Posts filed under: Q and A

Trigonometry does the trick here. Below is that line making a 42° angle with the positive x-axis. I’ve also drawn a dotted segment to make myself a neat little right triangle.

Remember that slope is rise over run—how high the line climbs divided by how far it travels right. In this case, the dotted segment labeled a is the rise and the bottom of the triangle labeled b is the run. And luckily for us, the tangent function calculates that a/b ratio! Remember your SOH-CAH-TOA. Tangent = Opposite/Adjacent.

Just use your calculator to evaluate tan 42°. You’ll get 0.90.

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You can make two equations here. First, you know the total number of marbles is 103, so:

The second equation is more complicated, so let’s do it in parts. First, he gives away 15 red marbles, so he should have r – 15 left. He gives away 2/5 of his blue marbles, so he should have b – 2/5b = 3/5b left.

So the ratio of red marbles he has left to blue marbles he has left (which the question tells us is 3/7) should be:

The question asks how many blue marbles he had originally, so let’s substitute and solve for b. First get r by itself in the first equation:

Now substitute that into the second equation and solve:

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This question comes from my own book, so my tips on how to deal with these can be found in the same chapter. The main key to getting it right is making sure you translate the words into math correctly.

Note that although the question tells you that Tariq makes brownies and Penelope makes cookies, in the end it only asks about “treats,” so we can lump cookies and brownies together.

Tariq makes 30 treats per hour and Penelope makes 48 treats per hour. Together, then, they make 78 treats per hour. We know they both worked for the same amount of hours.

The other key to getting this right is keeping track of the units of the numbers you know. In this case, we have treats and hours for units. We know the number of total treats, and we know the rate of treats per hour. We want the number of hours. How do we set up the equation we need to solve? We need to divide the total number of treats, 312, by the number of treats they made per hour, 78.

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One gallon of honey weighs approximately 12 lbs. If one gallon of honey is mixed with 5 gallons of water to make tea, how many ounces of honey will be in each 8 fluid ounce cup of tea?
Choices are 1, 2, 3 or 4. (Answer= 2) given: 16 oz = 1 lb, 128 fl oz = 1 gallon

Wow, that’s going to be some really sweet honeywater tea.

You’re making 6 gallons of “tea” (1 gallon honey + 5 gallons water = 6 gallons). Converting that to fl oz, you should have, in total, (128 fl oz/gallon)(6 gallons) = 768 fl oz.

Therefore, a cup with 8 fl oz is 8/768 = 1/96 of the mixture.

By weight in oz, the 12 lbs of honey you put in the mixture is (16 oz/lb)(12 lbs) = 192 oz. How may fl oz of honey will be in 1/96 of the mixture if 192 oz of honey are in the whole mixture?

(192 oz)(1/96) = 2 oz

PWN the SAT Parabolas drill explanation p. 325 #10: The final way to solve: If we are seeking x=y, since the point is (a,a), why can you set f(x) = 0? You start out with the original equation in vertex form, making y=a and x=a, but halfway through you change to y=0 (while x is still = a). How can we be solving the equation when we no longer have a for both x and y?

For everyone else’s context, here’s the problem:

Now, be careful! I am not changing to y = 0 in that algebraic solution in the back of the book; I am subtracting a from both sides! Note how the a term on the right changes from –22a to –23a.

 

Draw this out. Start with the two points you’re given.

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Now remember that the shape is a rectangle, and that you’re told that point B is on the x-axis. The only way that happens is if B is at (5, 0). Point D, by the same logic, must be at (–3, 2).

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Now draw the rectangle and measure the lengths. The long ends have length 8, and the short ends have length 2.

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Therefore, the perimeter is 8 + 2 + 8 + 2 = 20.

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Start with the second equation, which tells you that t = 4.

