David Q

Try solving a slightly easier question for a moment. Suppose the question read "This represents a 100% increase over the year 2005". You would infer that the number of 2005 tax returns was HALF of 74.2 million. To get that, you would have had to solve the...

17 years ago

You'll need to fill in the following table with numbers that add up to 900, because your Venn diagram is going to be composed of these seven mutually exclusive segments: Female / Non-Union / Non-Single Union / Non-Female / Non-Single Single / Non-Union /...

17 years ago

It sounds like the answer is 365 to me, unless I've misunderstood the question. Are you sure the question isn't something like "Out of how many permutations of three people randomly selected would you expect one to have all three of them sharing a birthda...

17 years ago

I assume that the original equation should read (t+3)/t = t², not (x+3)/t = t², yes? You can actually do it quite quickly by iteration if you've got access to a calculator with a cube-root function (or logs and antilogs). Rearrange it so that it reads: t...

17 years ago

I assume - although you haven't said so - that P and Q are the points at which the line you've given cuts the X and Y axes respectively. If so, then P is the point (2,0) and Q is (0,-3). (To get those, just put Y=0 and solve your equation for X, then put...

17 years ago

You need to find five numbers for which the mean is 8, the middle one is 6 and the difference between the biggest and the smallest one is 10. Let's guess: try some numbers between 4 and 14, to get the range right. So we know what three of the five numbers...

17 years ago

It usually helps answering questions like this - which are really all about definitions - by thinking up an example or two. Usually the answer isn't far behind: 1. Here's a fraction: 1/2. Can that be written as a decimal? 2. Is -5 a whole number? Is zero...

17 years ago

This is a tricky one. I eventually managed to get an answer out of it, but I have a feeling that the original problem might have been mis-specified (unless of course I've got it wrong myself). The zeros of this function are easy: all you need are the prod...

17 years ago

Correction: f(1) = 240 = 48*5. I got the factorization right, but copied down the original number wrong. Whoops.

17 years ago

You got it. Now see if you can invent a few more sets that also work, as asked for by the final part of the question. Obviously you'll need to have the median somewhere between the lower and upper limits that define the range, and you also won't be able t...

17 years ago

The first function takes the value zero at approximately X= -0.9 and again at around 4.63, with a minimum value between them somewhere around 3.46; also has a gradient of zero at X=0. Are you sure the equation is correctly written down, given that it's ap...

17 years ago

I suspected as much :) Your first function appears to have two zeros, but both are for positive X (approx. 0.5 and 5.0). The second one appears to have a couple of zeros somewhere around -1.4 and +1.4: would you like to check that out? (That's just from a...

17 years ago

Ah - you didn't mention that you were selecting the answer from a list of options - that would have saved a lot of time! Having said that, I've just checked all four, and none of them produces the f(X) values listed. Not surprisingly, because the function...

17 years ago

Try solving it iteratively: if g(X)=0, then X^3-3X+1=0, so rearrange the equation to read X = (3X-1)^(1/3) Not put X = 2, and evaluate the function. You'll get about 1.71. Feed that into the equation again, and you'll get about 1.60. Keep going for a few...

17 years ago

With no restrictions, just select any four from ten, which is: factorial(10) / (factorial(4) x factorial(6)) = (10x9x8x7x6x5x4x3x2x1) / {(6x5x4x3x2x1) x (4x3x2x1)} = (10x9x8x7) / (4x3x2x1). If you've got to have four women and two men, then you're choosin...

17 years ago

Okay, so it's slightly more dense than water, so the obvious answer is that it'll sink to the bottom of the cup. But wouldn't what the test liquid actually IS have a bearing on what happens? Suppose it ignites or vaporizes on contact with water? Suppose i...

17 years ago

Butterfly is a discipline of swimming, whereas dressage is a discipline of horseriding (I just googled "define dressage" to find that out). A ski-jumper skis down a mountain, so it ought to be that a diver dives down a ... what? Is it a swimming pool, bec...

