Basic Algebra Practice

Practice solving equations with 1 variable

1. A number is increased by 2 and then multiplied by 3. The result is 24. What is this number?

A) 4
B) 6
C) 8
D) 10

2. My father’s age divided by 5 is equal to my brother’s age divided by 3. My brother is 3 years older than me. My father’s age is 3 less than 2 times my age. How old is my father?

A) 34
B) 45
C) 56
D) 61

3. (x – 2) / 4 – (3x + 5) / 7 = -3, x = ?

A) 6
B) 7
C) 10
D) 13

4. 1 / (1 + 1 / (1 – 1/x)) = 4, x = ?

A) -3/4
B) 3/7
C) 4/7
D) 3/4

5. Jane is three times Peter’s age. When Peter is at Jane’s age, Jane will be 8 less than 2 times Peter’s age. Find the sum of Jane and Peter’s age at present.

A) 18
B) 21
C) 25
D) 32

 

Answer Key

1. B
Let us call this number by x:

This number is increased by 2: x + 2

Then, it is multiplied by 3: 3(x + 2)

The result is 24: 3(x + 2) = 24 … Solving this linear equation, we obtain the value of the number:

x + 2 = 24 / 3
x + 2 = 8
x = 8 – 2
x = 6

2. B
My age: x
My brother is 3 years older than me: x + 3
My father is 3 less than 2 times my age: 2x – 3
My father’s age divided by 5 is equal to my brother’s age divided by 3: (2x – 3) / 5 = (x + 3) / 3

By cross multiplication:

5(x + 3) = 3(2x – 3)
5x + 15 = 6x – 9
x = 24

My father’s age: 2.24 – 3= 48 – 3 = 45

3. C
There are two fractions containing x and the denominators are different. First, let us find a common denominator to simplify the expression. The least common multiplier of 4 and 7 is 28. Then,

7(x – 2) / 28 – 4(3x + 5) / 28 = – 3.28 / 28 … Since both sides are written on the denominator 28 now, we can eliminate them:

7(x – 2) – 4(3x + 5) = -84
7x – 14 – 12x – 20 = -84
-5x = -84 + 14 + 20
-5x = -50
x = 50/5
x = 10

 

4. B
We can follow outside to inside method to solve this type of problems. x is in the inner part of this fraction; then, we need to narrow the circle to reach x:

1 / (1 + 1 / (1 – 1/x)) = 4

This means that (1 + 1 / (1 – 1/x)) is equal to 1/4. Then,

1 + 1 / (1 – 1/x) = 1/4
1 / (1 – 1/x) = 1/4 – 1
1 / (1 – 1/x) = – 3/4

This means that 1 – 1/x = – 4/3. Then,

1 – 1/x = – 4/3
1 + 4/3 = 1/x
1/x = 7/3
So, x = 3/7.

5. D
Let us say that Peter is “a” years old today. Then, Jane is “3a”. When Peter is “3a”, 3a – a = 2a years will have passed and so, Jane will be 3a + 2a = 5a. Then, we are given that Jane will be 8 less than 2 times Peter’s age. So;

5a = 2.3a – 8
5a = 6a – 8
a = 8.

We are asked to find the sum of Jane and Peter’s age at present.
Today, Peter is a = 8 years old and Jane is 3a = 24 years old.
Their sum is 8 + 24 = 32.

Written by:
Modified: September 29th, 2017
Published: October 12th, 2016

Operations with Polynomials

Adding Polynomials

1. (x3 + 5x + 3x2 +2) + (4x3 + 3x2+ 14)

a. 5x+ 9x+ 16
b. 4x3+6x2+x+14
c. 5x3+6x2+5x+16

 

2. (2x3+ 5x4 + 3x2 +12) + (7x3 + 4x2+ 3)

a. 7x3+5x2+15
b. 5x3+5x4+12+x2
c. 5x4+9x3+7x2+15

 

Subtracting Polynomials

3. (-4x2 – 5y2 – 8) – (2x2 + 3y2 + 8)

a. -4x2-5y2+8
b. -4x2-4y2-16
c. -6x2-8y2-16

 

4. (x2 + 4x) + (x2 + 8x) – (3x – 7)

a. 2x2+9x+7
b. X2-3x+7
c. 4x2-12x-7

 

