Math for 14 year olds

    Most of the math on the GRE is set to what would be called junior high school level in the United States. It involves simple algebra and geometry. The irony is that most students could have done better on the GRE Math section ten years earlier when the material was fresher in their minds. To help you, we have math chapters dedicated to Algebra, Percentages/Ratios, Geometry, and Arithmetic.

    Test writers must only use pre-high school level math because if they used higher level math, it would create an advantage for math majors. The result is that there is no calculus or trigonometry on the exam. For test writers this creates a challenge, "How do you create questions using math for 14 year olds that Harvard grads will get wrong?" The answer is that they primarily use:
1) trick questions
2) extremely difficult Word Problems
3) questions that require extensive deductive reasoning (Geometry and Data Sufficiency)

     Word Problems are the focus of the test, and for that reason, the Math section is more of a reading test than strictly a "math" test (for those British/Commonwealth students "maths" is "math" in the U.S.). Test writers devise extremely complex and contorted Word Problems that can fool 95% of GRE students using only basic math principles. That is why we have a chapter dedicated exclusively to Word Problems strategy. Foreign students should pay particular attention to this section because Word Problems will give them the most trouble. We focus on Word Problems in chapter 7.

    Note: these chapters are particularly dense with material. Take them slowly and digest the information thoroughly before progressing to the next chapter.


Scrap Paper

     We discussed scrap paper earlier in the guide. Doing calculations in your head may increase errors, particularly under the pressure of test day. This is no-win situation because copying questions also invites errors. Because you cannot write on the screen, every question has to be partially recopied onto scrap paper. Students should always practice with unlimited scrap paper (just like on test day) in order to get used to this awkward process of recopying questions.

This chapter is broken into two parts:
I.   General Math Strategies
II.   Basics

I. General Math Strategies

A. Backsolving
B. Plug-In
C. Possible Range Strategy

A. Backsolving (otherwise known as "reverse-solving")

     This is the most important strategy for the math section! Backsolving involves inserting answer choices to solve problems. This is often preferable to trying to translate a complicated question into an algebraic equation. Inserting answer choices into a Word Problem can also make a complicated question more understandable. Backsolving helps you eliminate or choose a sample answer. Remember, math questions on the GRE use very simple principles, so the test writers have to make the questions as complicated and intimidating as possible. Some questions are actually written so that backsolving is the only effective way to solve the problem.     

How to Backsolve:

1. Decide if the problem is too complicated to solve algebraically (this should take only a few seconds).

2. Insert the middle answer--the one that would be in the middle of potential answers if it were on a number line.

3. If a smaller number would work, choose answer choices 1 or 2; if a larger number would work, choose 4 or 5. (The answer choices are usually arranged from lowest to highest value--answer choices 1 through 5).

4. Eliminate down to one answer and choose.

Try backsolving on this question:

When the positive integer Z is divided by 24, the remainder is 10. What is the remainder when Z is divided by 8?

a) 1
b) 2
c) 3
d) 4
e) 5

Solution
Notice that this question seems to defy a quick algebraic solution. The best way to address this problem is to backsolve, to take potential answers and feed them into the question until one answer works. Pick a number for Z such that Z/24 has a remainder of 10; that is, a number 10 greater than a multiple of 24. 24 is a multiple of 24, so let Z = 24 + 10 = 34. When 34 is divided by 8 you get 34/8 = 4 with a remainder of 2. Not satisfied? Try 58 (which is 2 × 24 with a remainder of 10). 58 divided by 8 gives a remainder of 2. It appears that (B) is the correct answer.

Strategy: Plug-in is an even more effective strategy for double checking your answers. When you arrive at an answer, plug in the answer to test that it works or plug in in more numbers. Remember that you have nearly two minutes to do each question on the GRE CAT. This gives you plenty of time to double check yourself; however, pacing is still extremely important, as you read in Chapter One.