17 years ago

Say the radius of the sphere is r. The sphere touches both sides of the cube, so the distance between two faces of the cube is 2r. But that also equals one edge, so one side of the cube has area 2r x 2r = 4r². But a cube has six such sides...

17 years ago

The area of a triangle is half its base times its vertical height. Call the base a. Now: you know that the other two sides are twice as long as that, so they're each 2a. What you need to find out now is the vertical height - so think of your isosceles tri...

17 years ago

The area of D will be (6/5)² times the area of C, which is (6/5)² x 47.6"² = 68.544"² (don't forget that you need to square that factor of (6/5), because it applies to both the base AND the height). The base of D is 6.72", and the area of a triangle is ha...

17 years ago

I'm afraid I don't recognise the equations in C, D and E you've written down - but I've just sketched out the problem on a piece of graph paper. If it helps you to check your working, the centre of the circle ought to be at (3, -2) and the radius is sqrt(...

17 years ago

Er... if the area is half the base times the height, then the height must be TWICE the area divided by the base. Sorry!

17 years ago

You need the same denominator for both fractions before you can subtract one from the other. Multiply top and bottom of the first one by the denominator of the second one, and multiply top and bottom of the second one by the denominator of the first one....

17 years ago

Rather than try to work out what the two brackets are directly by inspection, can we find a value of X that makes the above expression zero? How about X=4? Fine... so divide the original expression by (X-4), and you'll get the other factor.

17 years ago

Could you clarify what you mean by "A varies BC" please? I feel there's more to this question than reproduced here.

17 years ago

No, not really - any symbol would do, provided you made it clear somewhere that it represented a variable - after all, most people would understand the problem "Factorize ¿² + 6¿ + 5 = 0". But you'll need to find a way to represent them that you can gener...

17 years ago

I'd have thought - unless someone corrects me - that they both have the same meaning as they do in basic arithmetic. By a whole number you mean an integer, and by an irrational number you mean a number that can't be represented as the ratio of two integer...

17 years ago

Eleven seems rather a lot: according to Wolfram MathWorld (I can't post the address, it seems, but just google "field axioms" to find it) there are five: Associativity, Commutativity, Distributivity, Identity and Inverses. Algebraic definitions of these a...

17 years ago

Proof by contradiction: Suppose not. Then a negative number N divided by a positive number P would be positive or zero. (I've assumed incidentally that all numbers are positive, negative or zero, and that zero is neither positive nor negative.) Assume N/P...

17 years ago

For the first bit of the question, get all the terms over onto the left-hand side of the equation. For example, take the second one: X²-6X-7=0 Usually you can factorize it easily into the following form: (X+A)(X+B). (There's no guarantee that it's an easy...

17 years ago

In "Principia Mathematica", Alfred North Whitehead and Bertrand Russell succeed in proving that 1+1=2 on page 379 of Volume 1. I would imagine the proof of the above proposition can be found somewhere between the start of the book and page 378.

17 years ago

The following two paragraphs come from Wikipedia: <<In science a theory is a testable model of the manner of interaction of a set of natural phenomena, capable of predicting future occurrences or observations of the same kind, and capable of being tested...

17 years ago

I'd guess that it probably means a total of 14 dots - but that's a guess. There's probably more to the question than this, so what's the rest? Maybe we can work out what the answer is from that.

17 years ago

In the context of the other question you've posted, it's the gradient of a straight line. You work it out by dividing the vertical height between two points on the line by the horizontal distance between the same two points.

17 years ago

I'm extremely rusty at these, but I think it's possible you might be able to use the substitution x=2cosh(u), where cosh is the hyperbolic cosine. cosh²(u)-sinh²(u)=1, so sqrt(x²-4) = sqrt(4cosh²(u)-4) = 2sinh(u). dx/du=2sinh(u), so dx=2sinh(u)du, which c...

17 years ago

You'll find it very helpful to remember that sin(2x)=2sin(x)cos(x) here, since this means that you're trying to find all the solutions of 2sin(x)cos(x)=cos(x). These will be equal under two possible scenarios: firstly if cos(x)=0 (since both sides will be...

