School Made Easy

Sample Final Exam:Business Finance

noreply@blogger.com (An aspiring millionaire) — Thu, 16 Sep 2010 07:44:00 +0000

The tutorial for the following is available at SoF.
TRUE / FALSE

1. Financing through common stock is the most popular method for corporations.

2. Although maximization of the market value of a firm’s common stock is a valid objective of the firm, it is not without its drawbacks since the effects of financial structure

decisions are not reflected in this term.

3. If a foreign currency is expected to depreciate with respect to the home currency, the

holder of a net liability in foreign currency will profit.

4. The money market is usually thought of as dealing with long-term debt instruments

issued by firms with excellent credit ratings.

5. Sales occurring in the secondary markets increase the total stock of financial assets that exist in the economy.

SIMPLE MULTIPLE CHOICE (only provide answer)

6. Which of the following statements is correct?

a. Firms whose sales are very sensitive to changes in the business cycle are more

likely to rely on debt financing.

b. Firms with large tax loss-carry-forwards are more likely to rely on debt financing.

c. Firms with a high operating leverage are more likely to rely on debt financing.

d. None of the above statements is correct.

7. The break-even quantity of output results in an EBIT level equal to:

a. fixed costs.

b. contribution margin.

c. zero.

d. variable costs.

8. Which of the following could offset the higher risk exposure a company would face if it

were to engage in heavy short-term borrowing?

a. Its inventory would need to be mostly highly liquid consumer impulse items.

b. Its accounts receivable collection policy could increase the average collection

period.

c. It could offer no discounts for early payment by its customers.

d. It could buy back some of its shares in the open market in order to reduce its

equity.

9. Which of the following is a characteristic of an efficient market?

a. Small number of individuals.

b. Opportunities exist for investors to profit from publicly available information.

c. Security prices reflect fair value of the firm.

d. Immediate response occurs for new public information.

10. Which of the following is inconsistent with an optimal capital structure policy?

a. Lower the blended cost of debt and equity.

b. Maximize a firm’s common stock price.

c. Minimize the cost of capital.

d. Maximize EPS.

PROBLEM MULTIPLE CHOICE (must show work)

11. Your company is considering an investment in one of two mutually exclusive projects. Project 1 involves a labor intensive production process. Initial outlay for Project 1 is $1,495 with expected after tax cash flows of $500 per year in years 1-5. Project 2

involves a capital intensive process, requiring an initial outlay of $6,704. After tax cash

flows for Project 2 are expected to be $2,000 per year for years 1-5. Your firm’s discount rate is 10%. If your company is not subject to capital rationing, which project(s) should you take on based on the basic capital budgeting criteria (NPV and IRR)?

a. Project 1

b. Project 2

c. Projects 1 and 2

d. Neither project is acceptable.

12. A friend of yours is trying to determine whether to open a sandwich stand at the local mall based on the following data. She expects total fixed costs per year of $24,000, a sale price per sandwich of $3.00, and variable costs per sandwich of $1.80. What is the breakeven level of output for this endeavor:

a. 12,000.

b. 16,000.

c. 20,000.

d. 24,000.

13. Kimble Carpets, Inc. has asked you to calculate the company’s current ratio for 2001. All you have is a partial balance sheet and some assumptions. Using the information provided, calculate Kimble’s current ratio for 2001.

Gross profit margin = 50%

Inventory turnover (COGS/Inv) = 5

2001 sales = $3,000

Assets Liabilities & Equity

Cash 60 Accounts payable $50

AR $40 Accruals 200

Inventory 300 Long-term debt $400

Net fixed assets $500 Equity $250

Total assets $900 Total liab. & equity 900

a. 0.3

b. 0.8

c. 1.6

d. 2.2

14. Company ABC has a target capital structure of 50% debt and 50% equity. They are

planning to invest in a project which will necessitate raising new capital. New debt will

be issued at a before-tax yield of 12%, with a coupon rate of 10%. The equity will be

provided by internally generated funds. No new outside equity will be issued. If the

required rate of return on the firm’s stock is 15% and its marginal tax rate is 40%,

compute the firm’s cost of capital.

a. 13.5%

b. 12.5%

c. 7.2%

d. 11.1%

.12*(1-.4)*.5+.5*.15=11.1%

Use the following to answer questions 15-16. The information below describes a project with an

initial cash outlay of $10,000 and a required return of 12%.