B. Plug-In
     Sometimes the best approach to backsolving is not to put in sample answer choices, but to insert numbers that prove/disprove the question. The numbers you choose for backsolving should fit the question's parameters. For example, if the question asks for an integer, you should insert integers. Usually try plugging in a few different numbers (positive, negative, etc.).

1. Decide if the problem is too complicated to solve algebraically (this should take a only few seconds).

2. Insert sample numbers into the equation. Depending on the question, try negative numbers, positive numbers, odd/even and fractions.

3. Eliminate down to one answer and choose.

Plug-in Example

If n is an even integer, which of the following must be an odd integer?
a) 3n - 2
b) 3(n + 1)
c) n - 2
d) n/3
e) n/2

Solution
(B) Every time you have variables in the answer choices, you should plug in. Make n equal to 2. If n is 2, then 3(n + 1) = 9. Since our target is an odd integer, this answer choice works. Try a few more numbers to double check. For example, 2 may work with choice (e) to make an odd number (1), but it will not work with any other even numbers.

C. Ballpark Strategy (otherwise known as the possible range strategy)

This strategy allows you to answer questions quicker than doing the calculations, and it is an effective tool to double-check your answers. Using the Ballpark Strategy, you find an answer by what could reasonably be in the range of the answer. This is particularly useful when the possible answers are scattered over a large range.

Try the Ballpark Strategy here:

If 0.303z = 2,727, then z =

a)9,000

b)900

c)90

d)9

e)0.9

Solution
(A) Because the answer choices are so far apart, you can ballpark this problem. Think about it: .303 is close to 1/3. 1/3 of z = 2,727, then what answer could possibly be correct? You don't even have to do the math. 2,727 is about 1/3 of 9,000; therefore, the answer must be 9,000, according to the Ballpark Strategy (note that there are no other answers even in the 9,000 range. Or, you could multiply both sides by 1000 to eliminate the decimal points, then divide 2,727,000 by 303 and get the same answer.

Strategy: Ballparking is an effective strategy for double checking your answers. When you arrive at an answer, make sure it is in the ballpark of what the answer could be.

II. The Basics   

    This chapter contains a basic review of math concepts. Most students should skim through this section. It is vital that you know all the terminology in this section since the questions on the test will assume you know it.

A. Integers
B. Positive / Negative Rules
C. Fractions
D. Equivalent Fractions
E. Multiplying and Dividing Fractions
F. Adding and Subtracting Fractions
G. Decimals
H. Adding and Subtracting Decimals
I.  Multiplying and Dividing Decimals
J. Averages and Medians


A. Integers

A positive number is a number greater than zero, such as + 5 (usually written as 5).

A negative number is a number less than zero, such as -5.

A whole number is a number that does not include a decimal part or a fractional part. It is also called an integer.

The absolute value of a number, written as | -5 |, is the magnitude of the number; the absolute value of + 5 is equal to the absolute value of -5, written as |+5 | = |-5 | = 5.

Zero is an integer

B. Positive / Negative Rules

What happens when positive and negative numbers are added/subtracted/multiplied?

Multiplication/Division
Positive × Positive = Positive, for example 2 × 2 = 4
Positive × Negative = Negative, for example 2 × -2 = -4
Negative × Negative = Positive, for example -2 × -2 = 4

Addition/Subtraction
Adding a negative number is the same as subtraction.
      4 + (-5) = 4 - 5 = -1

Subtracting a negative number is the same as addition.
      4 - (-5)  = 4 + 5 = 9

An even number is an integer that is divisible by 2 (4, 6, 8, 10).

An odd number is an integer not divisible by 2 (3, 5, 7, 9).

A prime number is a positive integer that has exactly two different positive divisors, 1 and itself. For example, 2, 3, 5, 7. The number 1 is not a prime number since it has only one positive divisor.

Consecutive numbers are a set of numbers in which each member of the set is the successor of its predecessor.

