David Q/R

The way the question is phrased suggests that not all of it has actually been posted. "Evaluate the related series" doesn't mean anything on its own - at least, not to me. Has your teacher defined a "related series" to be the sum of the individual element...

15 years ago

Okay... you've got $M, and you can afford one qualified teacher for x days, so a qualified teacher costs M/x dollars per day. Alternatively, you can afford a relief teacher for y days, so a relief teacher costs M/y dollars per day. So, if you're paying fo...

13 years ago

f(x) = x^3 - 12x. You can find the maxima and minima by differentiating it (I'm assuming here that you can use calculus) to get f'(x) = 3x^2 - 12, and solving f'(x) = 0, so 3x^2 - 12 = 0, so 3x^2 = 12, so x^2 = 4, so x is either +2 or -2. But only one of...

13 years ago

Concentrate on a single room for a moment. It's got two bulbs, each of which has a 1/10 chance of failing, so at the end of the time interval in question the chance of a double failure is 1/10 x 1/10 = 1/100. It's only double failures we're interested in...

13 years ago

Oops! Actually, we want one minus that, don't we. The chances of at least one room having a double failure is one minus the chances of none of them having a double failure - and that's 1 - 0.97, which is 0.03. That's more like it.

13 years ago

Just to argue the point in 3), I think I'd say that neither is quite correct. What the writer actually means is a reduction in troop numbers. "Reducing troops" somehow implies that you're making them smaller, which doesn't seem very likely!

13 years ago

One quarter passed with an A - so that's 24/4 = 6. Twelve of them passed with a B - and that can't include any who passed with an A, so that's a total of 12 plus 6 = 18. Assuming that it's only possible to pass with either an A or a B, that's 18/24 = 3/4...

13 years ago

The equation you want is v = u + at, where u = initial velocity; v = final velocity; a = acceleration and t = time. v = 20. u = 10. a = 9.81 m/sec, though I'd guess from the numbers supplied that you may be expected to assume that a is approximately 10. Y...

13 years ago

It sounds like 13 flavors times 9 toppings times 4 types of nuts to me. Am I missing something here?

13 years ago

If you're working in complex numbers, don't forget that there are three solutions to the cubic equation x©ø = -1. x = -1 is one of them; the other two are x = (1 + ¡î3i) / 2 and x = (1 - ¡î3i) / 2, since if you cube either of these you'll get -1.

13 years ago

Sorry - I can't upload symbols, it seems. If you're working in complex numbers, don't forget that there are three solutions to the cubic equation x cubed = -1. x = -1 is one of them; the other two are x = (1 + sqrt(3)*i) / 2 and x = (1 - sqrt(3)*i) / 2, s...

13 years ago

You need to work out the standard deviation of the mean of the bill for 15 residents. That's $6//sqrt(15), which is 1.549. So the distribution of the mean of the bill for 15 residents will be Normal, with mean $72 and standard deviation 1.549. You have to...

13 years ago

Treat it like a long division. First, work out how many times 6b goes into 42b^3, which is 7b^2, and that will be the first term of your answer. Now multiply (6b+5) by 7b^2, and you'll get 42b^3 + 35b^2. Write that underneath the original expression and s...

13 years ago

I'm not quite sure I understand the second question, but I'll take a guess. You want to simplify {sqrt(x) - sqrt(y)} / {sqrt(x) + sqrt(y)}. Multiply the top and bottom of this expression by {sqrt(x) - sqrt(y)}. The numerator will then be {x - 2.sqrt(xy) +...

13 years ago

See my reply to your re-post: unless I've slipped up myself I think you may have made a typing error with the constant term of the numerator, since 40 would make the numerator exactly divisible by (6b+5), but 49 doesn't.

13 years ago

Use the binomial theorem. The probability that out of 7 stolen cars, the first three will be recovered is 0.89^3 x (1-0.89)^4 (because you also have to calculate the probability that the last four will NOT be recovered). That is 0.000103. But it's not jus...

13 years ago

Assuming that (i) there is a point B between A and C along one side of the rectangle, and that (ii) you mean "BC=10cm" instead of "C=10cm", then it seems to me you have a right-angled triangle ABE with the hypoteneuse BE which has length 13cm, and length...

