I am volunteering some of my time to help students understand some of the more difficult topics in school. In doing so, I hope that I can help make school much easier.
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.
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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 Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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!
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!
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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 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
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
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.
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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]
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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)
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
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
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
Maggie and Jeff
Sam and Jeff
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.
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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!
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
Combinationis 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!
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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.
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
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
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:
0 <= x <= 23 for the amount of gas in the tank can not go below 0 and above 15 gallons
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)
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
Question Jane Manning, an outside business woman, uses her car forboth 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 candeduct all highway travel, but no city travel, on her taxes.
How many miles should Jane be allowed as a business expense?
Solution
Letx=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 1to 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.
Please do not forget to leave a feedback in Allexperts.com AND a comment in this blog.
