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Tuesday, March 10, 2009

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Monday, March 9, 2009

Solving for unknowns

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

  1. x=4

  2. Test


  • |4-4|=0
  • |0| = 0
  • 0 = 0

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


    1. |x-4|=9-3

    2. |x-4|=6

    3. x=-2

    4. x=10

    5. 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 |

    1. x=1

    2. Test


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

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

    1. x=0

    2. Test


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

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    Solving unknowns

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




    Solution

    1. 3-4k+5k-30=2

    2. 5k-4k+3-30=2

    3. k-27=2

    4. k=2+27

    5. k=29

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    Perimeter of a rectangle

    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

    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







































    YearInvestmentInterestBalance
    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.
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    Friday, February 20, 2009

    How to compute for gain and loss?

    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?

    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

    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?

    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

    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?

    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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    Thursday, February 19, 2009

    Solving for permutations and combinations

    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

    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

    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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    Monday, February 16, 2009

    Calculating speed

    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

    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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    Friday, February 13, 2009

    How to solve for proportions?

    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

    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

    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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    [Image via TPUB]

    Solving for the missing number

    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

    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?

    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?

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

    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.
    1. 4/9 * 2/2 = 4*2/9*2 = 8/18
    2. 4/9 * 3/3 = 4*3/9*3 = 12/27
    3. 4/9 * 4/4 = 4*4/9*4 = 16/36
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    Calculate the shortest possible time to cross a bridge

    Question
    Maggie, Sam, Reed, and Jeff are on one side of bridge and they want to all make it to the other side in 17 minutes. It takes Maggie 10 minutes to cross, Sam 5, Reed 2, and Jeff 1. Only 2 people can walk across at a time and there is one flashlight. The flashlight has to be each person or group, and it has to be walked across, not tossed. Can it be done in 17 minutes?
    Solution
    First determine the fastest of the four. Jeff is the fastest who can cross the bridge in just a minute. Hence, use Jeff to walk back and forth of the bridge carrying the flashlight.
    Second, identify pairs.
    1. Maggie and Jeff
    2. Sam and Jeff
    3. Reed and Jeff
    Third, compute the time spent by each pair and the time spent by Jeff going back to the foot of the bridge
    Pair/Person Time (in minutes)
    Maggie and Jeff 10 ===========> Maggie's time*
    Jeff 1
    Sam and Jeff 5 ===========> Sam's time*
    Jeff 1
    Reed and Jeff 2 ===========> Reed's time*
    *Although Jeff can walk much faster, he can not leave any of his companions in the middle of the bridge with no light.



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    How to recover Nokia Mobile Phone Security Code?

    Question
    How could I get a possible combination and permutation for a 5 digit code from 1-6. Locked my phone and can't remember it. I really don't wanna loose all my contacts if Nokia has to reset it.
    Solution
    Whew! This is a problem I can relate with. I once forgot the security code of my luggage. It is a good thing that it had only 3 digits and that I remembered the first and third numbers - so I only had to try 10 times to get the correct security code, otherwise I would have to get a locksmith.

    So, how do we go about your problem? Unfortunately, it is very difficult because because there are 5 digits, however it would have been more difficult if you choose any number from 1 to 10 instead of just 1 to 6.

    Given your situation, there are 7,776 possible 5 digit combination. This is computed by 6*6*6*6*6 0r 6^5! However, if you only used a number ONCE in any of the 5 digits, then your choices get narrower. In this case you will only have 720 possible combinations, which is computed by 6*5*4*3*2.

    Now, if you can just remember the first 2 digits, then your choices will be scaled down to just 4*3*2 which is 24!

    I hope this helps and please let me know what happens. Good luck!




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    Thursday, February 12, 2009

    How many possible combinations can pennies be distributed?

    Question
    In how many ways can 3 girls divide 10 pennies if each must end up with at least one penny?




    Solution
    First, let us determine whether the above problem is an example of a permutation or combination problem OR the combination of both.

    Combination is the various ways that the 10 pennies can be distributed to the 3 girls given that each girl must end up with at least 1 penny.
    Permutation, on the other hand, is defined as the possible arrangements of the pennies.

    Now, for the above problem, it does NOT matter what penny goes to each girl - what matters is that each gets at least a penny. Hence, the problem is both a combination and a permutation problem.