13 years ago

The number of getting two winning tickets out of six is 6! / (2! x 4!) = 6x5/2 = 15. Consider the probability of getting one of those 15 outcomes - say just the first two are winners, and the final four are losers. The chances of that happening - remember...

13 years ago

The hypothesis is that the rate of lorries passing through the village on Thursday mornings is 1.5 every five minutes - or equivalently, 9 per half hour. We've just observed 15 lorries passing through the village in a half-hour period, and we want to know...

13 years ago

Let’s eliminate x: 4x + 10y = 28, so multiply both sides by 50% to get: 6x + 15y = 42, but we know that: 6x + 7y = 6, so subtract one from the other to get: 8y = 42 - 6 = 36, so y = 36/8 = 4.5, so 10y = 45, but 4x + 10y = 4x + 45 = 28, so 4x = 28 - 45 = -...

13 years ago

X has a Normal (i.e. a Gaussian) probability distribution with mean 42.5 and a variance of 103.71. (That in turn means that it's got a standard deviation of sqrt(103.71) = 10.2, so since 95% of the distribution lies within the mean plus or minus about 2 s...

13 years ago

I googled "bond issue price calculation", and found this example which may help: Qn: A company issues $500,000 of 9% bonds that pay interest semiannually and mature in 10 years. Calculate the bond issue price assuming the bond's market rate is 8% per year...

13 years ago

This sounds a bit badly worded to me, since strictly speaking there is NO way that eight cars can fit into two spaces that between them can accommodate at most five cars. However I assume that the question is intended to be "In how many ways can exactly f...

13 years ago

9 goats would have 36 legs, but Sue's got some ducks as well, i.e. at least two, so at least four legs belong to the ducks. That leaves 32 - which means 8 goats.

13 years ago

The number of pens is (6T - 4), where T is the number of teachers. But the number of pens is also (9T - 25), so (6T - 4) = (9T - 25). So 3T = 21, which means T = 7, i.e. there are 7 teachers. That means there are 38 pens. Check it: If 7 teachers received...

13 years ago

An average of 18 out of 25 is 72%. I need to increase that by 3% to 75% by the time the marks from my final exam are included. Each exam is scored out of 25. Then so far I've got 72% out of a possible 100 marks, which is 72 marks. I need to get 75% overal...

13 years ago

For the conclusion to be valid, the results of the study need to be representative of those that would have been obtained if the entire company had been surveyed. Are they? A major corporation could have thousands of enployees in any number of departments...

13 years ago

If the first digit must be 2, and digits can be repeated, you can forget about the leading digit and just look for every THREE-digit integer (INCLUDING leading zeros) whose last digit isn't zero. That's every integer between 000 and 999 (of which there ar...

13 years ago

Greater than 15-12=3, surely?

13 years ago

If X = 0.2323... then 100X = 23.2323..., so 99X = 23.2323... - 0.2323... = 23. So X = 23/99. 23 is a prime number, so you can't simplify that any more.

13 years ago

|x-5.25| < 1.75

13 years ago

Calculate 0.81 x 1.45 x 1000 / 28.35, where 1000 is the number of grams in a kilogram, and 28.35 is the number of grams in one ounce.

13 years ago

The number of men and women is irrelevant (unless they've got any rules that restrict the gender of any of those posts), so all that matters is that there are (16+19)=35 people from which to choose. The instruction to round the answer to five decimal plac...

13 years ago

Oops - I can't do permutations properly: it's 25 times too big. That last calculation should have been 9.25240 x (10^11). Sorry!

13 years ago

The first dice generates a random integer between 1 and 6, and the second one does likewise - which gives 36 possible outcomes, all of which are equally likely if you distinguish between the two dice. Of these, how many total 5 or 12? There's only one way...

13 years ago

Because there's no sample space diagram here to clarify the point, my first question is "Does an ordinary deck of cards contain a joker?", because that affects the answer to every subsequent question. Assuming that it doesn't: P(five of clubs): there's on...

