The SAT doesn’t just test math, it also tests critical thinking. The SAT’s main trick to achieve difficulty is to make questions appear more complicated than they really are. The techniques in this chapter, and in the examples throughout the Prep Guide, will give you the tools to overcome this trick.

Many of the techniques shown in this section are ways to find answers without doing all the calculations. You don’t need to solve each question. You need to “think before you act” to find the right answer. The “Aha!” moment occurs when you have found a question’s trick (that most students do not catch). Now the question is much easier to answer.

In addition to the math, you need to know various approaches and strategies to problem solving. Answering SAT questions depends as much on your reasoning as it does on your math skills.

Study your math. It’s foundational. But just studying math won’t tell you when not to use algebra.

Therefore, in addition to reviewing math, the 800score SAT Prep Course teaches you the methods necessary for getting the right answers. By gaining insight into SAT math, you will be able to solve problems more efficiently and more accurately.

The SAT assumes you can do the math, so it gives you questions that test your ability to think creatively. If you just work through SAT math problems on autopilot, you will not be working efficiently and you will not maximize your score. The techniques presented here will enable you to identify and defeat the SAT’s many tricks.

Below is an overview of the 7 techniques that will be explained in this section. They will also be used in examples throughout the 800score SAT Prep Guide.

1.Plow

2.Don’t Do That Math!

3.Backsolving

4.Plug-In

5.Ballpark

6.Experiment

7.Pattern

Just do the calculations.

Don’t be on autopilot and just start calculating.

See which answer meets all of the requirements.

Choose some numbers and test them

Look at the answers and see how exact your calculations need to be.

Use numbers to remind yourself of math rules and properties.

Make a list of values and look for repeating numbers.

Plow means you simply put marker to dry-erase board and grind out the calculations.

Sometimes you will read a problem and realize it is labor-intensive, but do-able. This is when plowing may be an effective strategy. But don’t always start calculating immediately. You should ask yourself, “Do I do the calculations, or do I choose another technique?”

You have 75 minutes to answer 37 questions, which means you have about 2 minutes per question, and you can’t use a calculator. Also, the dry erase board isn’t suited to writing out complex calculations. As a result, you simply don’t have the time or the means to do highly tedious calculations, so they should be avoided when possible.

Always look for a shortcut, using more reasoning and less calculation. The typical question is designed to have both a longer and a shorter way of getting to the answer, so be wary of spending too much time on calculations.

But when it’s the only option, you have to plow. The key is to stay focused. Make sure you keep all your numbers, variables and units straight.

The SAT is not an attempt to determine if you are a human calculator, so it doesn’t want you to waste 10 minutes doing calculations. It assumes that you can do the math and is more concerned with your critical thinking. Questions will almost always be designed with shortcuts that allow you to find an answer in a fraction of the time. A major part of preparing is honing your ability to spot these shortcuts.

## Example

123 × 341 = ?

A) 40,816

B) 41,913

C) 42,637

D) 42,755

E) 43,321

### Solution

Don’t just do the calculation. Stop and think.

Take a look at the answer choices. Multiply the last digit by the last digit. You get 123 × 341 or 3.

The SAT is a multiple-choice test, so pick the only answer choice ending in 3, which is (B).

## Example

3 × 324/162 × 4 = ?

### Solution

Always look to cancel. Use factoring.

3 × 324/162 × 4=3 × 4 × 9 × 9/3 × 6 × 9 × 4 =9/6= 1^{1}/_{2}

Instead of factoring all the way, you may notice that 324 = 162 × 2.

The SAT often provides these easy cancellation shortcuts to save you time, so you need to be able to spot them.

## Example

There are 90 students going on a field trip. A bus carries 36 students. How many buses are needed?

A) 2 B) 2.5 C) 3 D) 3.5 E) 4

### Solution

The direct calculation is 90/36 = 2.5, which is answer option (B).

But that is way too easy. Don’t just do the math. Stop and look for the trick.

The question asks for the number of buses. There is no such thing as half a bus! The correct answer is option (C).

Backsolving means inserting the answer choices into the variable(s) in the question. You can use this technique when there are variables in the question and numbers in the answer choices. (Backsolving does not apply to Data Sufficiency questions.)

By substituting the answers into the variables in the question, you can eliminate the answer choices that don’t work and skip the complicated algebra of solving for the variables. Backsolving is also an effective way to double-check your algebraic solution.

## 800score Tip:

A fundamental weakness of multiple-choice tests is that you can Backsolve questions. SAT questions are written with this in mind and are often designed so they cannot be backsolved. On the other hand, sometimes questions are designed so backsolving is the best technique. The SAT is testing your resourcefulness and expects you to have Backsolving in your toolkit.

Don’t jump to Backsolving too quickly. Try some calculations to decide if the problem is too complicated to solve algebraically. If after reading the question and the answer choices you are stumped, start plugging in answer choices.