17 years ago

Be careful here: the Imperial and U.S. Customary fluid ounces are different. In the Imperial system there are 20 fl oz in a pint, whereas in the U.S. system there are 16 fl oz in a pint. There are two pints in a quart in both systems. Which side of the At...

17 years ago

The speed in still water is X km/h, so the speed upstream is (X-3) km/h (slower than X because it's upstream), and the speed downstream is (X+3) km/h (faster than X because it's downstream). Travelling at (X-3) km/h for 12 km will take 12/(X-3) hours, and...

17 years ago

The easiest way is to get rid of the decimals before you start: multiply both numbers by 100 before you divide one into the other. If you do that, you'll end up with 18 divided by 30, which is a FAR easier long division sum to tackle.

17 years ago

You're welcome!

17 years ago

It means 4x4.

17 years ago

So 2x4² = 2x4x4=32.

17 years ago

From an algebraic point of view, no. From a practical point of view - if you're trying to measure the height and horizontal distance with a ruler, then yes - because the errors will be greater the closer the points are together. But I think you mean the f...

17 years ago

I imagine you want to express 120 as the product of as many factors as possible? If so, they'll be the prime factors of it. 120 equals 5x4x3x2, but that can be broken down further as 5x(2x2)x3x2. You can't break it down any more because all the factors ar...

17 years ago

Both 43 and 47 are prime. Which would you prefer?

17 years ago

Oops - so it is!

17 years ago

The fastest source of definitions is probably Google: just type in "define adaptation", you'll get back a list of places to search for the one you're interested in.

17 years ago

Was there another equation? What we're looking at is the equation of a line, not a point. You'd need a second line to define a point, which would be the intersection of the two (assuming there was an intersection, of course).

17 years ago

The probability that both will fail is 0.02 x 0.02. The probability that neither will fail is (1-0.02)*(1-0.02) = 0.98 x 0.98. The probability that one or the other will fail is (2 x 0.02 x 0.98). Why? Because there are two situations to consider here: Al...

17 years ago

From the description above, we know that: (W+2) x L = 140 (L+1) x W = 120 where W = the width and L is the length, so... WL + 2L = 140 WL + W = 120 Subtract the second equation from the first, you get... 2L - W = 20, so W = 2L - 20. So feed that back into...

17 years ago

(3x-4y)² = (3x-4y)(3x-4y) To get the xy term, just multiply each term in the first bracket by each term in the second, and add together all the terms in xy that you find. You should get... 9x² - 12xy - 12xy + 16y². So just add the two xy terms together.

17 years ago

On the assumption - and it IS an assumption - that the temperature is rising at roughly a constant rate, then it ought to be halfway between 19 and 24 degrees (since 1.75 minutes is halfway between 1.5 and 2 minutes), which would be 21.5 degrees. But the...

17 years ago

I think I'd read the general gist of it as something like this: "It's not as if we haven't been concerned about the British. They've been warned several times about their unwarranted attempts to impose their laws on us. We've reminded them of the circumst...

17 years ago

Science implies understanding - or at least an attempt to understand. Primative cultures don't understand why their crops grow, but that doesn't stop them developing a basic technology to help them with the harvest. That would be a case of technology aris...

17 years ago

A hundreth = 1/100 A thousandth = 1/1000 But 10 x (1/1000) = 1/100 so there are ten thousandths in a hundredth. Alternatively, what's a hundreth divided by a thousandth? It's (1/100)/(1/1000) = (1/100) x (1000/1) = 10

17 years ago

Aren't there some restrictions on the sort of function f(x) is? Suppose f(x)=4. Then f(-x)=4 also, and (f(x)-f(-x))/2 = (4-4)/2 = 0, and that ISN'T odd. Another one: put f(x)=x², then f(-x)=x², and (f(x)-f(-x))/2=0 again. Same problem: something's wrong h...

17 years ago

Also consider f(x)=sin(x). f(-x)=sin(-x)=-sin(x), so O(x)=(sin(x)+sin(x))/2=sin(x) - and for most values of x that isn't even an integer, let alone an odd one.

