Civil service exam prep!!!

Numerator

The whole number that appears on the top of a fraction.

?/D

Denominator

The number that appears on the bottom of the fraction.

N/?

Proper fraction

The numerator is smaller than the denominator.

Ex. 1/2

Improper fraction

The numerator is larger than the denominator.

Ex. 2/1

Mixed numbers

A mixed number consists of a whole number and a proper fraction. A improper fraction can be written into a mixed number.

How do you convert an improper fraction into a mixed number?

1. Divide the numerator by the denominator. The result will be a whole number, with or without a remainder.

2. Write the whole number.

3. If there is a remainder, write a fraction with the remainder in the numerator and the original denominator in the denominator.

Ex. Convert 17/5 to a mixed number?

17/5=3 with a remainder of 2

17/5=3 2/5

How do you convert a mixed number into a improper fraction?

1. Multiply the whole number in the mixed number by the denominator of the fraction.

2. Add the result to the numerator of the fraction.

Ex. Convert 5 1/6 to an improper fraction.

5x6= 30

30+1= 31

5 1/6= 31/6

When should you convert mixed numbers into improper fractions?

Whenever you are asked to multiply or divide mixed numbers

How do you reduce a fraction?

You reduce a fraction by dividing both the numerator and the denominator by a single value that divides evenly into both of them.

Ex. Reduce 5/10?

Both 5 and 10 are divisible by 5

5/5=1

10/5=2

5/10= 1/2

Key words for equal?

Is, are, has, was, were, had

Key words for addition?

Sum, together, more, total, greater, or older than

Key word for multiplication?

Product, times, of

Keywords of division?

Per, evenly

Key words of subtraction?

Difference, less than, fewer, or younger than, remain, left over

example of key words for subtraction. Jacob has 5 fewer than Leslie?

Would be Leslie-5=Jacob

Distance formula problems include key words such as?

speed, plane, train, boat, car, walk, run, climb, and swim

Distance Formula

Rate x Time = Distance

Key word in the distance formula provide 2 of the 3 elements

rate, time, distance

Plug those 2 elements into the formula and solve for the third

Rate and time have to be measured into common units

If the rate is measured in miles per hour, the time has to be measured in hours, not minutes or days

2 ways to solve word problems

1. Translate the problem into an algebraic equation and then solve for the missing information

2. Work backwards by plugging in one of the answer choices

A neat trick to see if a number is divisible by 3.

Add all of its digits. If the sum of the digits is divisible by 3, the number itself is divisible by 3.

Ex. Is 132 divisible by 3?

1+3+2=6 6/3=2

Which means that 132 is divisible by 3. This method also works for 9.

Raising fraction to higher terms

It allows you to rewrite a fraction with a larger numerator and denominator

When would you usually have to raise a fraction to higher terms

you often have to raise one or more fractions to higher terms in order to add or subtract them

How to add fractions that have common denominators

simply just add the numerators and write the result over the denominator. Do not add the denominators

Ex. 3/12+6/12= 9/12

How to add mixed numbers

add the whole numbers together, and then add the numerators together and place them over the denominators. Do not add the denominators.

Ex. 3 3/12 + 6 3/12= 9 6/12 Reduced equals 9 1/2

If the sum of the 2 fractions happen to be an improper fraction

You must convert the improper fraction into a mixed number

4/7 + 6/7 = 10/7 so converted into a mixed number it would be 1 3/7

when can you not add fraction?

You cannot add fractions unless it has a common denominator

how to find a common denominator

Look at the denominators of all the fractions in the problem. Do they all divide evenly into the largest denominator? If they do that number is your common denominator. If they don't, run through the multiplication table of the largest denominator until you find a number that all the other denominators divide into evenly.

Ex. 2/3 + 1/5 =

2x5=10 3x5=15 = 10/15

1x3=3 5x3=15 = 3/15

Now you have found the common denominator add it.

10/15 + 3/15 = 13/15

How to subtract fractions

subtracting fractions is just like adding them. when fractions share common denominators, simply subtract the numerators. Do not subtract the denominators.

Ex. 7/8 - 4/8 = 3/8

How to subtract mixed numbers

Subtract the whole number from the whole number and the fraction from the fraction

Ex. 5 3/4 - 1 2/4 = 4 1/4

When will have to make extra steps while subtracting fractions

If the fraction in the second mixed number is larger than the fraction in the first mixed number, you will have to borrow in order to rewrite the first number.

Ex. 12 1/5 - 7 3/5=

You can't subtract 3/5 from 1/5 because it would be a negative number. in the first fraction you'll need to borrow from the whole number to make it into an improper fraction.

11 6/5 - 7 3/5= 4 3/5

Since the first fraction is now bigger you are able to subtract. Remember do not subtract the denominator.

