Note that OC = 5 and OD = 5 because both of those are also radii. Note also that because chord CD is perpendicular to OB, it’s bisected by OB. In other words, it’s split into 2 segments each measuring 4.

Things are really coming together! Because we know our Pythagorean triples when we see them, we know that we’ve got 3-4-5 triangles here. Therefore, the sine of angle OCD is 3/5.

You know point C must be on the x-axis, and that AB = BC. Since B is on the y-axis, it turns out all we need to do here is reflect point A across the y-axis. Point C must be at (8, 0), so x = 8.

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.

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.

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.

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.

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.

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:

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.