If t = 4, then you can rewrite the first equation as follows (and solve):

4u – u = 18
3u = 18
u = 6

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Pilar is a salesperson at car company. Each car costs at least $15,000. For each car she sells, she gets 6% commission of the amount by which selling price exceeds $10,000. If Pilar sells a car at d dollars, which function gives her the commission in dollars on sale?
A) C(d)=0.06(d-10000)
B) C(d)=0.06(d-15000)
C) C(d)=0.06(10000-d)

Plugging in might help you think about this in a more concrete way. From what the question says, if Pilar sells a car for $18,000, for example, then we’d expect her to earn commission on $8,000—the amount of the car’s price above $10,000. A 6% commission on $8,000 is 0.06\times $8,000=$480. Which of the answer choices, when you plug in $18,000 for d, gives you $480?

Choice A is the only one that works.

The other way to think through this is to notice that all the choices have the same 0.06 in the beginning, so the 6% part of the problem is taken care of. Our job is to figure out which of the choices has the right thing in the parentheses. Which of those things will provide the amount that d, the selling price, exceeds $10,000? Well, translating the words into math, we’d have to say that “the amount d exceeds $10,000″ can be written as: d – 10,000.

Researchers in Australia experimented to determine if color of a coffee mug affects how people rate the flavor intensity of the coffee. Volunteers were randomly assigned to taste coffee in mugs: some white and some clear. If same type of coffee was used, researchers concluded that rating was significantly higher for those who drank coffee in clear mug. What can be concluded.
A) Color caused the difference and can be generalized to all drinkers
B)Same as A but cannot be generalized to all drinkers

Volunteers are not a random sample, so the results cannot be generalized to all coffee drinkers. There may be something different about people who would volunteer for a coffee drinking study. For example, people who would volunteer for such a study might be more likely to drink a lot of coffee and thus consider themselves able to discern subtle differences in taste.

Think of it this way: the g function is doing SOME AS-YET-UNKNOWN THINGS to (–x + 7) to turn it into (2x + 1). Of the simple mathematical operations probably at play here (addition, subtraction, multiplication, division) what could be going on?

First, the only way you go from –x to 2x is you multiply by –2. So let’s see what happens if we just multiply f(x) by –2.

–2(–x + 7) = 2x – 14

OK, so the first part’s good now, but how can we turn –14 into +1? Well, we don’t want to multiply or divide again because that would screw up the 2x we just nailed down, so why don’t we try adding 15?

2x – 14 + 15 = 2x + 1

Combine the two operations we just did (multiply by –2, add 15) and you have the g function. The function g will multiply its argument by –2, then add 15. Mathematically, we can write that like this:

g(x) = –2x + 15

Now, start from the top and make sure we’re right.

 g(f(x))
= g(–x + 7)             <– substitute (–x + 7) for f(x)
= –2(–x + 7) + 15   <– apply the g function to (–x + 7)
= 2x – 14 + 15
= 2x + 1

It works! Now all we need to do is calculate g(2).

g(2) = –2(2) + 15
g(2) = 11

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I have a question on practice question 7 in “Circles, Radians, and a Little More Trigonometry.”

I solved it a different way, but I’m not sure if I was just lucky to get the correct answer. Basically, I figured that, because one radian is when the arc and radius are the same length, that radians are like proportions. So if arc RQ were equal to 6, it would be 6/6, or one radian. So then I divided π by 6 and concluded that’s how many radians it was.

Does that actually work? Or was I just lucky?

Yes, that 100% works. Nice thinking!

Practice test 8 Calculator #13

First, you can plug in on this one, so if you feel rusty on your exponent rules at all, that’s a good move. Especially on the calculator section. Say, for example, that you plug in 4 for a. Just enter it all into your calculator (you may need to be careful with parentheses in the exponent depending on the kind of calculator you have):

    \begin{align*}4^{-\frac{1}{2}}&=x\\0.5&=x\end{align*}

Now that you know x, plug 0.5 into each answer choice to see which one gives you 4.