After-tax cash inflow

Year 1 $6,000

Year 2 $2,000

Year 3 $2,000

Year 4 $2,000

15. Which of the following statements is correct?

a. The project should be accepted since its NPV is $353.87.

b. The project should be rejected since its NPV is -$353.87.

c. The project should be accepted since it has a payback of less than four years.

d. The project should be rejected since its NPV is -$23.91.

16. This project:

a. has an IRR of 9.86%.

b. should be accepted based on the IRR criterion.

c. has an IRR of 12%.

d. both a and b.

This is the only acceptable choice since NPV is negative at 10%, hence IRR is lower than that.

DETAILED PROBLEMS (must show work)

Problem # 1:

Bailey and Brothers has applied for a loan from the Me Too Bank in order to invest in several potential opportunities. In order to evaluate the firm as a potential debtor, the bank would like to compare Bailey and Brothers to the industry. The following are the financial statements given to Trust Us Bank:

Balance Sheet 12/31/2005 12/31/2006

Cash $305 270

Accounts receivable 275 290

Inventory 600 580

Current assets 1,180 1,140

Plant and equipment 1,700 1,940

Less: acc depr (500) (600)

Net plant and equipment 1,200 1,340

Total assets $2,380 $2,480

Liabilities and Owners’ Equity

Accounts payable $150 $200

Notes payable 125 0

Bonds 500 500

Owners’ equity

Common stock 165 305

Paid-in-capital 775 775

Retained earnings 665 700

Total owners’ equity 1,605 1,780

Total liabilities and

owners’ equity $2,380 $2,480

Income Statement

Sales (100% credit) $1,100 $1,330

Cost of goods sold 600 760

Gross profit 500 570

Operating expenses 20 30

Depreciation 160 200

Net operating income 320 340

Interest expense 64 57

Net income before taxes 256 283

Taxes 87 96

Net income $ 169 $ 187

a) Compute the following ratios for 2005 and 2006:

2005 2006 Industry Norms

Current ratio 5.0

Acid test ratio 3.0

Inventory turnover 2.2

Average collection period 90 days

Debt ratio 0.33

Times interest earned 7.0

Total asset turnover 0.75

Fixed asset turnover 1.0

Operating profit margin 20%

Net profit margin 12%

Return on total assets 9.00%

Oper income return on investments 15.00%

Return on equity 10.43%

b) What are the firm’s financial strengths and weaknesses?

Be sure to discuss each ratio in relation to the industry norm. What does the

comparison mean?

Be sure to discuss each ratio in relation to the year-to-year trend. What does the trend

indicate?

What qualitative assessments can you make based on your analysis?

c) Should the bank make the loan? Why or why not?

The answer to this question should be a thoughtful response.

In your answer be sure to address trends in liquidity, inventory management,

operating profits and whether or not you believe the loan is needed and why.

A simple “yes” / “no” type answer will receive ZERO

Problem # 2: (15 pts)

ANSWER: Please see Excel file

Ducker Industries’ projected sales for the first six months of 2004 are given below:

Table 1

Jan. $200,000 April $400,000

Feb. $240,000 May $320,000

March $280,000 June $320,000

25% of sales is collected in cash at the time of the sale, 50% is collected in the month

following the sale, and the remaining 25% is collected in the second month following

the sale.

Cost of goods sold is 75% of sales.

Purchases are made in the month prior to the sale, and payments for purchases are

made in the month of the sale.

Total other cash expenses are $60,000/month.

The company’s cash balance as of February 28, 2004 will be $40,000. Excess cash

will be used to retire short-term borrowing (if any).

Ducker has no short-term borrowing as of February 28, 2004.

Assume that the interest rate on short-term borrowing is 1% per month.

The company must have a minimum cash balance of $25,000 at the beginning of each

month.

Round all Answers to the nearest $100.

a) Prepare a cash budget for Ducker and highlight (using color, borders or bold) the following:

_ total cash receipts (collections) for April 2004

_ total disbursement in May (not including interest on short-term borrowing)

_ ending cash balance (before borrowing) in March

_ projected cumulative short-term borrowing as of April 30, 2004

_ when Ducker pays off its short-term borrowing

b) Prepare a simple income statement and highlight (using color, borders or bold)

Ducker’s EBIT for March 2004

Problem # 3: (15 pts)

ANSWER: Please see Excel file

Use the following information to answer questions for Problem 3.