17 years ago

What sort of factors are you supposed to consider, and what sort of culture had you in mind?

17 years ago

It depends on whether the round brackets surrounding your interval of (6,13) indicate that the numbers 6 and 13 are not included in the interval. The notation I'm familiar with uses square brackets to indicate that the numbers ARE included, and round brac...

17 years ago

Is the ÷ symbol confusing you? This is a division sign: it's usually typed as a slash (/) these days because there's no ÷ symbol on most modern keyboards. What you're looking at is (9/3)x(8/4), which as Bob's already noted is just 6 times 2.

17 years ago

Advantage: charities provide a mechanism for the transformation of donations from many donors into practical benefits for the causes supported by those charities: something that well-meaning donors would individually find difficult. Disadvantage: existenc...

17 years ago

It's a question about factorials: the answer is the number of ways you can choose 3 from 5. You could enumerate them without too much difficulty, but the shortcut is to work out 5! / (3! x (5-3)!), where N! means N x (N-1) x (N-2) .... x 2 x 1.

17 years ago

No - you're distinguishing between the 1st, 2nd and 3rd, which you don't need to do. ABC is the same as BCA, which is the same as CAB etc. You've got to allow for all the combinations that are the same.

17 years ago

In fact, every triple can occur in six different ways - so the answer is actually the 60 you identified, divided by 6.

17 years ago

Kiana is absolutely right: 9 divided by 3 IS 3. How embarrassing - sorry.

17 years ago

Is your denominator ((e^t)-7) or (e^(t-7))? The difference is important: if it's the first of these, the function would be discontinuous at t=ln(7). If it's the second, then it's not discontinuous over the range given.

17 years ago

Do you mean mental/emotional properties as in objectivity, patience and attention to detail? Or do you mean physical properties? Or do you mean things that a scientist possesses? You need to be more specific here.

17 years ago

It's going to be something like this (but do check my algebra)... You've been told what the Energy, Protein, Vitamin and Price are for unit quantities of Sunflower, Sorghum, Maize and Wheat. This gives you four equations - one in each of Energy, Protein,...

17 years ago

Oh - and one more constraint, namely that the total quantity of Sunflower, Sorghum, Maize and Wheat equals one kilogram, since you're trying to find the optimal mix in that amount of feed. This isn't actually part of the problem you've been set, but if yo...

17 years ago

By "top ten", do you mean the ten most positive characteristics of such an environment, or just the ten characteristics that most strikingly differentiate a multicultural school environment from a non-multicultural one? And should these characteristics be...

17 years ago

Have you been given some data upon which to perform the hypothesis testing procedure? If so, we'll need to see it too in order to help.

17 years ago

A compelling argument or counterargument is one which it's very difficult to refute. Do you have a particular thesis in mind, or is this a general request for help in how to set about doing it?

17 years ago

Biased surveys generate answers that are unrepresentative of the population about which you want to draw general conclusions. In this case, the population consists of all the students in your school. As Ms Sue says, what would you conclude if only you sur...

17 years ago

The standard error of the mean (SEM) is the standard deviation divided by the square root of the sample size - which in this instance is 9 - so the SEM here is 16/sqrt(9)=5.33. The question actually contains a little clue that you're probably on the right...

17 years ago

We'll need to know rather more about what that "a" is supposed to be - but I don't think your answer is correct. Whatever "a" is, the answer is unlikely to be just 1-0.84, which I suspect is how you've calculated it.

17 years ago

Do you mean "Can y be expressed as a function of x?". If so then the answer is yes it can, if by "x^2y" you mean "x²y" and not "x raised to the power of 2y". The equation x²y - x² + 4y = 0 can be written as (x²+4)y = x², from which you can easily express...

17 years ago

I still don't understand what "a" or "á" is supposed to represent here. Can you define it please?

17 years ago

Fair enough, though you could presumably define a function by restricting the range to just zero plus either the positive or negative real numbers.

