Posts filed under: Q and A

In 200 ml of 30% saline, there’s (0.3)(200) = 60 ml of saline.

60 ml is 40% of what?

60 = (0.4)x
150 = x

So the saline will be 40% when enough water evaporates that there’s 150 ml of solution. Therefore, 50 ml of the original 200 ml must evaporate.

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A tank initially contains 7 gallons of water. A faucet is opened and water begins pouring into the tank at a rate of 1.5 gallons per minute untill the tank is full. Which of the following represents the volume v of water , in gallons, in the tank as a function of time t , in minutes, that has elapsed since the faucet was opened?
A) V(t) = 1.5+ t
B) V(t) = 8.5+ t
C) V(t) = 1.5t
D) V(t) = 1.5t -7
E) V(t) = 1.5t +7

See a previously posted solution for that one here.

Let’s work backwards, and use hours instead of distance for ease (assume the plane travels at a constant speed for our purposes).

Let’s say when she woke up, she had 1 hour left in her flight. That’s half of the time she was sleeping so she must’ve slept for 2 hours.

She first fell asleep halfway through the flight, so she was awake for the first 3 hours, then slept for 2, then was awake for the last 1.

She was asleep for 2/6 = 1/3 of the trip.

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Rearrange that equation:

x + 2y = y
x = –y
x/y = –1

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Note that the SAT doesn’t test properties of even and odd like this, although the old SAT (pre-2016) used to.

The product of two numbers will be even if and only if one or both of the numbers being multiplied is even. Therefore, you know any card showing an odd number must have an even number on the other side.

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A semicircle is exactly half a circle, so take the formula for circumference (C = 2πr) and divide by 2. Since r = 2, you end up with (2π(2))/2 = 2π. That covers the curved part.

You know the straight part has a length of 4 because the radius of the semicircle is 2.

4 + 2π is your answer.

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d = 30 + 2(40 – s)

The machine begins the day with $30 inside, so that’s the “30 +” part. Easy enough.

The variable s is defined as how many sodas the machine has in it, but what we really care about is how many sodas are sold. We know the machine begins the day with 40, so 40 – s should give us the number of sodas sold. (When s = 40, no sodas have been sold; when s = 35, 5 sodas have been sold…)

For each soda that’s sold, the machine should have $2 more, so that’s why “2(40 – s)” is in there.

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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.