Taking Vital Signs is a critical part of nursing and appears on many nursing and nursing assistant, nursing aide certification exams.

Nursing Assistant/Nursing Aide – CNA, NNAAP, NCMA
Medical Assistant – RMA, CMA, NCMA
First Responder, EMT

John, a 78 year old man with Dementia has been assigned to you. He fell out of his bed and sprained his wrist. He is diabetic and suffers from left-sided weakness due to a stroke. He requires total care and assistance with everything he does. He has dentures. He can no longer walk alone, feed himself, bathe or dress himself, and he is incontinent of urine and stool. His vital signs are to be monitored q 4 hrs.

1. The taking of John’s vital signs includes

1. Temperature, blood pressure, respirations. and pulse
2. blood pressure, respirations, pulse and ROM
3. temperature, I&O, respirations, pulse and blood pressure
4. All of the above

2. When taking John’s blood pressure, you should make certain that

1. The cuff is the correct size
2. John is lying on his left side
3. The cuff is positioned dependent to his elbow
4. The cuff is pumped to at least 20 mm above his baseline b/p

3. The most accurate temperature is obtained when taken

1. Under the arm
2. Rectally
3. At the groin
4. In the ear

4. When counting the pulse rate, you may use the pulse at what points?

1. The carotid artery
2. The radial artery
3. The apical area of the heart via the chest using a stethoscope
4. All of the above

5. While assessing John’s temperature, you note that it is 101.2 degrees. What is your FIRST action?

1. Notify your supervisor
2. Call the doctor
3. Wait the appropriate amount of time and take the temperature again
4. Have a co-worker check your work.

6. You can count respirations while

1. Taking John’s b/p
2. Counting John’s pulse
3. Taking his temperature
4. None of the above

7. When assessing John’s respirations you should

1. Count the number of times his chest rises and falls in one minute
2. Auscultate his chest
3. Observe whether John is breathing easily or seems to be having difficulty
4. All of the above

8. If John’s 3 year old granddaughter was visiting and was watching you  take her grandfather’s vital signs and became curious about how you would take hers, you might explain to her that

1. You would do it on the arm but with a smaller cuff
2. You would do it on her thigh but with a smaller cuff
3. You would do it on her ankle using a smaller cuff
4. All of the above.

Answer Key with Explanations

1. A
The four components of the vital sign assessment are temperature, blood pressure, pulse, and respirations.

2. A
If the cuff is not the proper width for your patient you will get a false reading on your blood pressure check.

3. B
The rectal temperature is taken within the body cavity and therefore yields the temperature which is closest to the actual body temperature.

4. D
The pulse can accurately be counted using any of these areas, although for most initial assessments the radial pulse is used.

5. C
Always double-check your vital signs to make certain there is a valid problem.

6. C
While the thermometer is registering John’s temperature, it is a good time to count his respirations. This is an example of good time management.

7. D
The respiratory assessment should ALWAYS include the number of respirations, the quality of them and accompanying breath sounds.

8. D
A child of this age could have her b/p assessed on any of the above mentioned areas using a child’s cuff.

Sample Biology Test Questions

Biology Questions are common on these exams:

High School – AHSGE, GHSGT, CRCT, BC Provincial,
Nursing Entrance  - HESI, PAX RN, TEAS
Medical Technicials – MLT, Histotechnology, PCAT, OAT, DAT

1.    A _______ _______ is the sequence of developmental stages through which members of a given species must pass.

1. Life cycle
2. Life expectancy
3. Life sequence
4. None of the above

2.    Life _________ are the ___________ and ___________ activities that all ________ systems must be able to carry out in order to maintain life.

1. Life sequences are the chemical and biological activities that all living systems must be able to carry out in order to maintain life.
2. Life expectancies are the biochemical and biophysical activities that all sentient systems must be able to carry out in order to maintain life.
3. Life cycles are the organic and inorganic activities that all living systems must be able to carry out in order to maintain life.
4. Life functions are the biochemical and biophysical activities that all living systems must be able to carry out in order to maintain life.

3.    Nutrition is the sum total of activities through which a living organism obtains food; what are the three processes included in nutrition?

