will reach 5,000 computers (per week) in week 49

Case Study 2.1 At a certain time a company produces 1,200 computers each week and then increases its production by 80 computers each week after that. (a) How many computers will it be producing per week in week 10?

(b) What will be the total number of computers the company has produced in week 10?

(c) Use Excel to find in what week production will reach 5,000 per week.


Solving the problem

Part (a) is fairly easy to calculate.

In week 1 the company is producing

In week 2 the company will produce .

In week 3

In week 8 the company will produce

Therefore, in week 8 the company will be producing computers.

To write this out a bit more mathematically:

Let A be the number of computers made (production value) in week 8.

Then, A =

Part (b) is also not too bad. Use Excel to complete the table.

Week Total

Computers

Part (c)

There is a mathematical formula for calculating the total number of computers produced, but here we will use Excel. Extend the table by increasing the week numbers until production exceeds 5,000 per week.

Answer

Self -Assessment Question (SAQ) At a certain time a company produces 1,500 computers each week and then increases its production by 90 computers each week after that. (a) How many computers will it be producing per week in week 8?


(c) Use Excel to find in what week production will reach 5,000 per week.


Graph of the data for Case Study 2.1

A very important thing to be able to do is to see on a graph this relationship between production value and number of weeks .

This is the graph of compute production against weeks for constant growth.

The graph looks like a straight line.

And in fact it is.

This is because production increases by a constant amount each week.


Solving the problem

Part (a)


In week 2 the company will produce 1200 + (1200 x 0.05) = 1200 + 60 = 1260 computers.

In week 3 the company will produce 1260 + (1260 x 0.05) = 1260 + 63 = 1323 computers.

In week 4 the company will produce 1323 + (1323 x 0.05) = 1323 + 66.15 = 1389.15 computers.

In week 5 the company will produce 1389.15 + (1389.15 x 0.05) = 1389.15 + 69.46 = 1458.61

And so on . . .

This time the pattern is easy to continue, but it is not easy to see a formula.

If we want to know how many computers are made in week 10, we have to do this calculation another 5 times.

We need a different approach this time and we will use some extra mathematics.

5% = 0.05

Case Study 2.2 Another company produces 1,200 computers each week and then increases its production by 5% each week after that. (a) How many computers will it be producing per week in week 10?


(c) Use Excel to find in what week production will reach 5,000.


Lets start again.


In week 2 the company will produce 1200 + 1200 x 0.05 = 1200(1 + 0.05)

= 1200(1.05) = 1260 computers.

In week 3 the company will produce 1260 + 1260 x 0.05 = 1260(1.05)

= 1200(1.05)(1.05)

= 1200(1.05)2 = 1323 computers.

In week 4 the company will produce 1323 + 1323 x 0.05 = 1200(1.05)3 = 1389.15 computers.

And so on . . . Keep the pattern.

In week 10 the company will produce 1200(1.05)9 = 1861.16 computers

= 1861 actual computers

Part (b) is also not too bad. Use Excel to complete the table.

Week 1 2 3 4 5 6 7 8 9 10 Total

Computers 1200 1260 1323 1389

Part (c) There is a mathematical formula for calculating the total number of computers produced, but here we will use Excel. Extend the table by increasing the week numbers until production exceeds 5,000 per week.

Production will reach 5,000 computers (per week) in week 31

Remember:

1260 = 1200(1.05)


In week 1 the company is producing computers.





Part (b) is also not too bad now.

Week Total

Computers

Part (c)

Extend the table by increasing the week numbers until production exceeds 5,000 per week.

SAQ Another company produces 1,500 computers each week and then increases its production by 4% each week after that. (a) How many computers will it be producing per week in week 8?


(c) Use Excel to find in what week production will reach 5,000.


Graph of the data for Case Study 2.1

A very important thing to be able to do is to see on a graph this relationship between production value and number of weeks .

This is the graph of compute production against weeks for percentage growth.

The graph looks like a straight line.

And in fact it is not.

This is because the effect of percentage increases do not become obvious in the initial stages.

This looks like a straight line, but later we will see that it definitely is not.

Plot the data from the SAQs in Excel in a similar way.


SAQ 2.1

The company produced 1,500 computers in week 1 and increased production each week by 90.

SAQ 2.2

The company produced 1,500 computers in week 1 and increased production each week by 4%.

