Algebra
Tackling Buncho Items
Buncho items are easy to spot, which is good news for you. Once you see an item loaded with variables, numbers, and equations/inequalities, use the following step method to tackle it:
Step 1: Look at the stem and determine which variable or term the item wants solved. Circle it.
Step 2: Look at the equation(s) and locate that same variable or term. Circle it as well. If the entire term isn’t present, isolate the variable needed to find the right answer.
Step 3: Write out the equation and start manipulating it—add, subtract, multiply, divide, square, take the square root of, whatever—until the circled variable or term is isolated on one side of the equation.
The reason we use the phrase variable or term is because the SAT often does not ask for the single numerical value of one variable. It often asks for two variables together or for one variable with some work done to it. In other words, an item won’t ask you the value of x. Instead, it will want the value of 5x. It does this to trip up students who find what x is, then look down at the answer choices and pick that value. They get the value of x correct, but they get the item wrong because it was asking for 5x.
Buncho in Slow Motion
Now that you know the three steps to tackling Buncho items, we’re going to take you through it in slow motion to demonstrate exactly how it works.
4. If , then what is the value of ?
(A) 378
(B) 243
(C) 3
(D)
(E) 1.92
Step 1: Look at the stem and determine which variable or term the item wants solved. Circle it.
In this case, you want to circle the term . That’s what the item wants, not x. Sometimes the test-makers do this solely to bust your chops. Other times, however, the unusual term (such as ) can provide a shortcut to answering the item.
Step 2: Look at the equation(s) and locate that same variable or term. If possible, circle it as well. If the entire term isn’t present, isolate the variable needed to find the right answer.
In this case, the equation does not have lurking within it. However, you should see quite clearly that the key to this item centers on finding out the value of x.
Step 3: Write out the equation and start manipulating it, until the circled variable or term is isolated on one side of the equation.
Here’s where things get interesting. You can solve for x like we do below, then determine what will be. The toughest step is realizing that . This illustrates how the new SAT is all about using exponents and other lesser known math functions to make things difficult. Now solve:
That gives you x, which you could then place into and come up with the correct answer. However, the strange term does provide a shortcut. Look at the second line of the equation above, . The term is 14 times less than , the thing we are looking for. So if we multiply both sides of the equation by 14 at that point, we find the answer we are looking for.
There’s your answer. There is no coincidence at all about the presence of this shortcut. It gives you an option if you encounter a Buncho item that does not come out and ask for a single variable. It means you can either solve for a needed variable or you can do a little tinkering with the equation and maybe find a shortcut to the correct answer. Either way, you end up with answer A.
Guided Practice
Try this one on your own:
2. In the equation , what is the value of x?
(A) –1
(B) 3
(C) 5
(D) 9
(E) 20
Step 1: Look at the stem and determine which variable or term the item wants solved. Circle it.
This item is a little more straightforward. Circling it helps you stay focused.
Step 2: Look at the equation(s) and locate that same variable or term. Circle it as well. If the entire term isn’t present, isolate the variable needed to find the right answer.
The variable is standing alone amidst a sea of numbers.
Step 3: Write out the equation and start manipulating, until the circled variable or term is isolated on one side of the equation.
If you get rid of the 3 first, you won’t have to distribute this equation.
Guided Practice: Explanation
Step 1: Look at the stem and determine which variable or term the item wants solved. Circle it.
Circling the x at the end of the stem is simple enough. It also alerts you to the fact that there’ll be no shortcut or hijinks on this item.
Step 2: Look at the equation(s) and locate that same variable or term. Circle it as well. If the entire term isn’t present, isolate the variable needed to find the right answer.
Because there’s only one variable, x, circling it in the equation should give you an idea of what you need to do to isolate it. The 3 needs to go, then the 12, and finally the 4 right next to it.
Step 3: Write out the equation and start manipulating, until the circled variable or term is isolated on one side of the equation.
Time to do some math. Write it all out so you don’t miss a step:
Fill in answer C and yell out, “Buncho!”
Independent Practice
After you complete the following item, look at the following page for the explanation.
10. Let x = 2y. For what value of z is ?
(A) x
(B) y
(C) 3
(D) 27
(E) 81
Independent Practice: Explanation
Step 1: Look at the stem and determine which variable or term the item wants solved. Circle it.
The item is asking you to solve for the value of z.
Step 2: Look at the equation(s) and locate that same variable or term. Circle it as well. If the entire term isn’t present, isolate the variable needed to find the right answer.
The z in the equation is attached to y3, so you know you’ll eventually have to get rid of the y3. But first you have to do some fancy footwork with the x and the y.
Step 3: Write out the equation and start manipulating, until the circled term is isolated on one side of the equation.
You’re dealing with three variables here, which is a lot to handle. Ideally, you want to get rid of one of those variables. We know that x = 2y, so why not change the x in our equation into a y term:
Much better. Now you’re left with just some pretty basic math:
How about that? The y3 cancels out, and you end up with 27 = z, answer D.
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