Start at the middle answer, (C). The answer choices are usually arranged in ascending value from (A) to (E). So if (C) doesn’t work you can move on to (D)/(E) or (A)/(B) depending on whether you need a higher or lower value.

Many SAT questions have two distinct solution strategies. The 800score Prep Guide examples will often feature two approaches: calculations or algebra and Backsolve or Plug-In.

## Example

When the positive integer

xis divided by 24, the remainder is 10. Ifxis divided by 8, the remainder is 2. What is the value ofx?A) 18 B) 34 C) 40 D) 49 E) 57

### Solution

There are variables in the question and numbers in the answer choices, so this is a good place to backsolve.

We can eliminate answer choice (C) because 40 divided by 24 has a remainder of 16, not 10. Moving to choice (B), we see that 34 divided by 24 does have a remainder of 10. Now check the second restriction. When 34 is divided by 8, it gives a remainder of 2. Choice (B) meets both restrictions, so it’s the correct answer. Backsolving turned a hard algebra question into an easy arithmetic question.

Plug-in means you choose your own numbers to insert into a question. This is useful when there are variables in both the question and the answer choices.

Translating the variables into numbers will also help you understand the question. Play with numbers to see if you notice any patterns or restrictions.

Make sure the numbers you choose fit the question’s parameters, and pick a variety of numbers to make sure you cover all reasonable possibilities.

Try positive and negative numbers, and zero. For example, you could plug in -2, 0 and 2.

If the question doesn’t specifically require “integers,” make sure to try some positive and negative fractions (a common trick on the SAT).

Many SAT questions have two distinct solution strategies. The 800score Prep Guide examples will often feature two approaches: calculations or algebra and Backsolve or Plug-In.

#### When to Use Plug-In

Use plug-in for questions that have a small and finite set of numbers to check.

How many primes between 11 and 30 satisfy this statement?

Start substituting a few prime numbers. This works well on small sets of numbers because you can test a few or even all the options without spending too much time.

Plug-in also works well in true/false questions, such as: Is *ab* > 0? For a question like this, it is logical to test the conditions using a variety of numbers.

Plug-in can be used very effectively in Data Sufficiency questions to test if the statements are sufficient. You should use plug-in regularly on Data Sufficiency questions, just as you should use Backsolve on multiple-choice questions.

## Example

If n is an even integer, which of the following must be an odd integer?

A) 3

n– 2 B) 3(n+ 1) C)n– 2 D)n/3 E)n/2

### Solution

This question has variables in the question and in the answer choices, so plug-in is the best technique. Try *n* = 2.

Check (A). If *n* = 2, then 3*n* – 2 = 4, which is not odd.

Check (B). If *n* = 2, then 3(*n* + 1) = 9. Since the target is an odd integer, this answer choice works.

You can try another even number and/or use the rest of the answer options to double-check. For example, *n* = 2 works with choice (E) to make an odd number, but (E) is even with any other even value for n.

Note: When plugging in, if you get a result that works for two or more answer choices, you must plug in another number. Use a number that is fundamentally different from the initial number, such as larger, positive or negative, zero or a fraction.

## Example

If

x>y, then when isx^{2}>y?I.

x< 0

II.x> 0

III.x> 1A) I only B) II only C) III only D) I and II E) I and III

### Solution

Quickly reading the question, it seems like *x*^{2} > *y* is always true. So there must be a trick. Plug-in is the best technique.

Check statement (I). Both *x* and *y* will be negative. Since any negative number squared is positive, positive *x*^{2} will be greater than any value of the negative *y*. So statement (I) is true.

Before checking statements (II) and (III), notice how similar they are. The only difference is that statement (II) includes fractional values between 0 and 1.

Check statement (II).

Let *x* = 4 and *y* = 3. Checking *x*^{2} > *y*, 16 > 3. So (II) seems to be true.

Now try a fraction. Let *x* = 1/2 and *y* = 1/3. Checking *x*^{2} > *y*, but 1/4 < 1/3, so statement (II) is false.

In checking statement (II) we checked numbers greater than 1, so we also checked statement (III) and know it is true.

Statements (I) and (III) are true, so the correct answer is option (E).

**How to avoid complex algebra by Picking Numbers**

Video Courtesy of **Kaplan SAT prep.**

Ballpark means estimating rather than doing the full calculation. Looking at the answers before doing any calculations will tell you how exact your calculations need to be and whether you can use ballpark.

**When to Use Ballpark**

- When the answer choices are spread over a wide range. You can use your rough estimate to eliminate answer choices that are out of range.
- Multiplying and dividing large numbers or dealing with fractions. Use rounding to get an estimate.
- On Data Sufficiency questions, to gauge whether a statement is sufficient.
- To double-check calculations. Make sure your solution is in the ballpark of your estimate.