1. Ingestion, digestion, and adsorptionIngestion, diffusion, and assimilation
2. Ingestion, digestion, and assimilation
3. Incorporation, digestion, and assimilation

4.    __________ is the taking in of food, __________ refers to the chemical changes that take place in the body, and ___________ involves the changing of certain nutrients into the protoplasm of cells.

1. Assimilation is the taking in of food, digestion refers to the chemical changes that take place in the body, and ingestion involves the changing of certain nutrients into the protoplasm of cells.
2. Ingestion is the taking in of food, digestion refers to the chemical changes that take place in the body, and assimilation involves the changing of certain nutrients into the protoplasm of cells.
3. Digestion is the taking in of food, ingestion refers to the chemical changes that take place in the body, and assimilation involves the changing of certain nutrients into the protoplasm of cells.
4. Ingestion is the taking in of food, digestion refers to the chemical changes that take place in the body, and diffusion involves the changing of certain nutrients into the protoplasm of cells.

5.    The movement of molecules other than water from an area of ____ concentration to an area of _______ concentration is ________.

1. The movement of molecules other than water from an area of high concentration to an area of less concentration is diffusion.
2. The movement of molecules other than water from an area of less concentration to an area of high concentration is diffusion.
3. The movement of molecules other than water from an area of high concentration to an area of less concentration is osmosis.
4. The movement of molecules other than water from an area of lesser concentration to an area of less concentration is dispersal.

6.    During _________, a solvent moves through a/an __________ membrane from an area with a_______ concentration of solvents to areas of ________ concentration.

1. During diffusion, a solvent moves through a semipermeable membrane from an area with a lesser concentration of solvents to areas of greater concentration.
2. During osmosis, a solvent moves through an impermeable membrane from an area with a lesser concentration of solvents to areas of greater concentration.
3. During osmosis, a solvent moves through a semipermeable membrane from an area with a greater concentration of solvents to areas of lesser concentration.
4. During osmosis, a solvent moves through a semipermeable membrane from an area with a lesser concentration of solvents to areas of greater concentration.

7.    _________ and ________ are forms of _______ transport by which materials pass through plasma membranes.

1. Diffusion and osmosis are forms of active transport by which materials pass through plasma membranes.
2. Diffusion and osmosis are forms of passive transport by which materials pass through plasma membranes.
3. Dispersal and osmosis are forms of passive transport by which materials pass through plasma membranes.
4. Diffusion and synthesis are forms of active transport by which materials pass through plasma membranes.

8.    The scientific discipline that studies the physiological aspects, structures, life cycles and division of cells is called __________.

1. The scientific discipline that studies the physiological aspects, structures, life cycles and division of cells is called physiology.
2. The scientific discipline that studies the physiological aspects, structures, life cycles and division of cells is called cell science.
3. The scientific discipline that studies the physiological aspects, structures, life cycles and division of cells is called biochemistry.
4. The scientific discipline that studies the physiological aspects, structures, life cycles and division of cells is called cell biology.

9. Which, if any, of the following statements about mitosis are correct?

1. Mitosis is the process of cell division by which identical daughter cells are produced.
2. Following mitosis, new cells contain less DNA than did the original cells.
3. During mitosis, the chromosome number is doubled.
4. A and C are correct.

10.  Which, if any, of the following statements about meiosis are correct?

1. During meiosis, the number of chromosomes in the cell are halved.
2. Meiosis only occurs in eukaryotic cells.
3. Meiosis is the part of the life cycle that involves sexual reproduction.
4. All of these statements are correct.

Answer Key  

1. A
A life cycle is the sequence of developmental stages through which members of a given species must pass.

2. D
Life functions are the biochemical and biophysical activities that all living systems must be able to carry out in order to maintain life.

3. C
The three processes included in nutrition are, ingestion, digestion, and assimilation.

4. C
Digestion is the taking in of food, ingestion refers to the chemical changes that take         place in the body, and assimilation involves the changing of certain nutrients into the protoplasm of cells.

5. A
The movement of molecules other than water from an area of high   concentration to an         area of less concentration is diffusion.

6. D
During osmosis, a solvent moves through a/an semipermeable membrane from an area with a lesser concentration of solvents to areas of greater concentration.