Millennium Corp, Inc. has a base level of sales of 150,000 units. Sales price per unit is

$10.00 and variable cost per unit is $6.50. Total annual operating fixed costs are $155,000, and the annual interest expense is $90,000. (For consistency with answers below, carry all calculations to two decimals.)

a) What is base level EBIT?

b) What is Millennium Corp.’s operating leverage? Provide an interpretation to describe

Millennium’s degree of operating leverage.

c) Assume that Millennium expects units sold to increase 20%. What will be the resulting percentage change in EBIT?

d) What will be the new level of EBIT given the expected 20% increase in units sold?

e) Calculate Millennium’s degree of financial leverage. Provide an interpretation to

describe Millennium’s degree of financial leverage.

f) Calculate Millennium’s degree of combined leverage. Provide an interpretation to

describe Millennium’s degree of combined leverage.

g) Assume that Millennium expects units sold to increase 20%. What will be the resulting percentage change in earnings per share?

h) Assume that Millennium’s current level of earnings per share is $2.00. What will be the new level of earnings per share given the expected 20% increase in units sold?

I am Published

noreply@blogger.com (An aspiring millionaire) — Wed, 11 Mar 2009 06:14:00 +0000

Solving for unknowns

noreply@blogger.com (An aspiring millionaire) — Tue, 10 Mar 2009 04:23:00 +0000

Question

Solve for x: | 4 - x | = 0
Solve for x: | x - 4 | + 3 = 9
Solve for x: | -3x + 4 | = | -x + 2 |
Solve for x: | x + 2 | = | 2 - x |

Solution

Solve for x: | 4 - x | = 0

x=4

Test


|4-4|=0
|0| = 0
 0 = 0

Solve for x: | x - 4 | + 3 = 9

|x-4|=9-3

|x-4|=6

x=-2

x=10

Test

x=-2
 | -2 - 4 | + 3 = 9
 | -6 | + 3 = 9
 6 + 3 = 9
 9 = 9
x=10
| x - 4 | + 3 = 9
| 10 - 4 | + 3 = 9
| 6 | + 3 = 9
 6 + 3 = 9
 9 = 9

Solve for x: | -3x + 4 | = | -x + 2 |

x=1

Test


| -3*1 + 4 | = | -1 + 2 |
| -3 + 4 | = | 1  |
| 1 | = | 1  |
1  = 1

Solve for x: | x + 2 | = | 2 - x |

x=0

Test


| 0 + 2 | = | 2 - 0 |
| 2 | = | 2|
2=2

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Solving unknowns

noreply@blogger.com (An aspiring millionaire) — Tue, 10 Mar 2009 04:18:00 +0000

Question
3-4k+5(k-6)=2
Solve for k

Solution


3-4k+5k-30=2

5k-4k+3-30=2

k-27=2

k=2+27

k=29

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Perimeter of a rectangle

noreply@blogger.com (An aspiring millionaire) — Tue, 10 Mar 2009 03:42:00 +0000

Question
Rectangular garden is 85 meter long and 57 meter broad. Trees are to be planted around the garden at a distance of 5 meter having a distance of 4 meter the start. How many such trees can be planted

Solution
Procedure 1

compute for the perimeter of the rectangle that is available for tree planting

L: 85 meter - 4 meter = 81 meter

W: 57 meter 4 meter = 53 meter
Procedure 2

The perimeter of a rectangle is computed by 2L + 2W

2 (81 meter) + 2 (53 meter)

162 meter + 106 meter = 268 meter
Procedure 3

How many meters can be planted?

268 meter / 5 meter = 53 trees

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Interest rate compounded continuous

noreply@blogger.com (An aspiring millionaire) — Mon, 09 Mar 2009 13:57:00 +0000

Question
If you invest $6,700 in an account paying 8.5% compounded continuously, how much money will be in the account at the end of 4.5 years?

Solution
Solution 1
Investment (1 + interest rate)time
$6,700 (1.085)4.5
$6,700 (1.44356)
$9,671.83

Solution 2






































Year Investment Interest Balance
1 $6,700.00 $569.501 $7,269.50
2 $7,269.50 $617.91 $7,887.41
3 $7,887.41 $670.23 $8,557.64
4 $8,557.64 $727.40 $9,279.04
4.5 $9,279.04 $394.36 $9,673.40


NOTE: The difference in the final answers are due to rounding off.

