Posts filed under: Q and A

Thomas is making a sign in the shape of a regular hexagon with 4-inch sides, which he will cut out from a rectangular sheet of metal. What is the sum of the areas of the four triangles that will be removed from the rectangle?

So it’ll look like this:

It’s helpful just to know that a regular hexagon’s interior angles all measure 120°, but you can also calculate that using (n-2)\times 180^\circ:

\dfrac{(6-2)\times 180^\circ}{6}=120^\circ

That means that the four triangles you’re cutting off the rectangle are each 30°-60°-90° triangles with 4-inch hypotenuses.

Those will have legs of 2 and 2\sqrt{3}, and therefore areas of \dfrac{1}{2}(2)(2\sqrt{3})=2\sqrt{3}. Since there are four such rectangles, the total area you’re cutting off is 8\sqrt{3}

After doing some practice problems on Rational Equations on Khan Academy, I was wondering would the equation be undefined if after factoring the equation we obtain cancellable factors on the denominator and the numerator? For ex: ( (2x-1) (x+4) ) / (x+4) would -4 be a plausible argument? I see the x+4 factor can be cancelled together with the one in the numerator.

I don’t believe you’ll see this tested on the SAT, but the way you’d typically handle this is to say that the expression is equivalent to 2x – 1 for all x ≠ –4. Does that help?

There are five houses on each side of a street, as shown in the figure above. No two houses next to each other on the same side of the street and no two houses directly across from each other on opposite sides of the street can be painted the same color. If the houses labeled G are painted gray, how many of the seven remaining houses cannot be painted gray?

Without knowing what the figure looks like, I can’t say, but the process to follow to answer this is first to cross out all houses that can’t be gray given the rules. That means any house next to a G or across from a G gets crossed out.

To make sure you aren’t missing any, do each G one at a time: pick a house with a G in it, then cross off the house to the left, the house to the right, and the house across. Then move to the next G, etc. Since the question asks how many houses cannot be gray, count all the houses you crossed out, and that’s your answer.

One note: the current SAT doesn’t really ask questions like this. Are you sure you’re studying from up-to-date materials?

Here’s a problem from Khan Academy’s SAT practice section. Please explain this one for me.

A marine aquarium has a small tank and a large tank, each containing only red and blue fish. In each tank, the ratio of red fish to blue fish is 3 to 4. The ratio of fish in the large tank to fish in the small tank is 46 to 5. What is the ratio of blue fish in the small tank to red fish in the large tank?

A ratio of 3 red to 4 blue means the total must be a multiple of 7, so it might help you to first multiply the totals in each tank by 7. The 46:5 ratio for large tank to small tank becomes 322:35. Let’s just pretend those are the actual numbers of fish in the tank.

In the small tank, 4/7 of the 35 fish in the tank are blue, so 20 fish are blue.

In the large tank, 3/7 of the 322 fish are red, so 138 fish are red.

Therefore, the ratio of blue fish in the small tank to red fish in the big tank is 20:138, which simplifies to 10:69.

Note that while multiplying everything by 7 was helpful (in my opinion) in making the math more intuitive, you don’t need to. You can simply enter the following in your calculator and ask it to convert to a fraction:

\left(\dfrac{4}{7}\times 5\right)\div\left(\dfrac{3}{7}\times 46\right)

I’m having a lot of trouble with this problem, could you please break it down for me? It is number 2 on the functions practice questions. “If f(x-1)=x+1 for all values of x, which of the following is equal to f(x+1)?”
A. x+3
B. x+2
C. x-1
D. x-3

In the solutions in the back of the book, I have two approaches. I’m assuming you’ve already looked at those and neither went all the way for you, so here’s another look.

We know that function notation means, generally, that the bit in parentheses goes through some series of mathematical operations, and a result is spit out. So, for example, stuff can be added or subtracted, or the argument could be multiplied, divided, raised to a power, etc.

In this particular problem, we know that f(x-1) goes through some process that spits out an end result of x+1. Here’s the crux: what could happen to (x-1) to make it equal x+1? No multiplying or raising to powers necessary! The only thing you need to do to turn (x-1) into x+1 is add 2!

That’s how we know that the function f just adds 2.



Therefore, we can figure out what f(x+1) must be:


Does that help?

The midpoint formula tells you that the a segment with endpoints (a, b) and (c, d) will have a midpoint at ((c)/2,(d)/2)

So we know that (+ 9)/2 = 6 and (5 + y)/2 = 3. We can solve those!

(+ 9)/2 = 6
+ 9 = 12
x = 3

(5 + y)/2 = 3
5 + y = 6
y = 1

Therefore, x + y = 4.