7.  B
Diffusion and osmosis are forms of passive transport by which materials pass   through plasma membranes.

8. D
The scientific discipline that studies the physiological aspects, structures, life cycles and         division of cells is called cell biology.

9. A and C are correct.
a)  Mitosis is the process of cell division by which identical daughter cells are produced.
c)  During mitosis, the chromosome number is doubled.

10. D
All of these statements are correct.

Basic Science Practice Test Questions

High School Exams
Nursing Entrance Exams – TEAS, PAX RN, DET,
College – CBASE, CPAT, GED, CHSPE
Medical – PCAT,
Armed Services – AFQT 

1.    A __________ is a naturally occurring assemblage of plants and animals that occupy a common environment.

1. Society
2. Biosphere
3. Community
4. Population

2.    Because of the ban on hunting wolves, their population increased exponentially; within a few generations, their habitat had exceeded its __________ __________.

1. Carrying capacity
2. Food source
3. Population limit
4. Supply capability

3.    Name the four states in which matter exists.

1. Concrete, liquid, gas, and plasma
2. Solid, fluid, gas, and plasma
3. Solid, liquid, vapor, and plasma
4. Solid, liquid, gas, and plasma

4. In the periodic table of the elements, elements are arranged in order of their atomic  ______, which is the number of _______ found in their nucleus.

1. Mass, protons
2. Number, neutrons
3. Mass, neutrons
4. Number, protons

5.  A ______ ______ is a graphical description of feeding relationships among species in an ecological community.

1. Food web
2. Food chain
3. Food network
4. Food sequence

6. The depletion in the Ozone layer is caused by _____?

1. Chlorofluorocarbons.
2. Nitrous oxide.
3. Carbon dioxide.
4. Methane.

7. Which of the following is not an important characteristic of the Green Revolution?

1. Mechanized agriculture
2. Hybrid seeds
3. Monoculture
4. Slash and burn

Answer Key

1. C  Community
Communities are usually named after a dominant feature such as characteristic plant species, e.g., pine.

2. A Carrying capacity
An area’s carrying capacity is the maximum number of animals of a given species that area can support during the harshest part of the year.

3. A
The four states in which matter exists are solid, fluid, gas, and plasma.
The state of matter is determined by the strength of the bonds between the atoms that make up matter.

4.  D
In the periodic table of the elements, elements are arranged in order of their atomic  number, which is the number of protons found in their nucleus.

5.  A
A food web is a graphical description of feeding relationships among species in an ecological community.

Note:  A food web differs from a food chain in that the latter shows only a portion of the food web involving a simple, linear series of species (e.g., predator, herbivore, plant) connected by feeding links. A food web aims to depict a more complete picture of the feeding relationships, and can be considered a bundle of many interconnected food chains occurring within the community.

6. D

There is widespread concern that the ozone layer is deteriorating due to the release of pollution containing the chemicals chlorine and bromine. Such deterioration allows large amounts of ultraviolet B rays to reach Earth which can cause skin cancer and cataracts in humans and harm animals as well.[i]

7. A 

Chlorofluorocarbons (CFCs) chemicals found mainly in spray aerosols heavily used by industrialized nations for much of the past 50 years are the primary culprits in ozone layer breakdown. When CFCs reach the upper atmosphere they are exposed to ultraviolet rays which causes them to break down into substances that include chlorine. The chlorine reacts with the oxygen atoms in ozone and rips apart the ozone molecule.[ii]

Anatomy and Physiology questions are common on many Nursing Exams.

Nursing Entrance - HESI, PSB, TEAS,
Medical Assistants – RMA, CMA  
Nursing – IBCLC
Technicians – CBET, CRES, CLES,
Medical – COMLEX

 Anatomy and Physiology Practice Questions

1) Anatomy breaks the human abdomen down into segments called _______________.

1. Regions
2. Districts
3. Quadrants
4. Areas

2) The quadrant that is largely responsible for digestion is _______________.

1. Left Upper
2. Right Upper
3. Right Left
4. Left Lower

3) The body organ that is NOT located within the Right Upper Quadrant is _______________.

1. Liver
2. Gall Bladder
3. Duodenum
4. Sigmoid colon

4) The organ that IS located in the Right Lower Quadrant is _______________.