Year	Investment	Interest	Balance
1	$6,700.00	$569.501	$7,269.50
2	$7,269.50	$617.91	$7,887.41
3	$7,887.41	$670.23	$8,557.64
4	$8,557.64	$727.40	$9,279.04
4.5	$9,279.04	$394.36	$9,673.40

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How to compute for gain and loss?

noreply@blogger.com (An aspiring millionaire) — Fri, 20 Feb 2009 10:41:00 +0000

Question
The selling price of 10 articles is equal to cost price of 11 articles. Find the gain or loss per cent.

Solution
Let P = selling price
C = cost

Therefore, the above word problem is translated to the following equation
10P = 11C
Isolating P results to
P = 1.1C

Gain or loss, in percentage, is calculated by
P-C
----x100%
C

Therefore,

1.1C - C
--------x 100%
C

Therefore the gain is 10%

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How to compute for the length and width of a rectangle?

noreply@blogger.com (An aspiring millionaire) — Fri, 20 Feb 2009 10:26:00 +0000

Question
A pond is enclosed by a wooden deck that is 3 feet wide. The fence surrounding the deck is 100 feet long. If the pond is rectangular and the length of the pond is to be three times its width, what are its dimensions?

Solution
Let x = width of the pond
3x = length of the pond
Perimeter of the rectangle = 100 ft
The formula for perimeter is 2x + 6x = 100

Hence,
8x = 100
x = 12.5

Therefore, the width of the pond is 12.50 feet and its length is 37.50 feet.

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Investment Allocation

noreply@blogger.com (An aspiring millionaire) — Fri, 20 Feb 2009 09:46:00 +0000

Question
Mike invested $2000 in bonds and certificate of deposits. the bonds paid 5% and the CD's paid 4.5% if his total interest earned was $93.75 for the year, how much did he invest in each type?

Solution
Let x = investment in bonds
2000-x = investment in certificate of deposits

Therefore, .05x + .045(2000-x) = 93.75

Hence, x = $750

Therefore, Mike needs to invest $750 in bonds and $1,250 in certificate of deposits.

You may also want to check Investment Allocation
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How to solve for a quadratic equation?

noreply@blogger.com (An aspiring millionaire) — Fri, 20 Feb 2009 09:33:00 +0000

Question
A company sells shoes for $40 per pair if less than 50 pair are ordered. If more than 50 are ordered (up to 600), the price is reduced at a rate of $.04 times the number oredered. How many pairs can a dealer purchase for $8400

Solution
Let x = number of shoes which can be purchased by $8,400
Therefore,
($40-$0.04x)x = 8,400
40x - 0.04x^2 = 8400

The above equation is a simple quadratic equation.

Divide both sides of the equation by 0.04 to get
1,000x - x^2 = 210,000
Multiply both sides by -1 to get
x^2 - 1000x = -210000
Transform the above quadratic equation to get
x^2 - 1000x + 210000 = 0
Solve for the roots of the quadratic equation by thinking of two numbers when multiplied equals 210,000 and when added equals 1000 --- 700 and 300!

Hence, (x - 700)(x - 300) = 0
Therefore,
x1 = 700
x2 = 300

If you test the above numbers by substituting them in the quadratic equation 40x - 0.04x^2 = 8400, both numbers arrive at $8,400. This means that if you buy 300 pairs of shoes, you will pay $8,400 AND if you buy 700 pairs of shoes, you will still pay $8,400!

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Investment allocation

noreply@blogger.com (An aspiring millionaire) — Fri, 20 Feb 2009 09:13:00 +0000

Question
Need to know how much of 4200 investment was at 15% and how much of 4200 investment was at 8% to equal profit of 448.

Solution
If you have available funds for investment and you want to create an investment plan in order for you to achieve a certain return on investment, then there is a need to allocate your investments.

For the investment problem above, the target return on investment of $4,200 is $448 and the investment portfolio is comprised of an investment with a return of 15% and another with a return of 8%.

So, how do we compute for the funds to be allocated to each investment option?

Let x = investment with a return of 15%
(4,200-x) = investment with a return of 8%

Therefore, the expected return on investment is .15x + .08(4,200-x) = 448

Simplify .07x + 336 = 448

Isolate x
x = $1,600

Therefore, the investor needs to invest $1,600 in the investment with an expected return of 15% and $2,600 in the investment with an expected return of 8% to achieve a portfolio rate of return of 10.67% or $448.