A) \sqrt{0.5}\approx 0.707

B) -\sqrt{0.5}\approx -0.707

C) \frac{1}{0.5^2}=4

D) -\frac{1}{0.5^2}=-4

Obviously, C must be the answer.

To solve this algebraically, first start by squaring both sides. Raising a power to a power is the same as multiplying the powers, so that’ll get rid of the 1/2 on the left:

    \begin{align*}\left(a^{-\frac{1}{2}}\right)^2&=x^2\\a^{-1}&=x^2\end{align*}

Now raise both sides to the –1 power to get a truly alone. Remember that a negative exponent is the same as 1 over the positive exponent, so you can transform the right hand side from x^{-2} to \frac{1}{x^2} to finish the problem.

    \begin{align*}\left(a^{-1}\right)^{-1}&=\left(x^2\right)^{-1}\\a&=x^{-2}\\a&=\frac{1}{x^2}\end{align*}

A triangle’s base was increased by 15%. If its area is increased by 38%, what percent was the height of the triangle increased by?

The easiest way to get this question is to plug in! Say the base and height of the original triangle are each 10. The formula for finding the area of a triangle is A=\frac{1}{2}bh, where b and h are the base and height, so the area of our original triangle is A=\frac{1}{2}(10)(10)=50.

Increasing the base by 15% brings it from 10 to 10\times 1.15=11.5. Increasing the area by 38% brings it from 50 to 50\times 1.38=69. Plug those back into the formula to solve for the new height:

    \begin{align*}69&=\frac{1}{2}11.5h\\138&=11.5h\\12&=h\end{align*}

If the original height was 10 and the new height is 12, then the height increased by 20%.

Hi Mike,

Here is a really confusing question from Applerouth’s SAT text:

a = 1.5 x + 1.50
b = 1.25x + 4.50

In the system of equations above, a and b represent the cost, in dollars, of buying x buffalo wings at two different restaurants. What amount of money will get you the same number of buffalo wings at both restaurants?

A) 12
B) 19.5
C) 20
D) 29.5

The answer is A. No idea how to do this.

You have to find this by looking for the number of wings that costs the same at both stores, so set a and b (the costs at each store) equal to each other and then solve for (the number of wings that will make each store’s cost the same).

1.5x + 1.50 = 1.25x + 4.50
0.25x = 3
x = 12

Therefore, buying 12 wings at each store costs the same amount of money. The question appears to ask HOW MUCH money, so to finish the problem you need to plug 12 back in for x in either of the equations. I’ll do the first one:

a = 1.5(12) + 1.50
a = 19.50

So the answer really should be B. The answer would be A if the question asked for the number of wings that cost the same at both stores, but that’s not what the question asks.

One other note: it might make it more clear what’s going on to graph each line. What the graph below shows is that the price (on the y-axis) is cheaper at store a for up to 12 wings, but store b becomes a better deal for 13 or more wings. At the intersection point—12 wings, $19.50—the same number of wings costs the same amount at both stores.

One way to make sure you get questions like these right is to plug in some values to see which equation makes sense. For example, you might choose to plug in 0 for h here because you know that at zero feet above sea level the boiling point should be 212° F.

Choices C and D don’t give you 212 when h = 0, so they’re definitely wrong!

Now plug in 1000 for h. We should expect the right equation to do what the question says—the boiling point should be (212 – 1.84)° F = 210.16° F. Which remaining choice, A or B, does that when you plug in 1000 for h?

Choice A gives you a crazy low number: 212 – 1.84(1000) = –1628.

Choice B does exactly what you want: 212 – (0.00184)(1000) = 210.16

So the answer is B.

To get this without plugging in, you should think about the elements of the language you’re translating into math. You want to start at 212, and subtract 1.84 degrees for every thousand feet (h/1000), so you might write this to start:

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From there, a little manipulation lands you on the right answer choice:

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My recommendation, though: plug in. With a little practice you’ll get very fast at it, and then questions like this go from head scratchers to gimmies.

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