It looks like increasing production by a constant amount is better than increasing by a percentage.

But look how long it takes the constant increase to get to 5,000 per week compared with the

percentage increase. (40 weeks for constant growth and 32 weeks for percentage growth)

Something is going on here!

After 8 weeks production is 2130

After 8 weeks production is 1974


Solving the problem

Part (a) is fairly easy.


In week 2 the company will produce 1200 + 80 = 1280 computers.

In week 3 the company will produce 1200 + 80 + 80 = 1200 + 80 x 2 = 1360 computers

And so on (Keep the pattern)

In week 10 the company will produce 1200 + 80 x 9 = 1920 computers

Therefore, in week 10 the company will be producing 1920 computers.

What we want is to be able to write this out mathematically.

Let A be the number of computers made (production value).

Then, A = 1200 + 80 x 9.

Taking this further, we can generalize.

Let n be the number of weeks.

Then, we can write, that in week n, the number of computers produced in the nth week is:-

An = 1200 + 80(n1)

Now, we can now work out the production value for any number of weeks.

The problem above is an example of a very important topic in mathematics.

Case Study 2.1 REVISITED. GENERALIZATION. At a certain time a company produces 1,200 computers each week and then increases its production by 80 computers each week after that. (a) How many computers will it be producing per week in week 10?


(c) In what week will production reach 5,000?


The Mathematics of Constant Growth

Arithmetic Series (or Arithmetic Progression)

The numbers :

1200, 1280, 1360, 1440, 1520, 1600, 1680, 1760, 1840, 1920

(taken from the production values for constant growth) form a sequence of numbers.

In mathematics a sequence is a list of numbers which follow a definite rule or pattern.

In this case:

the difference between each consecutive number is the same (constant) .

This is called the common difference of the sequence, and we use the letter d to stand for this.

This type of sequence is called an arithmetic sequence.

Also, there is a first number (in mathematics, we say first term), and we use the letter a to stand for this.

From the example we saw that if we wanted to find how many computers would be produced in the nth week was

An = 1200 + 80(n1)

Generalizing further If we let An be the value of the nth term.

If we let a be the first term,

and d be the common difference.

Then,

An = a + (n1)d

So, we can write the general arithmetic sequence as:-

a , a + d , a + 2d , a + 3d , . . . , a + (n1)d

And now we can use this formula to find any term of an arithmetic sequence.

nth term

1st term


Example 2.1

Find the 50th term of the sequence

25, 40, 55, 70, . . .

From the first four terms, we can write: a = 25, d = 15

So, if we let the 50th term be A50 A50 = 25 + 15(49) = 760

Example 2.2

Find the 10th term of the sequence 70, 50, 30, 10, . . .

From the first four terms, we can write:

a = 70, d = 20

So, A10 = 70 + 9(20) = 110 [Check this using Excel]

SAQ Find the 25th term of the sequence:

1, 11, 21, 31, . . . [Check using Excel]

SAQ Find the 10th term of the sequence:

132, 120, 108, 96, . . . [Check using Excel]


Part (b). What will be the total number of computers the company has produced in week 10? This needs more thinking about This time, we get a mathematical series of terms:

1200 + 1280 + 1360 + 1440 + . . . up to week 10. Or,

1200 + 1280 + 1360 + 1440 + 1520 + 1600 + 1680 + 1760 + 1840 + 1920

How can we calculate the sum?

This is no problem with a calculator and even easier with Excel.

But what if the company wanted to know the total number of computers produced in a year! (52 weeks)

Eventually we would want a formula.

In other words we would want to:-

The next step will take us towards a formula.

The numbers:

1200 + 1280 + 1360 + 1440 + 1520 + 1600 + 1680 + 1760 + 1840 + 1920

form a series of numbers.

A series is the sum of the terms of a sequence.

Since the difference between consecutive terms is the same (common difference)

This type of series is called an arithmetic series.

Generalize


Formula For The Sum Of An Arithmetic Progression. (AP)

The formula for the sum of an arithmetic series became well known after a young boy in a mathematics class in about 17** astounded his teacher by working out the answer to the question, What is the sum of the first 100 numbers? in his head!

His name was Leonard Euler (pronounced Oiler) and he became one the most important mathematicians of all time.

We will approximately follow his method .

Let the total number of computers produced be S (for sum).