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How to compute volume of a rectangle?

noreply@blogger.com (An aspiring millionaire) — Fri, 20 Feb 2009 09:00:00 +0000

Question
A rectangular piece of a plastic is half as wide as it is long from each corner a 3cm square is cut out, and the ends are turned up as to make a box whose contents are 90 cubic cms what would be the volume of this box if filled to a depth of 1/10 cms.

Solution
To compute for the volume of a rectangle, the formula is Side1*Side2*Side3 OR L*W*H
where
L = length
H = height
W = width
For the given rectangle, we have
L = L
H = H
W = 2L
Volume = 90 cubic centimeter
which is equal to L*2L*H
therefore, the height of the rectangle is
45
----
L^2
and the length of the rectangle is
the square root of 45/H

Given that the height is 1/10 centimeter, then
Length = 21 centimeters
Width = 42 centimeters

Therefore the volume of the rectangle is 88 cubic centimeters

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Solving for permutations and combinations

noreply@blogger.com (An aspiring millionaire) — Thu, 19 Feb 2009 15:24:00 +0000

Question
1) how many different ways can a teacher select 2 books from a possible 17 books?
2) how many different ways can be made from a test bank of 20 questions if the test consists of 5 questions?
3)how many different ways can 4 tickets be selected from 50 tickets if each tickets wins a different prize?

Solution
These are examples of permutations and combinations problems.
The problems can be solved in two ways depending on the assumptions.

Assumption 1 below is an example of a combination.
For example how many different committees of three students can be chosen from a group of 10?
Therefore,
10!
---
3!7!
which is equal to
10.9.8
------
3.2.1
to get 120.
Assumption 2 below is an example of a permutation.
For example, the permutation of three letters from the set a,b,c are 6 which are enumerated below
abc
bac
cab
acb
bca
cba
Which can also be computed by
3!
--
0!
which is equal to
6
-
1
to get 6.
Assumption 1: The order of the arrangements or combinations DOES NOT matter
Question 1
17!
---
2!
which is
17.16
-----
2.1
to arrive at 136
Question 2
20!
----
5!
which is
20.19.18.17.16
---------------
5.4.3.2.1
to arrive at 15,504
Question 3
50!
----
4!
which is
50.49.48.47
------------
4.3.2.1
to arrive at 230,300

Assumption 2: The order of the arrangements or combinations DOES matter
Question 1
17!
---
15!
which is 17.16
to arrive at 272.
Question 2
20!
----
15!
which is 20.19.18.17.16
to arrive at 1,860,480

Question 3
50!
----
46!
which is 50.49.48.47
to arrive at 5,527,200
You may also want to refer to
How many possible combinations can pennies be distributed?
How to recover Nokia Mobile Phone Security Code?
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Rewriting mathematical equations

noreply@blogger.com (An aspiring millionaire) — Thu, 19 Feb 2009 14:54:00 +0000

Question
8x-y=19
y+3x=7
How do you put the y under the y and the x under the x?
This question address how to rewrite equation.

Solution
The following steps teach students rewriting equations:
1. Isolate the variable you want to put the original equation under.
Equation 1: 8x-y=19
For y
-y = 19-8x
For x
8x = 19 + y

Equation 2: y+3x=7
For y
y = 19-3x
For x
3x = 7-y

2. Simply.
Equation 1: 8x-y=19
For y
-y = 19-8x
-1(-y = 19-8x)
y = -19+8x
For x
8x = 19 + y
-----------
8
x = 2.375 + 0.125y

Equation 2: y+3x=7
For y
y = 19-3x ===> already simplified
For x
3x = 7-y
--------
3
x = 2.33 - 0.33y

3. Rearrange
Equation 1: 8x-y=19
For y
-y = 19-8x
-1(-y = 19-8x)
y = -19+8x
y = 8x - 19
For x
8x = 19 + y
-----------
8
x = 2.375 + 0.125y
x = 0.125y + 2.375

Equation 2: y+3x=7
For y
y = 19-3x ===> already simplified
y = -3x + 19
For x
3x = 7-y
--------
3
x = 2.33 - 0.33y
x = -0.33y + 2.33

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Calculating density

noreply@blogger.com (An aspiring millionaire) — Thu, 19 Feb 2009 10:44:00 +0000

Question
A piece of metal has a mass of 86.24g and a volume of 13.4mL. What is its density?