Examples:
even consecutive numbers: 4, 6, 8, 10
odd consecutive numbers: 3, 5, 7, 9
prime consecutive numbers: 3, 5, 7, 11, 13.

Adding, Subtracting and Multiplying odd/even: The following is true of even and odd whole numbers (use example numbers in your mind to illustrate):

Even + Even = Even	exg. 4 + 4 = 8 (even)
Odd + Even = Odd,	exg. 3 + 4 = 7(odd)
Odd + Odd = Even	exg. 3 + 3 = 6 (even
Even × Even = Even	exg. 2 × 2 = 4 (even)
Even × Odd = Even	exg. 2 × 3 = 6 (even)
Odd × Odd = Odd	exg. 3 × 3 = 9 (odd)
Even - Even = Even	exg 16 - 8 = 8 (even)
Even - Odd = Odd	exg. 16 - 5 = 11 (odd)
Odd - Odd = Even	exg. 9 - 5 = 4 (even)

Even/Odd Example
If k is an odd integer, state whether each of the following is odd or even:
a) k + k + k
b) k × k × k
c) k + 2k
d) 2k × k

Solution
a) (k + k) is even. Thus (k + k) + k is an even plus an odd, which is odd.
b) k × k is odd. Thus (k × k) × k is an odd times an odd, which is odd.
c) k + 2k is an odd plus an even, which is odd.
d) 2k is even. An even times an odd is even.

Strategy 1: If you do not remember the rules, merely plug in by selecting an odd number (e.g., 3) or an even number and perform the required operation.

Strategy 2: Get used to plugging in numbers. On most math questions, you will have to either plug in strategy or backsolve.

A factor is an integer that divides another number resulting in a whole number. Consequently, a number can be expressed as a multiple of each of its factors. The number 24 has factors 1, 2, 3, 4, 6, 8, 12, 24. Note that 24 is a multiple of any one of its factors, i.e., 24 is a multiple of 8.

The least common multiple (LCM) of several numbers is the smallest integer which is a common multiple of the several numbers.

LCM example

Write the LCM (Least Common Multiple) of 6 and 12.

Solution
The factor 2 × 3 is used only once because it occurs in both numbers. Thus, the LCM is (2 × 3 × 2) = 12. Or, what is the smallest number that both 6 and 12 go into? 12. 12 is the lowest number divisible by both 6 and 12.

EXAMPLES

Example 1

Compute each of the following:
a) 2 - 3
b) 4 + 6 - 10
c) (-5 + 2)(-3)

Solution
a) 2 - 3 = -1
b) 4 + 6 - 10 = 10 - 10 = 0
c) (-5 + 2)(-3) = (-3)(-3) = 9

Example 2

State all the factors of 63.

Solution
The number 63 can be divided by 1, 3, 7, 9, 21, and 63. Hence, these are its factors.

 

Example 3

Write 63 as a product of prime numbers.

Solution
The number 63 is obviously not divisible by 2, but is divisible by 3. Hence,

63 = 3 × 21 = 3 × 3 × 7

Both 3 and 7 are prime numbers. We do not include 1 as a prime number.



Example 4

Find the LCM (Least Common Multiple) of the three numbers 20, 30 and 50.

Solution
Write each number as a product of prime numbers:

20 = 2 × 2 × 5

30 = 2 × 3 × 5

50 = 2 × 5 × 5

The factor 2 × 5 occurs in all three numbers; thus it is used only once.

The LCM is LCM = (2 × 5) × 2 × 3 × 5 = 300

 

C. Fractions

A fraction is one number divided by another number. It is division, such as 3/5.

The numeratoris the top number, and the bottom number is the denominator. The denominator represents the number of equal parts into which an entity has been divided; the numerator represents the number of parts that are selected. For example, if a garden is divided into 5 equal plots, 3 of the plots is 3/5 of the garden.