## 800score Secret:

Do you know how the geometry questions with drawings always say “not drawn to scale”? Actually they pretty much are drawn to scale. This means that you can double-check your geometry answers by seeing if they “fit” the geometry picture. It also means you can ballpark geometry questions by looking at the drawing. If a drawing looks like an equilateral triangle (three equal sides), and you came up with three sides of 3, 12, and 14, you need to double-check your math!

Deliberately making a drawing not look like the correct answer would just be going too far… even for the SAT.

## Example

What is 36,568 divided by 12,985?

A) 2.26 B) 2.816 C) 3.08 D) 4.23 E) 5.65

### Solution

Round the numbers.

For 36,568, use 36 or 37.

For 12,985, use 13.

36 / 12 = 3 so 36 / 13 is just under 3.

37 / 13 is also just under 3.

So the most reasonable answer is option (B) 2.816.

## Example

Which is greater: 24/51 or 26/49?

### Solution

It helps when dealing with fractions to simplify to a basic fraction like 1/4, 1/2 and 3/4.

24 is less than half of 50, so 24/51 is less than 1/2. 26 is greater than half of 50, so 26/49 is greater than 1/2.

So 26/49 is greater than 24/51.

## Example

If 0.303

z= 2,727, thenz=A) 0.9 B) 9 C) 90 D) 900 E) 9,000

### Solution

The answer choices are far apart, so use ballpark.

Round the decimal: 0.303 is close to 1/3, so 1/3 of *z* ≈ 2,727, and therefore *z* ≈ 3 × 2,727.

What answer could possibly be correct? You don’t even have to do the math. The answer must be 9,000. There are no other answers even in the thousands.

The correct answer is option (E).

Note: Don’t make the mistake of 1/3 of *z* = 2,727, so *z* ≈ 2,727 / 3. This common error is another SAT trick.

Under the intense pressure of test day, you can expect to forget some basic math rules. If this happens, you can take a moment to experiment with easy numbers in order to recall the rule.

Be careful — experiments can be very time consuming, so only use this technique if you are doing well on time.

## Example

Solve: 45

^{25}/ 45^{12}= ?

### Solution

You have forgotten the rule about division and exponents. Experiment using a base of 2.

2^{4} / 2^{2} = 16 / 4 = 4 = 2^{2}

Look at the exponents: 4 – 2 = 2

2^{6} / 2^{3} = 64 / 8 = 8 = 2^{3}

Look at the exponents: 6 – 3 = 3

So you subtract exponents when you divide.

45^{25} / 45^{12} = 45^{25 – 12} = 45^{13}

## Example

Solve:√144 + 225

### Solution

You don’t remember if splitting the square root works.

Is √144 + 225 = √144 + √225 ?

Experiment.

Is √4 + 9 = √4 + √9 ?

You can see that √13 ≠ 2 + 3.

So to simplify you need to add, then factor.

√144 + 225 = √369 = √3 × 3 × 41 = 3√41

A pattern is a repetition of numbers that results from repeating calculations. Sometimes you can find patterns in questions that otherwise seem impossible to solve.

ExampleWhat is the units digit of 3

^{801}?

### Solution

You know you don’t have time to calculate. This is a reasoning question. There must be a shortcut.

Try to find a pattern starting with smaller exponents. The pattern shows up pretty quickly.

3^{1} =3

3^{2} =9

3^{3} = 27

3^{4} = 3 × 27 = 81

3^{5} = 3 × 81 = 243

36^{6} = 3 × 243 = 729

37^{7} = 3 × 729 = 2187

38^{8} = 3 × 2187 = 6561

The units digit repeats after “4 powers.” The pattern is clearest for exponents that are multiples of 4.

Every exponent that is a multiple of 4 has a units digit of 1. Since 800 is a multiple of 4, the units digit of 3800 is 1. Looking at the pattern, the units digit of 3801 will be 3.

## 800score Tip:

When a question makes you scratch your head and think it will take an hour to solve, chances are there is a shortcut. Using Backsolve, Plug-In, Find Patterns or math shortcuts, you will likely find the easy way to solve it.

ExampleWhat is the sum of integers from 1 to 100, inclusive?

### Solution

Look for a pattern starting with smaller groups of numbers.

1 + 2 + 3 + 4 = 1 + 2 + 3 + 4 = 5 + 5 = 10

1 + 2 + 3 + 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5 + 6 = 7 + 7 + 7 = 21

First trick: Add first and last.

Second trick: How many pairs are there for an even number of integers?

You can check the pattern by doing one more batch of numbers.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = (1 + 10)(10/2) = (11)(5) = 55

So the sum of the integers from 1 to 100 is (1 + 100) (100/2) = (50)(101) = 5,050