1. Appendix
2. Heart
3. Left lung
4. Trachea

5) One reason that medical professionals should know the names and locations of the Quadrant  is _______________.

1. To keep the patient’s condition a secret from him.
2. To communicate about patients’ conditions with other doctors and medical  professionals.
3. For insurance purposes.
4. Not knowing the quadrants almost always results in death for the patient.

6) The stomach and colon are both in the _______________.

1. Left Upper Quadrant
2. Right Upper Quadrant
3. Right Lower Quadrant
4. Left Lower Quadrant

7) Commonly used abbreviations for the Quadrants of the abdomen are _______________.

A. QUR, QUL, QLR, QLL
B. ABC, DEF, GHI, JKL
C. RUQ, LUQ, RLQ, LLQ
D. RR, LL, QQ, RQ

The intestines are located in _______________.

1. LUQ
2. LLQ
3. RLQ
4. All of the above

Answer Key

1.  C
2.  A
The Left upper quadrant of the abdomen, is often abbreviated as LUQ, contains the stomach, spleen, left lobe of the liver, body of the pancreas, left kidney and adrenal gland.

3. D
The right upper quadrant of the abdomen, often abbreviated as RUQ, contains the liver, gall bladder, duodenum and had of the pancreas.

4.  A
The Right lower quadrant of the human abdomen, often abbreviated as RLQ, contains the appendix and ascending colon.

5.  B
Medical personnel divide the abdomen into smaller regions to facilitate study and discussion.

6.  A
The stomach and colon are both in the Left Upper Quadrant, together with, liver, spleen, left kidney, pancreas and large intestine.

7. C
The commonly used abbreviations for the Quadrants are, Right Upper Quadrant, RUQ, Left Upper Quadrant, LUQ, Right Lower Quadrant, RLQ, Left Lower Quadrant, LLQ.

8. D 
All of the above.  The Large Intestine passes through all of the quadrants.

Some High School Exams, particularly the HSPT have logic problems which are a type of syllogism.   Other types of High School Entrance exams have a type of logic question called Verbal Reasoning.    A third type of logic related question is Arithmetic logic, which are word problems.

Here are logic practice test questions for both types, HSPT Logic Syllogism style and ISEE, SSAT, SHSAT style Verbal Reasoning style.  

HSPT Logic Practice Test Questions

1.  The Silver fish can swim faster than the black fish.   The gold fish can swim faster than the black fish.  The gold fish can swim faster than the silver fish.  If the first 2 statements are true, then the third statement is:

2.  All rabbits have fur.  Some rabbits are pets.  Some pets have fur. If the first 2 statements are true, then the third statement is:

3. Deciduous trees drop their leaves in the fall.  Confers keep their leaves all year round.  Conifers are deciduous.  If the first 2 statements are true, then the third statement is:

4.  No homework is fun. Reading is homework.  Readingis not fun. If the first 2 statements are true, then the third statement is:

5. All informative things are useful things.    Some websites are not useful things.  Some websites are not informative. . If the first 2 statements are true, then the third statement is:

Answer Key 

1. Uncertain

2.  True

3. False

4. False

5.  True

See Wikipedia for a big tutorial on logic syllogisms 

High School Placement Test Study Guide and Practice Tests

Verbal Logic Practice Questions (ISEE, SSAT, SHSAT style)
1. Authority

1. Power
2. Rule
3. Knowledge
4. Age

2. Competition

1. Struggle
2. Team
3. Prize
4. Test

3. Experience

1. Facts
2. State
3. Event
4. Participation

4. Exchange

1. Gift
2. Reciprocation
3. Business
4. Transfer

 Answer Key 

1. A    An authority figure is someone who has the power to make rules, persuade, or influence people such as a government official, teacher, or parent.

2. D   A competition is a test of skills or abilities between two or more individuals or teams. The winner of a competition is one who can display better abilities in a certain skill or sport.

 3. D   An experience is the result of the observation or direct participation in a situation by an individual.

 4. B  An exchange is the reciprocation of gifts, ideas, or other goods; or the act of replacing one thing with another.

Independent Schools Entrance Test Study Guide & Practice Test 

Specialized High Schools Admissions Study Guide and Practice Tests

Secondary School Admissions Test (SSAT) Study Guide & Practice Test

Analogies are test questions where a pair of words are given, and you are asked to choose another pair with the same relationship.  For more help answering Analogies, or Verbal Analogies as they are sometimes called, see our Analogies Tutorial.