Then, S = 1200 + 1280 + 1360 + 1440 + 1520 + 1600 + 1680 + 1760 + 1840 + 1920

By reversing the order we get:

also S = 1920 + 1840 + 1760 + 1680 + 1680 + 1520 + 1440 + 1360 + 1280 + 1200

Now, add both sets of numbers together:

2S = (1200 + 1920) + (1280 + 1840) + (1360 + 1760) + . . . + (1920 + 1200)

So, 2S = 3120 + 3120 + 3120 + 3120 + 3120 + 3120 + 3120 + 3120 + 3120 + 3120

So, we can write:

2S = 3120 x 10

That is, 10

3120 156002

S

Therefore, the total number of computers produced at the end of 10 weeks is 15,600.

[Check this in Excel]

Notice that 3120 came from 1200 + 1920.

In other words, 3120 is sum of the first term and the last term.

There are 10 terms here


Lets look at the solution to the problem again. We found that the sum of the 10 terms was:

10

S 31202

which we can write as:

10

S 1200 19202

Generalizing Let n be the number of terms in the AP.

Let a be the first term

Let l be the last term Let Sn be the sum on n terms . (l is the letter ell)

then, nn

S a l 2

In words: But, suppose we dont know the last term.

Suppose we only know the first term and the common difference.

Can we still find the sum of n terms.

The answer is yes; we just need some algebra.

Step 1: the last term l is the nth term. Therefore, l = a + (n1)d Step 2: we can write:

n

n

nS a a n d

nS a n d

12

2 12

The sum of n terms of an arithmetic series is half the number of terms multiplied by the sum of the first term and the last term.

Recall: nth term of an

arithmetic sequence is:-

An = a + (n1)d


Example 2.3 Find the 15th term and the sum of the first 15 terms of the arithmetic series

36, 48, 60, 72, 84, . . . From the first four terms, we can write

a = 36, d = 12 Let A15 be the 15th term. Then, A15 = 36 + 14(12) = 204 Let S15 be the sum of the first 15 terms, Then

S ( ) ( ) , 15

152 36 14 12 1 800

2

Example 2.4 Find the sum of the first 12 terms of the arithmetic series

100, 75, 50, 25, . . . From the first four terms, we can write

a = 100, d = 25 So if we let S12 be the sum of the first 12 terms,

S ( ) ( ) 1212

2 100 11 25 4502

SAQ Find the sum of the first 20 terms of the arithmetic series

5, 20, 35, 50, . . .

SAQ Find the sum of the first 30 terms of the arithmetic series

125, 120, 115, 110, . . .

Always check with

Excel!


Part (c) In what week will production reach 5,000? This is more difficult and involves algebra. From part (a), we know that there is a simple formula to find the production after n weeks. The formula is :

An = a + (n1)d

Now in this problem we dont know how many weeks it will take to reach a production of 5,000.

In other words, the number of weeks is an unknown.

So, we use a letter from the alphabet in place of the number.

Let n be the number of weeks it takes to reach a capacity of 5,000.

In this case, a = 1200, d = 80

So, the equation becomes:

n

n

n

n

n

n

n

n

( )

( )

( )

( )

.

.

.

5000 1200 80 1

5000 1200 80 1

3800 80 1

80 1 3800

38001

80

1 47 5

47 5 1

48 5

Therefore, Production will reach 5,000 in week 49. The applications of arithmetic series play an important part in business mathematics.

We will visit arithmetic series again in this course.

It is important to know these formulas.


Part (c) In what week will production reach 5,000? From part (a), we know that there is a simple formula to find the production after n weeks. The formula is :

An = a + (n1)d





In this case, a = d =

So, the equation becomes:

Therefore, Production will reach 5,000 in week .

Self -Assessment Question (SAQ) At a certain time a company produces 1,500 computers each week and then increases its production by 90 computers each week after that.

(a) How many computers will it be producing per week in week 8?


(c) Use the formula for the sum of n AP to find in what week production will reach 5,000 per week.


Solving the problem

Part (a)

We saw that there is a method of calculating the 10th term


In week 2 the company will produce 1200 + 1200 x 0.05 = 1200(1 + 0.05)

= 1200(1.05) = 1260 computers.

In week 3 the company will produce 1260 + 1260 x 0.05 = 1260(1.05)

= 1200(1.05)(1.05)

= 1200(1.05)2 = 1323 computers.