13 years ago

When you're dividing something by a fraction, change it to multiplying the something by the reciprocal of the fraction instead: it's much easier. The reciprocal of a fraction (a/b) is just (b/a). Also you'll need to know that (a/b) x (c/d) = (a x c) / (b...

13 years ago

It sounds to me as if you need to work out the highest common factor (HCF) of 125 and 300, which I think is 25, since 125 = 5 x 25, and 300 = 12 x 25. If so, you'll have 25 seats in each row, with 5 rows in the first room, and 12 rows in the second.

13 years ago

Is the sun shining? If so, you could take a one-meter ruler, place it vertically on the ground, mark both the point on the ground where you placed the ruler and the point where the end of its shadow is, then take the ruler away. Next, use the ruler to mea...

13 years ago

You need to express the whole thing in terms of a common denominator (the denominator is the bottom of a fraction). You've got two fractions in the calculation: (9/10) and (3/4), so the best common denominator is probably 20, because 9/10 = 18/20, and 3/4...

13 years ago

The answer looks to me like 2x multipled by 1/2, which is just x.

13 years ago

Okay - this is the first time I've ever tried to use Bayes Theorem... P(A|B) = P(B|A).P(A) / Sum(P(B|Aj).P(Aj) Let A be the event "All five balls are white". Let Aj be the event "There are exactly j white balls". Let B be the event "I draw two white balls...

13 years ago

The answer is any number that isn't one of the following: 7, 11, 15, 19 etc, that is, any integer that can't be expressed in the form (4k-1) for k greater than or equal to 2 (as you've said that n has to be greater than or equal to 6). Having said that, I...

13 years ago

No problem - but you'll still need to be able both to measure a distance and to calculate an angle. Measure the distance between the bottom of the flagpole and a point some convenient distance away - it doesn't really matter how far, but say it's Y meters...

13 years ago

I imagine you'd want to keep the card with the largest number of factors. 24 has got the following factors: {2, 3, 4, 6, 8, 12}, so there are 6 of them. 30 has got {2, 3, 5, 6, 10, 15}, so there are 6 of them. 35 has got {5, 7} so there are just 2. 36 has...

13 years ago

Please tell us what Scenario 1-4 is.

13 years ago

The number is odd, and divisible by 5 - so the final digit must be 5. The largest digit is in the tens place, but isn't the largest digit, and no digits are repeated - so it must be 6, 7 or 8. The digit in the hundreds place is half the digit in the tens...

13 years ago

Already tackled - see your other post.

13 years ago

(I think 7485 will also do it: an explanation of the logic is supplied in response to Amy's posting of this question.)

13 years ago

Not quite, I think. Call the number of kilos of apples A, and the number of kilos of berries B. Then you know that: 4A + B = $5.75 2A + B = $3.25 so subtract the second equation from the first, and you'll get 2A = $2.50, so A = $1.25. Subtract the 2A = $2...

13 years ago

If the plane is travelling at 15 miles per minute, it'll take 500/15 = 100/3 minutes = 100/(60x3) hours = 5/9 hours to cover 500 miles. In that 5/9 hours, the car will travel (5/9) x 55 = 30.556 miles. (I've checked it three times because the numbers look...

13 years ago

It looks like a Venn diagram problem to me - which is a pity because I can't draw a Venn diagram here. Never mind: I'll try to manage with just the numbers. Pr(EuF) = P(E) + P(F) - P(EnF) = 0.55 + 0.55 - Pr(EnF) = 0.85, so Pr(EnF) = 1.10 - 0.85 = 0.25. Li...

13 years ago

There are 900 three-digit numbers in total (100 to 999 inclusive), and every odd-numbered one is even, i.e. 1st = 100, 3rd = 102, 5th = 104, 7th = 106 etc. There are as many odds and evens in 900 consecutive integers, so exactly half of them are even, i.e...

13 years ago

To get it right, make sure you distinguish between the first and second throw. If you do that, you'll see that there are 36 distinct outcomes: (1, 1), (1, 2), (1, 3), ... (6, 4), (6, 5) and (6, 6). They're all equally likely with probability 1/36, so how...