Analogies are a frequent item on Catholic High School tests, such as the HSPT, TACHS and COOP, as well as Private School Entrance Tests such as the ISEE and SHSAT.

Practice Analogy Questions

1.  Nest : Bird 

1. Cave : bear
2. flower : petal
3. window : house
4. dog : basket

2.  Teacher : School

1. Businessman : Money
2. Waitress : Coffee shop
3. Dentist : Tooth
4. Fish :  Water

3.  pebble : boulder

1. pond : ocean
2. river: rapids
3. fish : elephant
4. feather : bird

4. Poodle : Dog

1. shark : great white
2. dalmation : great dane
3. money : stock market
4. horse : pony

5.  fox : chicken

1. rat : mouse
2. cat : mouse
3. dog : cat
4. rabbit : hen

6.  lawyer : trial

1. plumber : pipe
2. businessman : secretary
3. doctor : operation
4. hairdresser : blow dryer

Answer Key
1. A

This is a Functional relationship.   A Bird lives in a nest, the way way a bear lives in a cave.

2 B

This is a functional relationship.  A teacher works in a school in the same way a Waitress works in a coffee shop.

3. A

This is a Degree relationship.   A boulder is a very large pebble – both are rocks, in the same way an ocean is a very large pond – both are very bodies of water.

4. A

This is a type relationship.  A poodle is a type of dog in the same way a great white is a type of shark.

5. B

This is a predator/prey relationship.  Foxes eat chickens in the sam way as cats eat mice.

6. C

This is a functional relationship. A lawyer defends a client in a trial in the same way a doctor performs an operation on a patient.

Wikipedia has a section on Analogies which is interesting but a little technical.
ThinkQuest has quite an extensive section on analogies with analogy practice questions and full commentary.
Math.com has a good section with practice analogy questions and commentary.

Word Problems are common on most High School exams and College Entrance Exams.  Generally, word problems can be classified into 12 types.  Below are examples of each type with a complete solution.

1.  Age

A girl is 10 years older than her brother. By next year, she will be twice the age of her brother. What are their ages now?

1. 25, 15
2. 19, 9
3. 21, 11
4. 29, 19

Solution: B

We will assume that the girl’s age is “a” and her brother’s is “b”. This means that based on the information in the first sentence,

a = 10 + b

Next year, she will be twice her brother’s age, which gives

a + 1 = 2(b+1)

We need to solve for one unknown factor and then use the answer to solve for the other. To do this we substitute the value of “a” from the first equation into the second equation. This gives

10+b + 1 = 2b + 2

11 + b = 2b + 2

11 – 2 = 2b – b

b= 9

9 = b this means that her brother is 9 years old.

Solving for the girl’s age in the first equation gives

a = 10 + 9

a = 19 the girl is aged 19. So the girl is aged 19 and the boy is 9

2.  Distance or speed 

Two boats travel down a river towards the same destination, starting at the same time. One of the boats is traveling at 52km/hr, and the other boat at 43km/hr. How far apart will they be after 40 minutes?

1. 46.67km
2. 19.23km
3. 20.4km
4. 14.39km

Solution 2:      C

After 40 minutes, the first boat will have traveled = 52km/hr x 40 minutes/60 minutes = 34.7km

After 40 minutes, the second boat will have traveled = 43km/hr x 40/60 minutes = 14.3km

Difference between the two boats will be 34.7km – 14.3km = 20.4km

3.  Ratio

The instructions in a cook book states that 700 grams of flour must be mixed in 100ml of water, and 0.90 grams of salt added. A cook however has just 325 grams of flour. What is the quantity of water and salt that he should use?

a) 0.41 grams and 46.4ml

b) 0.45 grams and 49.3ml

c) 0.39 grams and 39.8ml

d) 0.25 grams and 40.1ml

Solution 3: A

The Cookbook states 700grams of flour, but the cook only has 325. The first step is to determine the percentage of flour he has

325/700 x 100 = 46.4%

That means that 46.4% of all other items must also be used.