In week 4 the company will produce 1323 + 1323 x 0.05 = 1200(1.05)3 = 1389.15 computers.

In week 10 the company will produce 1200(1.05)9 = 1861.16 computers

= 1861 actual computers

What we want is to be able to write this formula out more mathematically.

Let G be the number of computers made (production value).

Then in week 10, G = 1200(1.05)9

Taking this further, we can generalize.

Let n be the number of weeks.

Then, we can write, that in week n, the number of computers produced in the nth week is:-

Pn = 1200(1.05)(n1)

Now, we can now work out the production value for any number of weeks.

Case Study 2.2 REVISITED. GENERALIZATION Another company produces 1,200 computers each week and then increases its production by 5% each week after that. (a) How many computers will it be producing per week in week 10?


(c) In what week will production reach 5,000?

5% = 0.05

Remember:

1260 = 1200(1.05)


The Mathematics of Percentage Growth

Geometric Series (or Geometric Progression - GP)

This problem is another example of a very important topic in mathematics.

The numbers:

1200, 1260, 1323, 1389, 1458, 1531, 1608, 1688, 1772, 1861

form a sequence of numbers also (but, a different sequence from that in Example 2.)

In this case: the ratio between each consecutive number is the same.

This means that: . , . , . , 1260 1323 1389

1 05 1 05 1 051200 1260 1323

and so on

This is called the common ratio of the sequence, and we use the letter r to stand for this.

This type of sequence is called an geometric sequence.

Also, there is a first number (in mathematics, we say first term), and we use the letter a to stand for this.

From the example we saw that if we wanted to find how many computers would be produced in the nth week was

Gn = 1200(1.05)(n 1)

Generalizing further If we let Gn be the value of the nth term.

If we let a be the first term,

and r be the common ratio.

Then, the nth term of the geometric sequence is:

Gn = a r(n 1)

So, we can write the general geometric sequence as:-

a, ar, ar2, ar3, ar4, . . . , ar(n1)

nth term

1st term

2nd term


Example 2.5

Find the 9th term of the geometric sequence: 2, 6 , 18, 54 , . . .

a = 2 , r = 6

2= 3

Let 9th term be G9. Then, G9 = 2 x 38 = 13,122

Example 2.6 Find the 7th term of the geometric sequence:

, , , ,1 1 1

12 4 8

which is the same as 1 , 0.5 , 0.25 , 0.125 , . . .

a = 1 , r =

112

1 2 which is the same as a = 1 , r = 0.5

So, G7 = 1 x

61 1

2 64 which is the same as G7 = 1 x (0.5)6 = 0.015626

SAQ Find the 5th term of the geometric sequence:

5, 1, , . . .

SAQ Find the 5th term of the geometric sequence:

6, 24, 96, 376, . . .


Part (b) The formula for the sum of series

1200 + 1260 + 1323 + 1389 + . . . + 1772 + 1861

is too difficult to prove at this stage, but we will give it and use Excel to verify the results. Let Sn be the sum of the first n terms of the geometric sequence.

Then Sn = na rr

1

1 This is can be proved, but the mathematics is more difficult than for arithmetic series. So, 1200 + 1260 + 1323 + 1389 + 1458 + 1531 + 1608 + 1688 + 1772 + 1861

is S10 = ..

101200 1 05 1

1 05 1 = 15,093

Therefore, the total production after 10 weeks 15,093 computers. [Use Excel to check this.]


Example 2.6 Find the 8th term and the sum of the first 8 terms of the geometric series

12, 36, 108, 324, . . . Since we are told that this is a geometric sequence.

a = 12 , r = 36

12 = 3

Let G8 be the 8th term of the sequence. Then, G8 = 12 x 37 = 26,244 Let the sum of the first 8 terms be S8

Then, S8 =

812 3 1

3 1 = 39,360

Example 2.7 Find the 6th term and the sum of the 6th term of the geometric sequence

5 , 1 , 0.2 , 0.04 , 0.008 , . . .

a = 5 , r = 0.2 Let G6 be the 6th term of the sequence. Then, G6 = 5 x 0.25 = 0.0016 Let the sum of the first 6 terms be S6

Then, S6 = .

.

65 0 2 1

0 2 1 = 6.2496

SAQ Find the sum of the first seven terms of the geometric sequence:

3, 15, 75,375, . . .