13 years ago

Let my age now be A, and let Mum's age now be M. Then we know that: A = M/3, so 3A = M A-8 = (M-8)/5, so 5A-40 = M-8 but M = 3A, so 5A-40 = 3A-8 so 2A = 40-8 = 32 so A = 16 so M = 48. Check it: A is one-third of M, and eight years ago I would have been 8...

13 years ago

You're welcome!

13 years ago

It should be 2 x 3 x 5 x 7, I think, i.e. the product of the first four prime numbers.

13 years ago

An obvious candidate is 5, so if you divide 555 by 5 you'll get 111. That's divisible by 3 (because the sum of the digits is divisible by 3, which is a useful quick check), so do that and you'll get 111/3 = 37. But 37 is a prime number, so you won't be ab...

13 years ago

Please tell us how many owners there are in total, and how many horses are owned by each of them.

13 years ago

a(x+1) = b, so divide both sides by a: x+1 = b/a, so subtract 1 from both sides: x = b/a - 1.

13 years ago

I agree with all your answers, if by "d" you mean the left hand side of the equation. Regarding no. 7, I suggest you do something like this: Each side of the equation inside the "absolute bars" could be either positive or negative. Try assuming that each...

13 years ago

Suppose he's got D ducks and C cows. Then we know that D + C = 22. Now: a duck has two legs, and a cow has four. So 2D + 4C = 56, which is the total number of legs. But D + C = 22, so 2D + 2C = 44. So if you subtract this equation from the one that tells...

13 years ago

X is exponentially distributed, so the probability density function f(x) = m.exp(-m.x), where E(X) = 1/m. You're told that the expected value is 10 mins, so 1/m = 10, so m = 1/10 = 0.1. So what you need to calculate is the integral of the pdf between 5 mi...

13 years ago

I think what you're being asked is this. There's some polynomial - let's call it f(x) - that when divided by (x + 10) equals (x^2 - 6x + 10) with remainder -1. What is f(x)? If I'm right, then we should be able to find it by multiplying (x^2 - 6x + 10) by...

13 years ago

That's what I get too. It doesn't look so much that way, does it!

13 years ago

Ask for what t is -16t^2 + 34t + 15 = 0? Use the usual quadratic formula for that one: x = (-b +/- sqrt(b^2 - 4ac)) / 2a sqrt(b^2 - 4ac) = 46, so the answer is either a negative number or 80/32 = 10/4 = 2.5. So that's the answer: two and a half seconds.

13 years ago

The temperature will drop on an exponential decay curve, i.e. a curve with the equation F = 53 + (185-53)exp(-a.t) = 53 + 132.exp(-a.t) for some parameter a which we need to determine. We know that because at t = 0, F = 185, and as t increases without lim...

13 years ago

Asking for the gcf implies that there ought to be at least two things that you're trying to factorize (because the gcf is the "greatest common factor"), whereas here there's only one. I think part of the question must have been lost somewhere.

13 years ago

x + y = 5.3 x - y = 1.7 So add the equations together: 2x = 7.0 x = 3.5 But x + y = 5.3, so y = 1.8 Check it: x times y = 6.3. Correct!

13 years ago

Suppose the number is ABC. Then if C is twice A, we know that A can't be any greater than 4. So the number must be one of the following: 1B2, 2B4, 3B6 or 4B8. When rounded to the nearest hundred it's 500, so it must be the last of those four, i.e. 4B8. If...

13 years ago

Add 1 to both sides, and then add (T-3) to both sides. That gets you 6 = 3(T-3). Divide both sides by 3 and you'll have 2 = (T-3). Then add 3 to both sides, and you've got T, which presumably is what you want, yes?

13 years ago

Permutations and combinations? xP1 is the number of ways in which you can take one item out of x items - and that's just x. So for example the number of ways in which you can take one item from 10 is going to be 10. It doesn't matter whether order is impo...

13 years ago

I'm going to try to infer a lot here, because the question is so obscure. I assume you're being asked to work out what the denominator of a fraction is when you're presented with the equivalent decimal. So suppose you're given a decimal number like 0.42,...