If the fraction that you're subtracting does not share common denominators

Its like adding fractions you must find the common denominator

Ex. 3/4 - 1/6 =

The common denominator is 12:

3x3= 9 4x3= 12 = 9/12

1x2= 2 6x2= 12 = 2/12

9/12 - 2/12 = 7/12

Remember do not subtract the denominators when they're common

Multiplying fractions

Multiplying fraction is much easier than adding or subtracting. all you have to do is multiply the numerators and the denominators.

Ex. 2/3 x 5/7 =

2x5= 15 3x7= 21= 15/21

If you can reduce go ahead and reduce

3/7

How to multiply mixed numbers?

1. Change any mixed numbers to an improper fraction.

2. Then multiply as normal.

Ex. 4 2/3 x 5 1/2=

4 2/3= 14/3

5 1/2= 11/2

14/3 x 11/2= 154/6 = 25 2/3

Ex. 8,400 fans attended a college football game 3/4 of the fans rooted for the home team. the rest rooted for the visiting team. How many fans rooted for the home team?

To solve, multiply 8400 by 3/4

8400x3= 25,200

1x4= 4

25,200/4

A fraction is one way of writing a division problem. When you see a fraction with a numerator larger than the denominator, you can divide (that's also how we convert improper fractions to mixed fractions).

25,200/4= 6,100

There was 6,100 fans rooting for the home team

How do you divide fractions?

Dividing fractions is just like multiplying fractions, but with one extra step. In order to divide 2 fractions, use the reciprocal of the second fraction and turn the division sign into a multiplication sign.

Ex. 1/2 / 3/5?

you would take the reciprocal of 3/5 which would be 5/3.

which would make the answer, 5/6.

How to divide a fraction by a whole number?

First change the whole number to a fraction by putting the whole number in the numerator of a fraction with a denominator of 1.

Ex. 3/5 / 2?

3/5 / 2/1 = 3/5 x 1/2= 3/10

How to divide mixed numbers

Rewrite them as improper fractions

Ex. 3 1/5 / 1 1/2?

3 1/5= 16/5

1 1/2= 3/2

take the reciprocal of 3/2 which would be 2/3

16/5 x 2/3 = 32/15 = 2 2/15

If the problem ask for an answer in the form of a mixed number, use the conversion technique to convert.

0.1

1 tenth, 1/10

0.01

1 hundredth, 1/100

0.001

1 thousandth, 1/1000

0.0001

1 tenth thousandth, 1/10000

can zero be be added to the right of the decimal without it changing the value

yes, .01 is the same value as .01000. this is helpful when adding, subtracting or dividing by decimals.

Mixed decimal

A decimal with numbers on both sides of the decimal point

Ex. 32.30

A decimal with numbers to the right of the decimal point only Is called

A decimal not a mixed decimal

how to write a fraction into a decimal

divide the numerator by the denominator

Ex. Write 3/4 as a decimal

3/4= .75

How to change a decimal into a fraction

write the digits of the decimal as the numerator of a fraction and write the decimals name as the denominator of the fraction. reduce if possible.

Ex. rewrite 0.018 as a fraction?

1. write 18 as the numerator of the fraction

2 write the name of the fraction in the denominator. 0.018 is eighteen one-thousandth, so you should write 1,000 in the denominator of the fraction.

3. Reduce. Both 18 and 1000 are divisible by 2, so 18/1000 can be reduced.

18/1000= 9/500

1/5 converted to a decimal?

0.2

1/4 converted to a decimal?

0.25

1/3 converted to a decimal?

0.33

2/5 converted to a decimal?

0.4

1/2 converted to a decimal?

0.5

3/5 converted to a decimal?

0.6

2/3 converted to a decimal?

0.66

3/4 converted to a decimal?

0.75

What is the bar over the last digit in some decimals?

It indicates that the decimal repeats infinitely. decimals are not the most precise way to represent fractions whose denominators are not factors of a power of 10 ( powers include 10; 100; 1000; 10000, etc.)

Comparing decimals

Ex. Compare0.08, and 0.1? when compared 0.1 is the largest between the 2

Percentages

The word percent comes from the Latin words meaning "for every one hundred." A percent expresses the relationship between a part and a whole in terms of 100. When we say that 10% of the customers at the grocery store pay with cash, we are saying that 10 out 100 customers are paying with cash. you encounter percentages every day. Sales tax, interest, discounts, and tips, on meals are just a few of the percent calculations people make on a daily basis.

Changing a percent to a fraction and vice versa

A percent can easily be written as a fraction. Simply take the percent and put it in the numerator of the fraction. Then, write 100 in the denominator.

Ex. Convert 75% to a fraction?

75/100

How to write a fraction as a percent

Multiply the fraction by 100/1

Ex. Convert 1/4 to a percent?

1/4 x 100/1= 25 which means it converts to 25%

another way to convert a fraction to a decimal

Divide the numerator by the denominator, and then move the decimal point 2 places to the right in your quotient.

