SAT Prep: A Crate Full of Ninjas, or, Translating Words to Algebra

Kyle Hausmann is a Content Developer at Knewton, where he helps students with their SAT prep.

Translating words to algebra is hugely important on the SAT. The test contains trickily worded problems that are crafted specifically to test this skill. Fortunately, it’s something you can easily improve upon with a little bit of practice!

As you go through practice SAT math problems,  focus on phrases which signify an operation, a fraction, an equality or inequality. Is there a “half as,” “five less than,” or “six times as many?” Write out all of the expressions these phrases signify. The goal is to get everything written out so do not need to look at the wording again. You can practice and re-practice on the same problem — don’t bother solving the equations if that isn’t your problem area. Then, move onto a new problem and see if your speed in translating has improved. Here’s a bit of a long example problem to get started. If you can handle this, you’re in pretty good shape.

A large crate containing statuettes of ninjas, pirates, robots, and flying monkeys fell off a loading dock, and half of the statuettes break. Of those unbroken, one third are ninjas, three times as many are pirates as are robots, and half as many are flying monkeys as are pirates. If there are 20 unbroken robot statuettes, what was the total number of statuettes in the crate before it fell?

After reading through the problem, you can see that we are looking for the total number of statues before the crate fell. But all of the information is about the unbroken statues that are left after the fall. So, let’s call the total t, and the number of unbroken statuettes u.

The first thing we learn is that half the statuettes broke. That means half remain unbroken. So, we can write:

1/2  t = u

Next, we learn, “Of those unbroken, one third are ninjas.” We can understand the word “of” to mean multiplication, but normally there is a noun before the word “of.” We can rewrite the sentence as, “One third of those unbroken are ninjas.” This is easier to translate. Calling the number of ninjas n, we turn “of” into a multiplication sign, and “are” into an equal sign:

1/3 u = n

Next, we are told, “Of those unbroken… three times as many are pirates as robots.” We need some new variables here. Let’s call the unbroken pirate statuettes p and the unbroken robot statuettes r. Now when we see “three times as many,” we have to be careful. There are more pirates than robots, so we should write 3r = p, not 3p = r.

Then, we learn, “Of those unbroken… half as many are flying monkeys as are pirates.” We’ll call flying monkeys m. Again, we must be careful when we see “half as many.” Since there are more pirates than flying monkeys, we have to write 1/2 p = m, not 1/2 m = p.

Lastly, we learn that there are 20 unbroken robot statuettes, so we write r = 20.

And, since there are no other kinds of statuettes, we can write n + p + r + m = u.

Now we put all of our equations together to figure out how many statuettes there were all together.

1/2  t = u
1/3 u = n
3r = p
1/2 p = m
r = 20
n + p + r + m = u

Since we know r, we can plug it into 3r = p, giving us 3(20) = p, or 60 = p. Then, with p, we can plug into 1/2 p = m, giving us 1/2 (60) = m, or 30 = m.
So, now we know p, r, and m. We can plug them into our big equation, n + p + r + m = u, to get:

n + 60 + 20 + 30 = u
or,
n + 110 = u

Since we have another equation with just n and u, we should be able to solve:

n + 110 = u
1/3 u = n

If we substitute 1/3 u for n in the first equation, we get:

1/3 u + 110 = u
110 = 2/3 u
(3/2) 110 = u
165 = u

We could also solve for n, which would be a third of u, so 55. But u is what we want, because with it we can find t and answer the question:

1/2 t = u
1/2 t = 165
t = 330

***

Unrelated to solving this problem, a statue of Chuck Norris also fell off the loading dock. The ground broke and the loading dock sank down, with the statue landing on top of the loading dock again. Even statues of Chuck Norris don’t fall.

(Also, Chuck Norris knows who would win in a fight of ninjas, pirates, and robots. The answer: Chuck Norris.)