46.4% of 100 = 46.4ml of water

46.4% of 0.90 = 0.41 grams of salt

 
4. Percent

An agent received $6,685 as his commission for selling a property. If his commission was 13% of the selling price, how much was the property sold for?

1. $68,825
2. $121,850
3. $49,025
4. $51,423

Solution 4:      D

Let’s assume that the property price is x

That means from the information given, 13% of x = 6,685

Solve for x,

x = 6685 x 100/13 = $51,423

5. Sales & Profit

A store owner buys merchandise for $21,045. He transports them for $3,905 and pays his staff $1,450 to stock the merchandise on his shelves. If he does not incur further costs how much does he need to sell the items to make $5,000 profit?

a) $32,500

b) $29,350

c) $32,400

d) $31,400

Solution 5: D

Total cost of the items is $21,045 + $3,905 + $1,450 = $26,400

Total cost is now $26,400 + $5000 profit = $31,400

6. Tax/Income

A woman earns $42,000 per month and pays 5% tax on her monthly income. If the Government increases her monthly taxes by $1,500, what is her income after tax?

1. $38,400
2. $36,050
3. $40,500
4. $39, 500

Solution 6:      A

Initial tax on income was 5/100 x 42,000 = $2,100

$1,500 was added to the tax to give $2,100 + 1,500 = $3,600

Income after tax left is $42,000 – $3,600 = $38,400

7. Interest

A man invests $3000 in a 2-year term deposit that pays 3% interest per year.  How much will he have at the end of the 2 year term?

a) $5,200

b) $5,020

c) $5,110

d) $7,000

e) $6,400

Solution 7:

This is a compound interest problem.  The funds are invested for 2 years and interest is paid yearly, so in the second year, he will earn interest on the interest paid in the first year.

3% interest in the first year = 3/100 x 3,000 = $90

At end of first year, total amount = 3,000 + 90 = $3,090

Second year = 3/100 x 3,090 = 92.7.

At end of second year, total amount = $3090 + $92.7 = $3,182.7

 
8. Averaging

The average weight of 10 books is 54 grams. 2 more books were added and the average weight became 55.4. If one of the 2 new books added weighed 62.8 g, what is the weight of the other?

1. 44.7g
2. 67.4g
3. 62g
4. 58g

Solution 8:      C

Total weight of 10 books with average 54 grams will be=10×54=540g

Total weight of 12 books with average 55.4 will be=55.4×12=664.8g

So total weight of the remaining 2 will be= 664.8 – 540 = 124.8g

If one weighs 62.8, the weight of the other will be= 124.8g – 62.8g = 62g

9. Probability

A bag contains 15 marbles of various colors. If 3 marbles are white, 5 are red and the rest are black, what is the probability of randomly picking out a black marble from the bag?

a) 7/15

b) 3/15

c) 1/5

d) 4/15

Solution 9:  A

Total marbles = 15

Number of black marbles = 15 – (3 + 5) = 7

Probability of picking out a black marble = 7/15

10. 2 Variables 

A company paid a total of $2850 to book for 6 single rooms and 4 double rooms in a hotel for one night. Another company paid $3185 to book for 13 single rooms for one night in the same hotel. What is the cost for single and double rooms in that hotel?

a) single= $250 and double  = $345

b) single= $254 and double = $350

c) single = $245 and double = $305

d) single = $245 and double = $345

Solution 10: D

We can determine the price of single rooms from the information given of the second company. 13 single rooms = 3185.

One single room = 3185 / 13 = 245

The first company paid for 6 single rooms at $245. 245 x 6 = $1470

Total amount paid for 4 double rooms by first company = $2850 – $1470 = $1380

Cost per double room = 1380 / 4 = $345

11. Geometry

The length of a rectangle is 5 in. more than its width. The perimeter of the rectangle is 26 in. What is the width and length of the rectangle?

a) width = 6 inches, Length = 9 inches

b) width = 4 inches, Length = 9 inches

c) width =4 inches, Length = 5 inches

d) width = 6 inches, Length = 11 inches

Solution 11: B

Formula for perimeter of a rectangle is 2(L + W)

p=26, so 2(L+W) = p

The length is 5 inches more than the width, so

2(w+5) + 2w = 26

2w + 10 + 2w = 26

2w + 2w = 26 – 10

4w = 18

W = 16/4 = 4inches

L is 5 inches more than w, so L = 5 + 4 = 9inches.