10% converted to a fraction?

1/10

20% converted to a fraction?

1/5

25% converted to a fraction?

1/4

30% converted to a fraction?

3/10

33 1/3% converted to a fraction?

1/3

40% converted to a fraction?

2/5

50% converted to a fraction?

1/2

60% converted to a fraction?

3/5

66 2/3% converted to a fraction?

2/3

80% converted to a fraction?

4/5

100% converted to a fraction?

1/1 = 1

Changing a percent to a decimal

Its the same way as converting a decimal to percent, but instead of moving the decimal point to the right 2 places you move the % sign to the left 2 places and change the sign to a decimal.

Solving for % word problems generally take one of three forms

1. find the percent of a whole

2. find what percent one number is of another number

3. find the whole when the percent of a number is given

Ex. for find the percent as a whole

what is 30% of 40?

translates to 30/100 x 40=

30/100 x 40/1= 1200/100 = 12

Ex. for finding what percent one number is of another

12 is what percent of 40?

translates to 12 = x/100 x 40

12 = x/100 x 40/1

12= 40x/100

Times both sides by 100

12 x 100 = 40x/100 x 100/1

1200 = 40x

Divide both sides 40

1200/40 = 40/40

X = 30

Ex. finding the whole when the % of a number is given

12 is 30% of what number?

translate to 12 = 30/100 x y

12 = 30/100 x y/1

12 = 30y/100

Times both sides by 100

12 x 100 = 30y/100 x 100

1200 = 30y

Divide both sides by 30

1200/30 = 30y/30

y = 40

Techniques for questions that ask about percent increase and percent decrease

Remember that after you calculate the percent, you must add (increase) or subtract (decrease) in order to determine the final result.

Ex. A merchant usually sells hats for $20. He decides to have a sale during which he decreases the price of the hats by 25%. What are the price of the hats during the sale?

25/100 x 20

25/100 x 20/1

500/100

= 5

25% 0f $20 is $5. The merchant plans to decrease the price of the hat by 25%, so he plans to decrease the price by $5. That means the sale price of the hats will be $20 - $5, which equals $15

The most common questions in reading comprehension.

1. Describe the main idea the passage

2. Identify a specific fact or detail in the passage

3. Draw an inference or conclusion based on the information in the passage

4. Define a vocabulary word that appears in the passage

The main idea of a paragraph or passage?

Is what the paragraph or passage is mostly about

When can an answer to a main idea question not be the right one?

When the answer choice only applies to one detail or a small portion of the paragraph or passage.

Topic sentence

The sentence that expresses the main idea. Sometimes the answer to a main idea question is stated clearly in the passage, frequently found in the first or last sentence.

Other times that the main idea is not stated in the topic sentence

it is usually implied by the overall content of the passage. In such circumstances you will need to deduce the main idea from the passage.

The main idea of the passage describes

the entire passage or paragraph. if it only describes one prat or one detail, it is not the correct answer.

Just because an answer refers to an entire passage doesn't mean that it is correct

Although such answers are usually correct, occasionally an incorrect answer will refer to the entire passage or paragraph.

What makes an answer that describes a passage or paragraph incorrect in a main idea question

It describes the content in the passage or paragraph inaccurately. Most often, it states a position stronger than the one the writer takes.

Make sure that the answer to the main idea accurately describes?

The content of the passage or paragraph

What do detail question ask you to identify

They ask you to identify a specific piece of information from the passage

Detail questions usually contain one of the following phrases

According to the passage...

The passage states....

The correct answer to a detail question

It will be a direct quote or a paraphrase of information in the passage

What cannot be the correct answer on a detail question

Any answer stating information that does not appear in the passage

Why are not all answers that repeat information from the passage are necessarily correct

The answer must also be relevant to the question being asked. Some incorrect answers will contain irrelevant information from the passage.

Steps to answer a detail question.

First read the question to determine its subject. Next, determine where in the passage. the subject of the question is discussed. reread the portion of the passage before answering the passage.

What do inference questions ask

They ask you to find information in the paragraph, and then use it to draw an inference that is necessarily, or deductively, true.

Inference questions often begin with one of these phrases

It can be inferred from the passage...

The passage suggest...

Which of the following conclusions is best supported..

When answering an inference question

Do not choose an answer that requires you to make a major assumption or to use outside knowledge. Look for answers that are fully supported by information in the passage.

Questions that are designed to test vocabulary

Are really trying to measure how well you can figure out the meaning of a unfamiliar word from its context

Context refers to

the words and ideas surrounding the vocabulary word

If the context is clear enough in a vocabulary question

you should be able to substitute a nonsense word for the one being sought and still be able to find the correct answer, because you will be able to determine meaning strictly from the sense of the sentence.

The tested word itself may contain a context clue

Look for familiar prefixes and suffixes. Look to see if the word shares a common root with some other word you know