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Practice Test 6 Calculator Section # 29

All you need to know about average speed is that it’s equal to \dfrac{\text{distance traveled}}{\text{time it took to travel that distance}}.

You’re given a formula for the distance traveled: when the car’s been traveling t seconds, it’s gone a distance of 16t\sqrt{t}. That formula contains all you need to calculate the average speed.

\text{distance traveled}=16t\sqrt{t}

\text{time it took to travel that distance}=t

\text{average speed}&=\dfrac{\text{distance traveled}}{\text{time it took to travel that distance}}=\dfrac{16t\sqrt{t}}{t}=16\sqrt{t}


I wanted to ask for your assistance on an SAT Practice Test question? It’s a question on systems of equations. The question I’m referring to is question 11 on the no-calculator math section of Practice Test #4.

Thank you in advance :’)

Sure thing. I’ve actually already posted a solution to that one here.

(You can check which official questions I’ve posted solutions for here.)

Could you do Test 8 section 4 question 24 for me?

I can’t understand why the answer is D and not C.

It’s not A because we have no way of knowing what the people who didn’t vote thought.

It’s not B because we know nothing about the age of the people voting.

It’s not C because the votes are worth the same whether they come in via social media or text message. If you change the method by which some people voted, you don’t change their votes.

(By the way, we know contestant 2 won 70% of 30% of the votes, and 40% of 70% of the votes. That means she lost: 0.7\times 0.3 = 0.21 and 0.4\times 0.7 = 0.28, so contestant 2 won 21% + 28% = 49% of the vote.

It is D because we know contestant 2 won 70% of the social media vote, and choice D is basically asking whether that’s true. If 70% of social media voters voted for contestant 2, then social media voters were more likely to prefer candidate 2, period.

On a question like this, look for reasons to eliminate choices rather than looking for reasons to accept choices. Otherwise, sometimes it’s easy to get tricked by a choice that sounds pretty good at first (like C) but doesn’t hold up under scrutiny.

When a system of linear equations has no solution, that means you have parallel lines, which means the lines have the same slope. So put both equations into slope-intercept form (y = mx + b) first:


In order for those lines to be parallel, their slopes must be equal, which means 2/5 = -4/k. That means k must be equal to –10.

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A logarathim question:

pH = – log h

The pH of a solution is dependent on the concentration of hydronium ions, h, and can be calculated by the equation above. If the concentration of hydronium in solution A is 100 times the concentration of the hydronium in solution B, what is the absolute value of the difference in their pH values?

A. –2
B. 2
C. 3
D. 10
E. 100

Hmm…Is this ACT? Subject Test? Definitely not SAT with logs (not to mention 5 answer choices). No matter—I got you.

Easiest way to go here is just to plug in a couple values for the concentrations of solutions A and B. Let’s say A has a concentration of 1,000,000 and B has a concentration of 10,000. That meets the requirement of the question: the concentration of A is 100 times more than the concentration of B. Now just calculate both pH values (using log base 10 in case that’s not your calculator’s default):

\text{pH}_A=-\log \text{1,000,000} = 6

\text{pH}_B=-\log \text{10,000} = 4

And there you have your answer—the pH values are 2 apart. Don’t be fooled by choice A: the question asks for the absolute value of the difference so you definitely don’t want a negative number.

Just do those things and simplify:


Looks like the volume of the second cylinder is half the volume of the first! 🙂

The shortcut here is to recognize that the r value gets squared but the h value doesn’t, so you can say the volume is divided by 4 then multiplied by 2.

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60° is 1/6 of the circle (which has 360° of arc in total), so the length of the minor arc will be 1/6 of the circumference. 1/6 of 12π is 2π.

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Huge shortcut here if you just know that for a parabola in standard ax^2 + bx + c form, the x-coordinate of the vertex will be at –b/(2a). In this case, that means it’s at –3/(2(–6)) = 3/12 = ¼.

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The way to think about this (for me, anyway) begins with understanding that 2[something] + 3 = 8x – 1, and our job is to figure out what that something is.

Since this question gives us answer choices, all we really need to do is try each one as the something to see what works. Since they’re not in numerical order, start with A.

2(4x – 2) + 3
8x – 4 + 3
8x – 1

Oooh, look at that! we’re already done! 🙂

(Also, for those worrying along at home, note that 5 answer choices means this isn’t an SAT question. Could be SAT Subject Test, though.)

If you didn’t have answer choices, you’d still start in the same place, but then think through logically. Thought process:

2[something] + 3 = 8x – 1

Hmm..there’s gotta be a 4x in the something otherwise i’ll never get 8x. But if it’s just 4x in there then I’ll end up with 8x + 3, not 8x – 1. How do i take 4 away? By adding a –2 inside the something!

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