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A function will only have that property if it’s a line that passes through the origin. For example, f(x) = 5x has that property:

You can try the same with other linear functions to see why they won’t work. For example, if f(x) = 5x + 2:

Nonlinear functions also won’t work. For example, if f(x) = x^2:

Anyway, now that we know we’re dealing with a linear function through the origin, we can figure out that if f(6) = 12, then the function we’re dealing with must be f(x) = 2x. Therefore, f(2) = 4.

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First, a circle inscribed in a square looks like this:

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If that square has an area of 2, that means each of its sides has a length of √2. And THAT means that the radius of the circle is √2/2.

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The area of a circle is πr^2. Plug √2/2 in for r and you’ve got your answer:

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If the actual width is W, then the actual length must be 2W. We know this because in the scale drawing, the length is twice as many inches as the width.

The area of the living room floor, therefore will be length times width:

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If you remember that the square of i is –1, you can reduce these to purely real numbers before you even FOIL.

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In a + bi form, 5 is just 5 + 0i, so a + b will be 5 + 0 = 5.

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\sqrt{x-k}=k

None. The square root of something will never be negative, so if k < 0, it can’t be all alone on the right side of that equation.

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Note that f(x) is a line in slope-intercept form, where a is the slope and b is the intercept.

Once you recognize that, you just need to know that, notationally, the –2 in front of f(x) means you multiply the whole thing by –2. So:

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That’s still a line, but now the slope is –2a.

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Note that f(x) is a line in slope-intercept form, where a is the slope and b is the intercept.

Once you recognize that, you just need to know that, notationally, the –2 in front of f(x) means you multiply the whole thing by –2. So:

image

That’s still a line, but now the slope is –2a.

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None. The square root of something will never be negative, so if k < 0, it can’t be all alone on the right side of that equation.

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If the actual width is W, then the actual length must be 2W. We know this because in the scale drawing, the length is twice as many inches as the width.

The area of the living room floor, therefore will be length times width:

image

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Best way to go if you’re not sure is to solve for n.

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Now all you need to do is find 5/7 of 98.

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Another way to go is to recognize that if the denominator of 7 is common to both fractions, all you really need to do is divide by 3 and multiply by 5:

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In this question, a boy can either be a junior or a senior. To figure out the minimum number of senior boys, maximize the number of junior boys!

Say all 30 juniors are boys. If that’s true, then of the 40 boys in the class, 30 are juniors and the other 10 are seniors. (In this configuration, all 60 girls would also be seniors.)

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A question like this on the SAT will always have answer choices, and in this case would be easy to backsolve. Don’t rob yourself of easy points—remember that answer choices are often a helpful tool! Use them to your advantage. In this case, you’d simply need to plug (2x – 1) in for x in each answer choice until one gave you cx.

Since you didn’t provide answer choices, though, we’ll have to do this the hard way.

The way I like to think about questions like this is pretty procedural. You’ve got some f(x) expression. You swap out the x for (2x – 1) and what happens? First, the 2x means there’s a doubling. We know we want to land on cx, so f(x) must have a c/2 in the x term that gets doubled. Make sense?

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But what about those question marks? Well, we know that the end result we want is that f(2x – 1) = cx, so that question mark must cancel out the –c/2 we get from multiplying c/2 by (2x – 1). What cancels out –c/2? Easy: +c/2! So let’s see if making f(x) = (c/2)x + c/2 works:

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Yep, that works. Again, it probably would have been easier to just start with the answer choices and see which one landed you on cx. Your work would end up looking just like that second bit of math above, but without so much abstract thinking required.

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Say x is the number of weeks from this Monday.

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That means that at the start of the 18th week, they’ll still be OK, but by the end of the 18th week they should have less than 15 bundles. They will order new paper on the Monday of the 19th week.

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Draw this to make sure you don’t get messed up. Let’s use X for all the people in line who aren’t Bill and B for Bill.

X X X X B X X X X X X X X X X X

Above, Bill is the 5th person from the left and the 12th person from the right–that means there are 4 people to the left of him and 11 people to the right of him. 4 + 11 + 1 for Bill is 16 people in line, total.

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Well, 1975 is a multiple of 25 and 24 is a multiple of itself, so 1975+24 = 1999. 🙂

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