17 years ago

You're welcome! :)

17 years ago

The answer is yes: you can write y as a function of x. Damon and I were debating whether you can write x as a function of y, which wasn't what you were asked.

17 years ago

Think of the various definitions of these functions in terms of the height, base and hypoteneuse of a right-angled triangle. You're told that tan(x)=8/3, which means that the ratio of the height to the side is 8 to 3. So you can work out the hypoteneuse b...

17 years ago

Whatever the number of friends is, it has to divide into $8.41 (i.e. 841 cents) without a remainder - so you need to find out what integers (because there can only be an integer number of friends) divide into 841. As Bob showed, that's easy: you can just...

17 years ago

It's referred to as the "mark-and-recapture" technique: you can find it under "Mark and Recapture" in Wikipedia, together with examples of the math involved.

17 years ago

Hi again Laura: it's putting into words the consequences of one of the assumptions that we made when we worked out the number of animals in that conservation park, namely that the density of animals in a small section of their habitat (in that example tha...

17 years ago

I don't believe there's a smaller one than that (I checked it in Excel). The next one is 5039, which is twice the last plus 1. (Incidentally, the one before 2519 corresponds to N=-1.) To work out which number gives a remainder of 1 for all the integers fr...

17 years ago

1 googol = 10^100, i.e. the digit 1 followed by one hundred zeros. The company name Google is a misspelling of the word "Googol".

17 years ago

Assuming "waiting street" is supposed to read "Watling Street", Wikipedia says that this was originally Wæcelinga Stræt, and is now the A2 from Dover to London followed by the A5 from London to Wroxeter. (Somehow it doesn't have the same ring to it, does...

17 years ago

Consider a very simple example in which you've only got two numbers. If they're not the same number, the mean won't be included in the set of data. Another reason why it might not be is when all the elements in the data are integers, but the mean is not a...

17 years ago

The correlation coefficient (r) is just the square root of r² ... but there are two square roots of r², namely +r and -r. However, there's a clue: the line of best fit has a negative slope - so that tells you the sign of r.

17 years ago

SINGER , Isaac Bashevis (1904-1991): Zlatek the Goat and Other Stories (Cuentos judíos de la aldea de Chelm) Okay - I've found it. What's the question please?

17 years ago

Google is usually a good place to start when it comes to definitions. Type in "define:surrealism" and see what you get.

17 years ago

I'll guess that "New York" is in there somewhere. Try removing those letters and then feed the remainder through an online anagram solver. I did that and didn't see anything I recognised, but you might be more lucky.

17 years ago

"New York - deny work"?

17 years ago

Your answer looks correct to me, but I don't see the relevance of the two intermediate steps. Each tile is one foot square; you've got 10x10=100 of them in a square, so it looks like a 10x10 chessboard. Half of those 100 squares will be black. (Just one t...

17 years ago

5.688 x 10^12 is in standard form already.

17 years ago

Do you mean "25 feet/second"? It's almost certainly a measure of speed, expressed as the number of feet travelled divided by the number of seconds it took to travel it.

17 years ago

Say J = no. of jellybeans and G = no. of pieces of gum then J + G = 28 (items) and 5J/3 + 2G = 50 (cents) so 5J + 6G = 150 and 5J + 5G = 140 You've now got two equations which differ by exactly 1 x G. Can you finish it off?

17 years ago

They come from multiplying the previous two equations by 5 and 3 respectively. What you're trying to do is get two equations that you can use to eliminate one of the two variables, so after I multiplied the equation that reads 5J/3 + 2G = 50 by 3 to clear...

17 years ago

What I *could* have done instead would be to just multiply the first equation by 2 to make the first one have the same number of Gs as the second: J + G = 28 (items) 5J/3 + 2G = 50 (cents) Multiplying the first one by 2 would give me: 2J + 2G = 56 5J/3 +...

17 years ago

I can't actually post a website address here, but if you google "latin phases quotes uk" you should find fairly high on the list a site at the University of Liverpool that has a lot of expressions covering an assortment of subjects. With a little ingenuit...

17 years ago