5. Simplify (2x 2 + 3x)(5x – 7)

a. 10x3-14x2-21x
b. 10x3+x2-21x
c. 10x2+15x-21

 

6. Simplify (- 3x 3 +3)(- 4x 2 – 6)

a. -12x5+12x2-18
b. 12x5+18x3-12x2-18
c. 12x5+18x4-x2-18

 

Answer Key

 

  1. C
  2. C
  3. C
  4. A
  5. B
  6. B

Written by:
Modified: September 29th, 2017
Published: September 19th, 2016

US History Practice Questions

US-HistoryUS History Practice Questions

1. Which one of these was not part of the Intolerable Acts?

a. The Stamp Act
b. The Townshed Acts
c. The Tea Act
d. The Foreign Trade Act

2. The British American colonists revolted from Great Britain because

a. most colonists wanted to be free from British rule
b. most colonists wanted to be represented in Parliament
c. most colonists did not want to be taxed
d.All of the Above

3. Why did the British Parliament impose the first of the Intolerable Acts?

a. To raise revenue for British Parliamentary salaries
b. To pay for the British military in the American Colonies
c. To make trade with non-British countries more expensive
d. To fund the British military’s ongoing war in India

4. What was the last plea to Great Britain to avert the American Revolutionary War?

a. The Olive Branch Petition
b. The Dove Accords
c. Jefferson’s Plea
d. Adam’s Plea

5. Why did the First Continental Congress meet?

a. Respond to the Intolerable Acts
b. Establish a military
c. Establish a federal government
d. Address issues of states rights

6. Why did the Second Continental Congress meet?

a. Establish a federal government
b. Respond to the Intolerable Acts
c. Ask for aid from the French
d. Establish a military

7. What document announced a state of war between the British American colonists and Great Britain?

a. The Articles of Confederation
b. The Declaration of Independence
c. The United States Constitution
d. The Declaration of Hostilities

8. Which of these was NOT considered part of Manifest Destiny?

a. American people and institutions are special
b. America had a mission to turn the west into an agrarian landscape
c. America’s destiny was to settle the west
d. All of the above are correct

9. What was the start of western expansion?

a. The Clermont steamboat
b. The Mexican American War
c. The invention of the telegraph
d. The discovery of gold in California

10. When did the Gold Rush start in California?

a. 1849
b. 1848
c. 1855
d. 1851

11. Which was the largest land acquisition for the United States?

a. The Louisiana Purchase
b. The Treaty of Guadalupe Hidalgo
c. The Texas Annexation
d. The Gadsden Purchase

12. Which State was once a nation with its own president before being added to the Union during westward expansion?

a. Delaware
b. California
c. Alaska
d. Texas

13. What most shortened the time it took for people to get across the United States during westward expansion?

a. The telegraph
b. The Clearmont steamboat
c. The Transcontinental Railroad
d. The Northwest Passage

14. What marked the end of western expansion?

a. Arizona statehood
b. The end of the Mexican American War
c. The Completion Speech
d. The President William Mckinley’s Assassination

15. What was the Trail of Tears?

a. The last bloody battle/retreat of the Sioux Nation westward
b. The forced removal of Irish immigrants from New York
c. The relocation of many Native American tribes to “Indian Territory”
d. Part of the Transcontinental Railroad that claimed the most lives in its building

 

Answer Key

1. D
The Foreign Trade Act did not take place until after (centuries later) the American Revolutionary War, all the Intolerable Acts took place in the lead-up to the American Revolutionary War.

Choice A is incorrect, the Stamp Act imposed a tax on the colonies by forcing them to use paper produced in Britain for many printed materials. This was the first of the Intolerable Acts or taxes that were placed on British American subjects without their consent.

Choice B is incorrect, the Townshed Acts were also part of the Intolerable Acts that were meant to raise revenue in the colonies for the employment of judges and governors. These were also used to punish colonists and force compliance with the numerous other taxes being levied against the colonies.

Choice C is incorrect, the Tea Act allowed the British East India Company to sell tea directly to the colonies, thus making it cheaper. Many colonists hated this act because a small tax was added on without the consent of the colonists, the Boston Tea Party was the result.