A fraction that has 0 as its denominator (e.g., 5/0) is infinitely large and undefined (not a real number or an integer). If the numerator is zero, then the fraction equals zero (e.g. 0/5 = 0). If the fraction has a numerator equal to the denominator (e.g., 5/5), the fraction is equal to 1.

A mixed number is a number that is an integer plus a fraction. The number 4 2/3 is the integer 4 plus the fraction 2/3. Any mixed number can be written as a fraction, and any fraction greater than 1 can be written as a mixed number. To express 4 2/3 as a fraction we multiply 4 × 3, add the numerator to this product, 12 + 2 = 14, and divide by the denominator: 4 2/3 = 14/3. To convert the fraction 17/5 into a mixed number, divide by the denominator (17 divided by 5 is 3 with 2 remaining), and add the remainder over the denominator: 17/5 =  3 2/5.

Example 5

Convert 4 5/7 into a fraction.

Solution
We multiply 4 × 7 and obtain 28. Add 5 to this and obtain 33. Put this over 7, and we find 4 5/7 = 33/7

Example 6

Convert 79/9 into a mixed number.

Solution
Divide 79 by 9 and obtain 8 with 7 remaining. Now add to the 8 the fraction 7/9 (8 7/9).

 

D. Equivalent Fractions

     A fraction that has a common factor in both numerator and denominator is equal to the fraction with the common factor canceled. The fraction 6/10 is equivalent to the fraction 3/5 since the common factor 2 occurs in both numerator and denominator of 6/10. In fact, the following fractions are all equivalent: 3/5 = 6/10 = 9/15 = 12/20. A fraction that has no common factors in the numerator and denominator is said to be expressed in lowest terms.

    A fraction with a negative numerator or denominator is equivalent to a negative fraction, that is, -3/5 = 3/-5 = - (3/5). If both numerator and denominator are negative, the fraction is positive, that is, -3/-5 = 3/5.

Example 7

Express 26/16 as a mixed number in lowest terms.

Solution
The mixed number is found by dividing by 16 giving 1 with 10 remaining. Hence

26/16 = 1 10/16 = 1 5/8

E. Multiplying and Dividing Fractions

    To multiply fractions, cancel out any common factors that appear in both numerators and denominators. Then multiply all numerators to form one numerator and all denominators to form one denominator. This final fraction may then be written as a mixed number, if desired. To divide fractions, say (x/y)/(a/b), invert the divisor (the fraction a/b) and multiply the two fractions, i.e., (x/y)/(a/b) = (x/y) × (b/a).

For instance,

(5/6) × (7/2) = (7× 5)/(6 × 2) = 35/12

 

F. Adding and Subtracting Fractions

     To subtract one fraction from another, we simply add a negative fraction to a second fraction. Consequently, the rules for adding and subtracting are the same. The first step is to write the fractions such that each fraction has the same denominator. Then add or subtract the numerators. Then simplify.

1/3 + 1/4 = 4/12 + 3/12 = 7/12

     To write all fractions with the same denominator, a quick choice is to multiply all denominators together. Hoever, this may give a rather large denominator. To avoid a large denominator, we could find the least common denominator (LCD); it is the least common multiple (LCM) of all the denominators.

G. Decimals

      A decimal fraction is a fraction whose denominator is a power of 10 but not a factor of the numerator. For example, the fraction 3/100 is written in decimal form as 0.03; the fraction 223/10,000 is 0.0223. Note that the number to the right of the decimal point is the numerator of the decimal fraction, and the denominator is 10 raised to the power n where n is the number of places to move the decimal point. To write 0.0031 as a decimal fraction, 31 is the numerator, and because we moved the decimal point 4 places, the denominator is 10 = 10,000; consequently, the decimal fraction is 31/10,000.

Rule #1: If you multiply a decimal by 10, you would move the decimal 1 to the right; 100, move 2 to the right; 1000, move 3 to the right, and so on.

For instance, 100,000 × 0.0054 = 540.

You see that there are 5 zeros in 100,000; therefore, we moved the decimal 5 places to the right.