13 years ago

You won't be able to determine x and y uniquely from just a single equation: you would need two equations to do that. The best you can do with just one equation is to work out x in terms of y, or y in terms of x. The way you do that is to get all the x's...

13 years ago

Let the number of servings of beef be B, and let the number of servings of mash be M. Then we know that: 20B + 2M = 29 11B + 25M = 61 Multiply the first equation by 25, and the second one by 2 to get: 500B + 50M = 725 22B + 50M = 122 Subtract the second e...

13 years ago

Let's assume the numbers are large enough so we can use a Normal approximation to the binomial distribution here. The mean and standard deviation of the binomial distribution will be Np and sqrt(Np(1-p)) respectively, where in this example N = 128 and p =...

13 years ago

The standard error of the mean (SEM) is a measure of how well the sample mean is estimated - so if you increase the sample size, the standard error of the mean should get smaller. Specifically, if S is the standard deviation of the sample, then the SEM =...

13 years ago

The sum of the angles of a triangle is 180 degrees. Yours is an isosceles triangle, and the vertical angle is four times the size of a base angle - so you've got two base angles, plus four more base angles in the vertical angle. That's six base angles in...

13 years ago

The standard error of the mean of a sample of N is the population standard deviation divided by sqrt(N), so if the second parameter of your N(188, 41) is the population standard deviation, the SEM should be 1.297. So you need to find out how many standard...

13 years ago

It's just 0.05 x 0.05 = 0.0025. The fact that they were both sold during a given hour is irrelevant to the calculation, BUT you're assuming when you perform that calculation that the returning of one item is independent of whether or not another one is re...

13 years ago

Because all four parts of the question involve the same set of calculations, I suggest you first work out all sixteen of the various probabilities, check that they add up to one (because if they don't, the chances are that you'll get several parts of the...

13 years ago

You don't even need a set of Normal tables to answer this one. Just remember that the total area under the curve is 1; note that you've been asked to find the proportion of the distribution in the larger section, and then look at the four options. How man...

13 years ago

More information is needed please. How does x in the above equation relate to either a coin or a pouch?

13 years ago

(x-5)/3 = 1, so multiply both sides of the equation by 3, and you'll get (x-5) = 3. Now add 5 to both sides of the equation and you'll get x = 3 + 5 = 8. And that's your answer.

13 years ago

a) This one should state that the mean of the observed values will be approximately Normal, not that the distribution will be approximately Normal. b) I don't know what the 68-95-99.7 rule actually is, but it'll probably say something about 68%, 95% and 9...

13 years ago

You're welcome!

13 years ago

I'm going to assume that you mean the average annual salary is $30k, not $300k - because $300k sounds a bit excessive to me, even for Americans! If so, the proportion of Americans who earn below $22k should be the area to the left of ($22k-$30k)/$7k = -1....

13 years ago

I think you're asking whether a t-test is appropriate when your data consists of a binary response, i.e. a Success or a Failure, and you've got a very small number of responses. If so, the answer is no it isn't: it seems to me you ought to be looking at s...

12 years ago

Assuming that's 1, 1, 2, 3, 5... it's the Fibonacci sequence, in which X(i) = X(i-1) + X(i-2), where X(1) = X(2) = 1. So 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13 etc etc.

12 years ago

I think there's posibly a piece of information missing in the question as you've stated it, which is that you're only supposed to be looking at the function between the values X = -1 and X = +1, because the function is discontinuous at both of those value...

12 years ago

Four will ensure you've extracted at least two of the same color - but unless you turn the light on there's still no guarantee that you'll get two of the same color on you feet. Is that why it's a riddle?

12 years ago

With no knowledge of the trigonometry of triangles, argue it like this. The question wouldn't have been asked if there wasn't a unique answer, so the answer must be the same regardless of whereabouts above the base the apex is - including the special case...

12 years ago

The distance between the X coordinates is 2 minus -5 = 7. The distance between the Y coordinates is 2 minus 1 = 1. Those two distances are the base and height of a right-angled triangle, so to get the hypoteneuse, use Pythagoras: that gives you that the d...

12 years ago