Solution
The formula for density is
m
--
v
where
m = mass
v = volume

Therefore,
86.24g
-------
13.4mL

is equal to 6.44g/mL

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Calculating speed

noreply@blogger.com (An aspiring millionaire) — Mon, 16 Feb 2009 14:28:00 +0000

Question
Mary left by plane to visit her mom in LA, 420km away. Fifteen minuntes later, her mom left to meet her at the airport. She drove 20km to the airport at 40km per hr, arriving just as the plane taxied in. What was the speed of the plane?

Solution
First, speed is calculated by D
----
t

where
D = distance
t = time

Second, determine the distance traveled by the airplane. 420 km

Third, determine the time spent by the airplane to travel such distance
1. 15 minutes after take off, Mary's mother set out for the airport
2. It took 30 minutes for Mary's mother to travel the 20 kilometer distance to the airport at a speed of 40 kilometers per hour.

Total travel time for the airplane therefore is 45 minutes.

Lastly, calculate for the speed 420 kilometers
-----------------------
45 minutes

The speed of the airplane is 9.33 kilometers per minute OR 560 kilometers per hour.

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How to solve for the length and width of a rectangle given its perimeter

noreply@blogger.com (An aspiring millionaire) — Mon, 16 Feb 2009 14:18:00 +0000

Question
If the perimeter of a rectangular block of land is 46 meters, find the lenght and width of the block.

Solution
The formula for a rectangle's perimeter is 2L + 2W where L is length and W is width.
Hence the above problem can be written as 2L + 2W = 46 which is also equal to L + W = 23, however it can not be solved yet given the information. The relationship of the rectangle's width and length must be stated in rder for the problem to be solved.

You may also want to look at previous posts discussing the same topic:
How to solve for the length of a rectangke given its perimeter
Calculate the perimeter of a quadrilateral

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How to solve for proportions?

noreply@blogger.com (An aspiring millionaire) — Fri, 13 Feb 2009 13:57:00 +0000

Question
Jordan can solve 12 math problems in 3 minutes. How long will it take him to solve 45 problems?

Solution
Write the word problem into an equation.
Let x = how fast Jordan can solve 45 problmes
12:3 = 45: x
Therefore, 12x = 135
x = 11.25

Jordan can solve 45 problems in 11 minutes and 15 seconds!

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How to solve for the length of a rectangle given its perimeter

noreply@blogger.com (An aspiring millionaire) — Fri, 13 Feb 2009 13:46:00 +0000

Question
If the length of a rectangular parking lot is 10 meters less than twice the width, and the perimeter is 100 meters, find the length of the parking lot

Solution
First, write the word problem as an equation.
Let x = width of the rectangle
Then 2x - 10 = length of the rectangle
Then its perimeter is 2x + 2(2x - 10) = 100

The perimeter of a rectangle is computed as 2L + 2W

Simplify the above formula 2x + 4x - 20 = 100 ===> 6x = 120
Therefore x = 20

The length of the rectangle is 30 AND its width is 20.

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Calculate the perimeter of the quadrilateral

noreply@blogger.com (An aspiring millionaire) — Fri, 13 Feb 2009 11:13:00 +0000

Question
The perimeter of the quadrilateral is 120cm. Find the lengths and sides. (x+5)(x^2-3)(3x-8)(x+3).

Solution
Let us first define how to measure the perimeter of a quadrilateral. The perimeter of any quadrilateral, or any shape with four sides for that matter, is measured by adding the lengths of its sides. Hence, the quadrilateral on the right has a perimeter of 30 which is computed by adding the measure of its sides - 6 + 9 + 7 + 8.

Now, let us go back to our problem. The four sides of the quadrilateral are (x+5), (x^2-3), (3x-8) and (x+3). Hence, its perimeter is computed by (x+5) + (x^2-3) + (3x-8) + (x+3).

To solve for the length of the sides of the quadrilateral:

(x+5) + (x^2-3) + (3x-8) + (x+3) = 120
x^2 - 3+ x+5 + 3x - 8 + x + 3 = 120
x^2 + 5x - 3 = 120 ===> add like terms
x^2 + 5x - 123 = 0 ===> right side of the equation is set to zero
(x + ?) (x - ?) = 0 ====> to arrive at the root of the quadratic equation, think of two numbers that when you multiply them yields -123 and when you add them gives 5. Hmmm...difficult, however nothing prevents us to looking at a solution which is NOT a whole number.

Now, I got 13.87 and -8.87. Their multiple is -123.02 [close enough] and their sum is 5.