Rule # 2: When you divide by 10, you move the decimal once to the left; 100, move 2 to the left; 1000, move 3 to the left.

Example 8

Write 31.2/(100,000) as a decimal.

Solution
When we divide by 100,000, which is 10, we move the decimal point 5 places to the left. Then 31.2 = 0.000312.

A mixed decimal is the sum of an integer and a decimal fraction, much like the mixed fraction. The integer 5 added to the decimal fraction 35/100 is 5 + 35/100 = 5 + 0.35 = 5.35. The mixed decimal, or a decimal fraction, is usually simply called a decimal. (Note: the leading 0 in 0.35 is simply convention; it does not have mathematical significance: 0.35 = .35.)

Example 9

Write the fraction 649/100 as a decimal.

Solution
The numerator 649 can be written as the decimal "649.0". When we divide by 100, we move the decimal point 2 places to the left so that

649/100 = 6.49

It is not necessary to write any zeros after the 9; that is, 6.490 is equivalent to 6.49.

Example 10

Express 0.075 as a fraction in lowest terms.

Solution
The decimal is expressed as a fraction .075 = 75/1000 The denominator and numerator are factored resulting in 75/1000 = (25 × 3) / (25 × 40) = 3/40

H. Adding and Subtracting Decimals

     To add or subtract decimals, we write the decimals in a column with the decimal points aligned vertically. First, combine the decimals with a plus sign. Next, combine the decimals with a minus sign. Then subtract the sum of the negative decimals from the sum of the positive decimals. In performing these tasks, we add zeros to the right of the decimal point so that each number has an entry in each column. For example, if we subtract 3.021 from 5, we write 5 as 5.000 so that there is an entry in each of the 3 places to the right of the decimal in both numbers.

Example 11

Add 5 + 2.783 + 3.04.

Solution
Write the decimals in a column, with zeros added if none exist:

   5.000
   2.783
+ 3.040
 10.823

Example 12

Compute 6.98 + 3.217 + 3 - 3.637

Solution
Add the positive decimals and the negative decimals:

   6.980
   3.217
+ 3.000
13.1973

Subtract the sum of the negative decimals from the sum of the positive decimals:

 13.197
-  3.637
   9.560

I. Multiplying and Dividing Decimals

     Multiply two decimals just like you would multiply two integers. The number of decimal places in the product is then equal to the total of the decimal places in the two decimals.

     To divide two decimals, move the decimal point in the divisor (the number doing the dividing) to the right so that the divisor is an integer. Move the decimal point in the dividend to the right the same number of places. Now perform the division, placing the decimal point in the answer directly above the decimal point in the dividend.

Example 13

Compute 0.05 × 12.

Solution
Set up the multiplication as though the decimals were integers:

12 × .05 = .60

The answer must have a total of 2 decimal places: .60.

J. Averages and Medians

     The average, or arithmetic mean, is the sum of a set of numbers divided by the total number of elements in the set. Arithmetic mean is used on the GRE because it is a more precise term than average.

     If all the numbers in a set are arranged in ascending or descending order, the middle number is the median. The median is different from the arithmetic mean. Half of the people in a country earn more than the median income, and half earn less. The average income does not split the people into a top half and bottom half. For example, if 5 people have weekly incomes of $200, $300, $500, $13000 and $6000, the median is $500, but the average is $4000. If a large number of people earn very little and a few earn a huge amount, the average would be quite impressive, but the median would be surprisingly low.

Example 14

Ten students on an exam scored 20, 30, 30, 25, 30, 35, 80, 60, 40, and 90. Calculate the average and the median.

Solution
The average is the sum of all the numbers divided by 10: average = (20 + 25 + 3 × 30 + 35 + 40 + 60 + 80 + 90) / 10 = 44. The median is, for the case of an even number of entries, the average of the two middle numbers when arranged in order: median = (30+35)/2 = 65/2 = 32.5

 

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