12. Totals and fractions

A basket contains 125 oranges, mangos and apples. If 3/5 of the fruits in the basket are mangos and only 2/5 of the mangos are ripe, how many ripe mangos are there in the basket?

1. 30
2. 68
3. 55
4. 47

Solution 12:   A

Number of mangos in the basket is 3/5 x 125 = 75

Number of ripe mangos = 2/5 x 75 = 30

Number series questions appear on most High School exams and on the HSPT.  An example is:

Consider the following series: 26, 21, , 11, 6.  What is the missing number?

a. 27
b. 23
c. 16
d. 29

Looking carefully at the sequence, we can see right away that each number is 5 less than the previous number, so the missing number is 16.

We can re-write this sequence in mathematical notation as, a₁, a₂, a₃,… a_n, where n is an integer and a_n is called its n^th term.  And we can write the sequence in the form of a formula, where an integer is substituted in the place of the variable in the formula and the terms are obtained.

For example, let us consider the sequence 5,10,15,20,…

• Here, a_n = 5n. The formula a_n = 5n.
• The n^th term of a sequence can be found by plugging n in the explicit formula for the sequence.  So for example if we wanted to find the 100^th number in this sequence, we would substitute n=100 in the formula and get 500.

Types of Number Sequence problems

1. Simple addition or subtraction – each number in the sequence is obtained by adding a number to the previous number.

For example,   2, 5, 8, 11, 14

Each number in the sequence is obtained by adding 3 to the previous number, which we could write as, a_n+1 = a_n + 3.

2. Simple multiplication – each number in the sequence is obtained by multiplying the previous number by a whole number or fraction.

For example,   3, 6, 18, 54

Or,

20, 10, 5, 2.5

Each number in the first sequence is obtained by multiplying the previous number by 3, which we could write as, a_n+1 = a_n X 3.

In the second example, each number in the series is the previous number divided by 2, or multiplied by ½, or a_n+1 = a_n X 1/2.

3. Prime Numbers – each number in the sequence is a prime number.

For example,

23, , 31, 37

Answer:  29

4. Operations on the previous two numbers

For example,

8, 14, 22, 36, 58

Here the sequence is created by adding the previous 2 numbers.

5. Exponents

The number sequence is created by each number squared or cubed.

For example,

3, 9, 81, 6561,  where each number is squared.

6. Combining Sequences

2, 7, 13, 20, 28, 37

Here the sequence starts with 2, and each element is added to another sequence starting with 5.  So, 2 + 5 = 7, 7 + 6 = 13, 13 + 7 = 20 and so on.

A variation is a sequence with a repeating element.  For example,

1, 2, 3, 5, 7, 9, 12, 15

Here the sequence is, for each n, +1, +1, +1, +2, +2, +2, +3, +3,

7. Fractions

For example,

16/4, 4/2, 2/2, ½, ?

Fractions are often meant to confuse.  If fractions don`t have an obvious relationship, reduce them to lowest terms or to whole numbers.   Reducing these to whole numbers, gives,

4, 2, 1, ½

Right away we can see the numbers are half the previous number, so the next in the series is ¼.

In this example, the answer is a fraction, however, you may have to reduce fractions to see the relation, and then convert back to get the answer in the correct form.

Strategy for Answering Number Series Questions

Answering number series questions is a skill of recognizing patterns, and the  best way to improve is to familiarize yourself with the different types, and to practice.

Here is a quick method that will help you answer number series.

For example:

2, 5, 6, 7, 8, ?

Step 1 – glance at the series quickly and see if you can spot the pattern right away.

Step 2 – Start analyzing.

Take the different between the first 2 numbers and the different between the second 2 numbers.

2,  (+3)  5, (+1)   6, (+1)  7, (+1)   8,

No clear patter with a simple analysis.  There is no addition, subtraction, multiplication, division, fractional or exponent relationship.

The relation must be a higher order or a second series.