    Second, the easiest way to solve a combination and permutation problem is by using the nPr function of your calculator.
    • Turn on your calculator
    • Key in '10' for the total number of pennies
    • Press nPr
    • Key in '3' for the total number of ways
    • Press equal or enter key
    • You get 720 possible combinations!
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    How to Solve for Least Common Denominator (LCD) of A Series of Numbers

    Question
    Can you show me a step by step example of how you find the LCD of 24, 30, and 60?


    Solution
    First, let us define what LCD means.

    LCD or least common denominator is the lest common multiple of a set of numbers whethe they be fractions or whole numbers. It is also called the lowest common denominator.

    To solve the problem ...

    First, take the multiples for each of the numbers.
    For 24 = 24, 48, .... 120
    For 30 = 30, 60 .... 120
    For 60 = 60, 120, 180

    Therefore, the LCD of 24, 30 and 60 IS 120.

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    BDV-999330-BDV

    Tuesday, February 10, 2009

    Ratios

    Question
    An area 2,500 square feet of grass produces enough oxygen for a family of 4. What is the area of grass needed to supply a family of 5 with oxygen?




    Solution
    The above world problem is translated to equation form below:
    2,500:4 = x:5

    Where x is equal to the area, in square feet, of grass needed to supply a family of 5 with oxygen.

    Hence, (2500)(5) = 4x
    Then, 4x = 12,500
    Therefore, x = 3,125

    A 3,125 square feet of grass is needed to supply a family of 5 with oxygen.

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    Computing median

    Question What is the median of the following numbers?
    340 540 710 720 1290 1980 2060 2390 2760 2790 2820 2860 2860 3000 3620 4230 4230 4270 4570 4770



    Solution Median is number in the middle of a set of numbers.
    There are two ways to answer the question.

    1. using the formula function of excel as shown below
    Note: If you want a copy of the Excel file for the above, please leave your email in the comment box below.
    2. doing it the hard way by counting the number of observations, then dividing that number by 2, then counting from left or right beginning from zero to that number to get the median.Hence, there are 20 numbers in the set, the median number then is the average of the 10th and 11th numbers in the set which are 2790 and 2820. The median then is (2790 + 2820)/ 2 = 2805

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    Total miles ridden

    Question
    Nicole rides her bike every day. If she rides at a constant speed of 10 mph every time she rides, how many miles will she ride if she rides for 30 minutes on Monday, 45 minutes on Tuesday, and 20 minutes on Wednesday?




    Solution
    Monday 30minutes x 10mph = 5.0 miles
    Tuesday 45minutes x 10mph = 7.5 miles
    Wednesday 20minutes x 10mph = 3.3 miles

    Total miles ridden: 15.8 miles

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    writing equations and determining range of functions

    Question
    John's old '87 Lebaron has a 15 gal gas tank and gets 23 mpg. The number of miles he can drive is a function of how much gas is in the tank. Require (a) write this relationship in equation form and (b) determine the domain and range of the function in the context.




    Solution
    Let x = gallons of gas in the tank
    y = number of miles

    Therefore, y = 23x which is bounded by the following ranges:
    1. 0 <= x <= 23 for the amount of gas in the tank can not go below 0 and above 15 gallons
    2. 0 <= y <= 345 for the number of miles John can travel can not go below 0 and above 345 miles (23 mpg multiplied by 14 gallons)

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    How many miles should Jane be allowed as a business expense?

    Question
    Jane Manning, an outside business woman, uses her car for both business and pleasure. Last year, she traveled 30,000 miles using 900 gallons of gas. Her car gets 40 miles per gallon on the Highway, and 25 mpg in the city. She can deduct all highway travel, but no city travel, on her taxes.

    How many miles should Jane be allowed as a business
    expense?



    Solution

    Let x = gas spent in the Highway

    y = gas spent in the City

    Therefore,
    Equation 1: x + y = 900

    Therefore,

    Equation 2: 40x + 25y = 36,000

    Transform

    Equation 1 to x = 900 – y


    Substitute the above equation in Equation 2


    Which is equal to 36,000 – 40y + 25y = 30,000


    Simplify the above equation to 6,000 = 15y


    Hence, y = 400 AND x = 500

    Therefore, Jane can deduct 500 gallons OR 10,000 miles on her taxes.


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