Notice that

S6 = = 6.2496 also.

also.

SAQ Find the sum of the first 5 terms of the geometric sequence:

10, 5, , . . .


Part (c) Once again, the solution to this problem is too difficult to solve mathematically at this stage, but we can solve it using Excel.

To find out when production exceeds 5,000 we can use two methods in Excel.

Method 1.

We can call this the Drag and See method.

What we do here is to set up the formula for the geometric progression that gives us the production value in each week and copy the formula down until the production value reaches 5,000 or greater.

Therefore, production reaches 5,000 in week 31.

Production reaches 5,000 in week - .

=1200*(1.05)^(A2-1)

Dragging the formula in cell B2 down we finally reach the cells which contain 31 (weeks)and 5186 (production value).

SAQ Another company produces 1,500 computers each week and then increases its

production by 4% each week after that. (a) How many computers will it be producing per week in week 8?


(c) Use the drag and see and then the solver method in Excel to find in what week production will reach 5,000.


Method 2 This method uses Excels way of solving equations.

From part (a), we know that there is an formula to find the production after n weeks.

The equation is Gn = ar(n1)





In this case, a = 1200, r = 1.05

So, the equation becomes: 5,000 = 1200(1.05)(n1)

And we want to know what n is. Using solver , with a = 12000 and r = 0.05, the solution is: 30.25

Therefore, production reaches 5,000 in week 31 .

Formula: =1200*(1.05)^(G7-1)

SAQ Using solver , with a = 15000 and r = 0.04, the solution is:

Therefore production reaches 5,000 in week .


Lets summarize what we have done so far.

The problems Case Study 2.1 and Case Study 2.2 were about a company producing computers.

In Example 2, the company increased its production by a constant amount each week, while in Example 3 the company increased its production by a percentage of its previous weeks amount each week.

We can write the results down. There are some important results here.

1. A constant increase in production means more computers in week 10 than percentage increase.

2. A constant increase in production means more total number of computers produced after week 10 than percentage increase.

3. But, a constant increase in production takes longer to reach a value of 5,000 than percentage increase.

THE QUESTION IS WHY?

WHAT IS HAPPENING?

The results become clear when we plot the graphs using Excel.

We will plot both sequences of numbers on the same graph. Exercise 1. Plot the graph of the production values for both examples up to week 10. 2. Plot the graph of the production values for both examples up to week 20. 3. Plot the graph of the production values for both examples up to week 50. What do you notice?

Case Study 2.1 Constant Increase

Result:

1. Arithmetic sequence

2. Formula for nth term:

An = a(n1)d

where a = first term, d = common difference

3. Production in week 10 = 1920

4. Formula for sum of n terms

5. Total after week 10 = 17,600

6. No. of weeks to reach 5,000 = 49

Case Study 2.2 Percentage Increase

Result:

1. Geometric sequence

2. Formula for the nth term:

Gn = ar(n1)

where a = first term, r = common ratio

3. Production in week 10 = 1861

4. Formula for sum of n terms

5. Total after week 10 = 15,093

6. No. of weeks to reach 5,000 = 31


Graph of Production Values for Week 1 to Week 10

Graph of Production Values for Week 1 to Week 20


Unit 1 Section 1.2 Constant Growth and Percentage Growth Exercises


Constant Growth and Percentage Growth Applications Exercises

1. At a certain time a company manufactures 600 TVs each week and then decides to increase its production by 25 TVs each week after that.

Part (a)

(i) How many TVs will it be manufacturing per week in week 8?

(ii) What will be the total number of TVs the company has manufactured in week 8?

(iii) In what week will production of TVs reach 1,200?

Part (b)

If the company had decided to increase the number of TVs manufactured by 3% per week, answer the following questions.

(i) How many TVs will it be manufacturing per week in week 8?

(ii) What will be the total number of TVs the company has manufactured in week 8?

(iii) In what week will production reach TVs 1,200?

Part (c)

(i) Which would be the better decision up to week 8? Explain your answer.

(ii) Use Excel to plot the graphs of TVs manufactured from week 1 to week 8 on the same axes.

(iii) Estimate the week in which the number of TVs produced would be the same. What happens after this week?

2. At a certain time a company is producing 5,600 greetings cards per day. The company decides to increase its output by 6% each day.