Next look at the relation between the 1^st number and the 2^nd and the 1^st and the 3^rd.     We see that,

1^st + 3 = 5,  1^st + 4 = 6.  That’s it!   The number 2 is added to the sequence, 3, 4, 5, 6, so the next number will be 2 + 7 = 9.

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Most students find math word problems difficult.   Word problems are found on must High School Exams, College Exams and a few Nursing Entrance exams like the PAX.  Tackling word problems is much easier if you have a systematic approach which we outline below.

Here is the biggest tip for studying word problems.

Practice regularly and systematically.   Sounds simple and easy right?   Yes it is, and yes it really does work.

Word problems are a way of thinking and require you to translate a real word problem into mathematical terms.

Some math instructors go so far as to say that learning how to think mathematically is the main reason for teaching word problems.

So what do we mean by Practice Regularly and Systematically?   Studying word problems and math in general requires a logical and mathematical frame of mind.   The only way you can get this is by practicing regularly, which means everyday.

It is critical that you practice word problems everyday for the 5 days before the exam as a bare minimum.

If you practice and miss a day, you have lot lost the mathematical frame of mind and the benefit of your previous practice is pretty much gone.   Anyone who has done any amount of math will agree – you have to practice everyday.

Everything is important

The other critical point about word problems is that all of the information given in the problem is has some purpose.  There is no unnecessary information! Word problems are typically around 50 words in 1 to 3 sentences.  If the sometimes complicated relationships are to be explained in that short an explanation, every word has to count.   Make sure that you use every piece of information.

Here are 9 simple steps to help you resolve word problems.

Step 1 – Read through the problem at least three times. The first reading should be a quick scan, and the next two reading should be done slowly with a view to finding answers to these important questions:

1. What does the problem ask of me? (Usually located towards the end of the problem)
2. What does the problem imply? (This is usually a point you were asked to remember).
3. Mark all information, and underline all important words or phrases.

   Also – try to pick out Key Words – see Purple Math for more information 

Step 2 – Try to make a pictorial representation of the problem such as a circle and an arrow to indicate travel. This makes the problem a bit more real and sensible to you.

A favorite word problem is something like, One train leaves Station A travelling at 100 km/hr and another train leaves Station B travelling at 60 km/hr.  …

Draw a line and the two stations and the two trains at either end.  This will help solidify the situation in your mind.

More on Modelling Word Problems  

Step 3 – Use the information you have to make a table with a blank portion to indicate information you do not know.

Step 4 – Assign a single letter to represent each unknown data in your table. You can write down the unknown that each letter represents so that you do not make the error of assigning answers to the wrong unknown, because a word problem may have multiple unknowns and you will need to create equations for each unknown.

Step 5 – Translate the English terms in the word problem into a mathematical algebraic equation. Remember that the main problem with word problems is that they are not expressed in regular math equations. You ability to correctly identify the variables and translate the word problem into an equation determines your ability to solve the problem.

Step 6 – Check the equation to see if it looks like regular equations that you are used to seeing and whether it looks sensible. Does the equation appear to represent the information in the question?  Take note that you may need to rewrite some formulas needed to solve the word problem equation. For example, word distance problems may need you rewriting the distance formula, which is Distance = Time x Rate. If the word problem requires that you solve for time you will need to use Distance/Rate and Distance/Time to solve for Rate. If you understand the distance word problem you should be able to identify the variable you need to solve for.

Step 7 – Use algebra rules to solve the derived equation. Take note that the laws of equation demands that what is done on this side of the equation has to also be done on the other side. You have to solve the equation so that the unknown ends up alone on one side. Where there are multiple unknowns you will need to use elimination or substitution methods to resolve all the equations.

Step 8 – Check your final answers to see if they make sense with the information given in the problem. For example if the word problem involves a discount, the final price should be less or if a product was taxed then the final answer has to cost more.

Step 9 – Cross check your answers by placing the answer or answers in the first equation to replace the unknown or unknowns. If your answer is correct then both side of the equation must equate or equal. If your answer is not correct then you may have derived a wrong equation or solved the equation wrongly. Repeat the necessary steps to correct.

 Extensive Word Problem Resources here

Strategies for Solving Word Problems (ThinkQuest)

Video on Solving Word Problems