(x + 13.87) (x - 8.87) = 0
x = -13.87 ===> this is NOT an option because it is NEGATIVE.
x = 8.87

Now to calculate the lengths of the sides of the quadrilateral:
(x+5) ===> 13.87
(x^2-3) ==> 75.68
(3x-8) ===> 18.61
(x+3) ===> 11.87

Now to check whether the sum of the above is 120. 13.87 + 75.68 + 18.61 + 11.87 = 120.02 [close enough]

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Solving for the missing number

noreply@blogger.com (An aspiring millionaire) — Fri, 13 Feb 2009 10:15:00 +0000

Question
32 is 40% of what number?

Solution
The problem is very straightforward.

To be able to solve it, let x = missing number
The the word problem is written as .40x = 32

Then solve for x.

x = 32
----
.40
x = 80

Lastly, double check. 40 % of 80 = 32!

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Computing for probabilities

noreply@blogger.com (An aspiring millionaire) — Fri, 13 Feb 2009 10:00:00 +0000

Question
When Paula played softball, she struck out twice, walked 3 times and had 6 hits. The probability she will get a hit the next time she is at bat is ____ out of ______.

Solution
First, count the number of times Paula played or she was at bat. ANSWER: 11 times
Second, count the number of times she had a hit. ANSWER: 6 times
Hence, the probability that Paula will have a hit next time she is at bat is 6/11 or 0.55 OR 55 out of 100 OR 55 per cent probability.

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How to simplify complex fractions?

noreply@blogger.com (An aspiring millionaire) — Fri, 13 Feb 2009 09:44:00 +0000

Question
Note: I had a difficulty understanding the equation. Below is my understanding of the equation.

(m + 2)* (m - 2)
--------------------
(2m + 4) * (m^2 - 4)

Solution
First, breakdown the equation to its simplest form. Hence

(m + 2)* (m - 2) (m + 2)* (m - 2)
-------------------- = ------------------------------
(2m + 4) * (m^2 - 4) 2 * (m + 2) * (m -2) * (m-2)

Now, you can cancel like numbers. Meaning if a number is in the numerator AND denominator, then they cancel each other. Hence, (m + 2) and (m -2) are canceled in the numerator and denominator.

Therefore,
1
------------------
2* (m-2)

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How to solve for and find a missing fractions?

noreply@blogger.com (An aspiring millionaire) — Fri, 13 Feb 2009 09:02:00 +0000

Question
If a certain fraction is multiplied by 4 and the result subtracted from twice the reciprocal of the original fraction there remains 1/2 of the original fraction. find the original fraction.

Solution
Let x = missing fractions
Then, the above word problem can be states as 2*(1/x) - 4x = (1/2)*x
Solve for x by isolating all variables with x.
Hence, 2/x - 4x - x/2 = 0.
Simplify 2/x - 4x - x/2.
(2/x - 4x - x/2)*2x = 0
(4 - 8x^2 - x^2)/2x = 0
(4 - 9x^2) = 0
(-3x + 2) * (3x +2) = 0
Then, look for x
(-3x + 2)= 0 =====> -3x = -2 =====> x = 2/3
(3x +2) = 0 =====> 3x = -2 =====> x = -2/3

Now, test the validity of the two solutions above by substituting them in the formula 2*(1/x) - 4x = (1/2)*x.

x = 2/3 ======> 2*(3/2) - 4*(2/3) = (1/2)*(2/3) ====> 2/6 = 2/6
x = -2/3 ======> 2*(-3/2) - 4*(-2/3) = (1/2)*(-2/3) ====> -2/6 = -2/6

Now, there is actually only ONE answer the above problem. The answer is 2/3 BECAUSE fractions are usually POSITIVE.

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The Easy Guide: How to write fractions

noreply@blogger.com (An aspiring millionaire) — Fri, 13 Feb 2009 08:47:00 +0000

Question
In Mark's collection of antique bottles, 4/9 of the bottles are dark green. Write three equivalent fractions for 4/9.

Solution
First, we need to write 4/9 in three different forms.
Second, multiple the fraction with 1 or its equivalent, such as 1/1, 2/2, 3/3, 4/4 and so on, to arrive at three different fractions which, when simplified, are still equal to 4/9.

4/9 * 2/2 = 4*2/9*2 = 8/18
4/9 * 3/3 = 4*3/9*3 = 12/27
4/9 * 4/4 = 4*4/9*4 = 16/36

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