Part (a)

(i) How many greetings cards per day will it be on day 12?

(ii) What will be the total number of greetings cards per day the company has produced by day 12?

(iii) On what day will production of greetings cards per day reach 12,000?

Part (b)

If the company had decided to increase the number of greetings cards per day produced by 450 greetings cards per day

(i) How many greetings cards per day will it be producing per day on day 12?

(ii) What will be the total number of greetings cards per day company has produced by day 12?

(iii) On what day will production reach greetings cards per day 12,000?

Part (c)

(i) Which would be the better decision up to day 12? Explain your answer.

(ii) Use Excel to plot the graphs of greetings cards per day produced from day 1 to day 12 on the same axes.

(iii) Estimate the day on which the number of greetings cards produced would be the same. What happens after this day?


3. The government of a certain oil-producing country is trying to predict the oil requirements in the year 2020. In 2011, the consumption was 10.4 millions of barrels per year. Two consultant agencies are asked to forecast the countrys oil requirements. They are:

Forecast 1. Increases of 350,000 barrels per year.

Forecast 2. Increases by 6% each year.

Use series to calculate the forecasts from each agency for the following years:

(a) The number of barrels of oil required in 2015, 2020, 2030.

(b) The total number of barrels used from 2011 to 2030.

4. An e-market company is planning to expand into two new markets. Based on extensive market research, the projections for each market are as follows:

Market 1: the number of customers will increase by 800 per year.

Market 2: the number of customers will increase by 7% each year.

If the number of customers at the start of 2007 is 7600 in Market 1 and 5200 in Market 2, use series to calculate for each market.

(a) The number of new customers in 2020.

(b) The number of customers in 2020.

(c) What will happen in the long run?


Arithmetic Progressions and Geometric Progressions Skills

Exercises 1. (a) State whether the following sequences are APs or GPs or neither.

If they are APs or GPs, state the value of the first term and the common difference or common ratio.

(i) 5, 8, 11, 14, . . .

(ii) 2, 8, 32, 128, . . .

(iii) 4, 4.5, 5, 5.5, . . .

(iv) 100, 98, 96, 94, . . .

(v) 0, -15, -30, -45, . . .

(vi) 100, 50, 25, 12.5, . . .

(vii) 2, 9, 16, 22, . . .

(viii) 1, 4, 9, 16, . . .

(b) For each of the APs or GPs in part (a):

(i) Calculate the value of the 10th term of the sequence.

(ii) Calculate the sum of the first 10 terms.

(iii) In parts (i) and (iii) calculate the first term which is greater than 100.

2. Find the sum of the first 12 terms of the series: 50 + 45 + 40 + 35 + . . .

3. Find the sum of the first 8 terms of the series: 1 3 5 7

4 4 4 4

4. Find the sum of the first 10 terms of the series: 1 1 1 1

3 9 27 81

5. Find the value of the 25th term of the sequence: 180, 162, 145.8, 131.22 , . . .

6. Find the sum of the first 11 terms of the series: 10 + 8 + 6 + 4 + . . .

In questions 7 9 use Excels drag and see function, or the Equation Solver, or algebra, to answer the questions.

7. Given that 1 + 2 + 3 + 4 + . . . = 78, find the number of terms we need to give 78.



10. Use Excel to investigate the sum of the series below for more and more terms.

1 1 11

2 4 8

Make a guess at the sum of the series as you add more and more terms.


COL112


Unit 1


Section 1.3

Large and Small Numbers Department of Mathematics and Statistics University College Zayed University


Unit 1 Section 1.1.3 Large and Small Numbers


1.1.3 Large and Small Numbers Look in any newspaper, watch any current affairs TV program, look at your bank statements(!?) and you will soon become aware of the number of times the words million, billion, and trillion are mentioned.

If you want to be able to understand these numbers and calculate with them, then you must know about how our number system works.

Dealing With Large (And Small) Numbers

Number Systems

Over the thousands of years of human cultural and scientific development many types of number systems evolved. Almost all of them have died out and some of the ones that remain are there only because historical accident and they are of limited use. (Roman, Babylonian, to name but two)

The system which has proved the most useful and powerful is the place value system.

Place Value System

The place value system is based on 10 and its powers. Using the place value system, only the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 (zero) are needed.

The value of a digit depends on its place in the number itself.

This means that the digit 7 can be represent the number:

7 tens; 7 thousands; 7 trillion; 7 hundredths; 7 millionths;

depending on where it is in the number.

The position of the digit 7 gives it its value in the number.

SAQ In the numbers below, write down the place value of the digit 7.

70 ; 57,625 ; 7,500,000,001 ; 0.0735 ; 0.105007.


Usually we do not write the + symbol for

positive exponents

But we always write the symbol for negative

exponents

Exponential Notation (Scientific Notation) When dealing with very large numbers we can use the fact that our number system is based on the powers of ten to shorten the way we write - and speak them. For example

3,000 =

(3 x 1000) + (0 x 100) + (0 x 10) + (0 x 1) =

3 x 1000 = 3 x 103

When we write 1000 as 103 we are using the exponential form.

The small 3 raised to the right of the 10 is called the exponent and the 10 is called the base.

An easy way to think of the exponential form is to think of numbers to the left of the decimal point as multiples of ten while numbers to the right of the decimal point as divisions by ten:

100000 = 10 x 10 x 10 x 10 x 10 = 10+5 = 105

10000 = 10 x 10 x 10 x 10 = 10+4 = 104

1000 = 10 x 10 x 10 = 10+3 = 103

100 = 10 x 10 = 10+2 = 102 10 = 10 = 10+1 = 10 1 = 1 = 100 = 1 0.1 = 1/10 = 10 1

0.01 = 1/100 = 10 2

0.001 = 1/1000 = 10 3 0.0001 = 1/10000 = 10 4 0.00001 = 1/100000 = 10 5 Some terms for numbers that come up many times in business and finance are: One Million = one thousand thousand = 106

One Billion = one thousand million = 109

One Trillion (US) = one million million = 1012

Look at these website:-

www.onlineconversion.com/large_numbers.htm

http://www.onlineconversion.com/finance.htm

Add them to your favorites.


Calculator Mathematics How Calculators Deal With Large and Small Numbers. In-class activity 1. Large Numbers Use your calculator to perform the following calculations and record the results. Punch in: 5000 X 1000000 and record the result.

Now, punch in 5000 X 10000000000 and record the result. On the Casio fx-82MS scientific calculator, the output display for that calculation is:

(Yours might be slightly different.) This is because 50,000,000,000,000 (the full answer) cannot fit into the 10 spaces in the calculator display.

The calculator is displaying the result in what is called scientific notation or exponential form.

It is a shorthand way of writing the result.

That is: 5.0 x 1013 = 5.0 x 10000000000000 = 50,000,000,000,000

Say that in words and then write it in words.

5. x10 1 3


Next: Punch in 2.567 x 500000000 and record the result. Now, punch in 2.567 x 5000000000 and record the result. Write the number out in full. Now say it and then write it in words. Punch in 25.67 x 5000000000 and record the result. Write the number out in full. Now say it and then write it in words. What do you think will be the result if you punch in 256.7 x 5000000000 ? Write down what you think and then use your calculator to check. Nice and simple. But, beware of calculator surprises.


Your calculator does well in the first calculation; the difference is only 35. But on the second calculation the difference is 35,000. If the numbers represented dollars or dirhams, a lot of money (to you and me) has disappeared! Things can get worse. The more multiplications and divisions that are involved, the more numbers disappear.

SAQ

On your calculator multiply 2,478,454,321 by 365 and write down the digits in the display.

On Excel multiply 2,478,454,321 by 365 and write down the answer. (The correct answer!)

Now write down the calculator answer in full.

Now subtract the answer from Excel and the full answer from the calculator .

SAQ

This time,

On your calculator multiply 2,478,454,321 by 365000 and write down the digits in the display.

On Excel multiply 2,478,454,321 by 365000 and write down the answer. (The correct answer!)

Now write down the calculator answer in full.

Now subtract the answer from Excel and the full answer from the calculator .


2. Small Numbers Use your calculator to perform the following calculations and record the results. Punch in: 1 2 and record the result.

Now, punch in 1 200 and record the result. On the Casio fx-82MS scientific calculator, the output display for that calculation is:

(Yours might be slightly different.) This is because the Casio displays decimal answers in exponential form.

It is a shorthand way of writing the result.

It means 5.0 x 10-3 , w

col112 unit 1 mathematical thinking and modeling in business(1)

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