2. B
Choice A is incorrect, even when the American Revolutionary War was occurring most of the colonists did not decisively want to be free from British rule, instead they either wanted to stay under British rule or they were undecided.

Choice B is correct, the British imposed taxes through the Intolerable Acts either without representatives from the colonies at all, or with representatives that effectively had no say in the matter.

Choice C is incorrect, while Colonists and people everywhere do not like to be taxed, taxes were what allowed government to function and the colonists made no statements against ALL taxes, just some of them.

Choice D is incorrect, A and B are incorrect

3. B
After the Seven Years War with France, Great Britain had a lot of debt and Parliament wanted the colonies to pay for some of the military expenditures and the cost of maintaining a military presence in the colonies.

Choice A is incorrect, the none of the Intolerable Acts were never used to raise revenue for the British Parliament salaries. The Intolerable Acts were used to pay off war debt, punish the colonies, and a variety of other things.

Choice C is incorrect, the Navigation Acts kept most trade in the British Colonies strictly between the colonies and Great Britain.

Choice D is incorrect, while the East India Company was fighting in India at this time the British military was not involved there yet.

4. A
The Second Continental Congress tried to avert war even though the battles of Lexington and Concord had already been fought. King James would not receive the petition because it came from what he perceived as an illegal and illegitimate assembly.

Choice C is false because while Thomas Jefferson wrote the first draft of the petition, most of it was redrafted, and it was never called Jefferson’s Plea.

Choice D is false because neither Samuel or John Adams thought the Plea would help and contributed nothing to the process. The plea was never called Adams Plea.

5. A
The First continental Congress met to discuss various ways of getting rid of the Intolerable Acts and petition the King of Great Britain for a redress of grievances.

Choice B is incorrect, this is the purpose of the Second Continental Congress.

Choice C is incorrect, the federal government was set up with the Articles of Confederation.

Choice D is incorrect, all the “states” at this point were not states yet, instead they were colonies that firmly belonged to Great Britain.

6. D
While many other things were achieved the main purpose of the Second Continental Congress was to establish a Continental Army.

Choice A is incorrect, the federal government was set up with the Articles of Confederation

Choice B is incorrect, Responding to the Intolerable Acts was the purpose of the First Continental Congress

Choice C is incorrect, asking for French aid was officially done after the Second Continental Congress was over, though Benjamin Franklin and others hinted at this possibility with many French officials.

7. B
With the Declaration of Independence the colonies disavowed the rule of Great Britain, effectively announcing a state of war between the colonists and Great Britain.

Choice A is incorrect, the Articles of Confederation Established a federal government and occurred after there was a state of war between Great Britain and the Colonies.

Choice C is incorrect, the United States Constitution was a document that strengthened the federal government and provided more effective ways of raising revenue.

Choice C is incorrect, while a declaration of hostilities does announce a state of war there was no such document called the Declaration of Hostilities.

8. D
All the reasons listed were part of Manifest Destiny,

Many Americans believed that they were special and could use the land better than anyone else.

The agrarian ideal was a very popular idea at the time, many dreamed that America should be populated by self-sufficient farmers. With all the “open” land in the west this seemed like a real possibility.

Many Americans believed that God had set America a destiny to bring civilization to the west, and that America would one day stretch across the continent.

9. A
The Clermont steamboat’s maiden voyage between New York and Albany took 32 hours, much less time than the four days it would take the average sloop. This showed that distances between areas were shrinking time-wise.

Choice B is incorrect, while the Mexican American War brought many people west it was not the start of western expansion.

Choice C is incorrect, the invention of the telegraph greatly helped communication across the country, but it was not invented until long after western expansion had started.

Choice D is incorrect, gold in California brought more people west faster than ever before, but western expansion had already started decades before.

10. B
Choice A is incorrect, while the miners were called 49ers after the year that most people started mining for gold in California this was not when the Gold Rush started.

Choice B is correct, this was the year that Gold was discovered in California and the Gold Rush commenced.

Choice C is incorrect, this was the year that the Gold Rush ended in California.

Choice D is incorrect, this was the year that California was admitted to the Union, not when the Gold Rush started.

11. A
The Louisiana Purchase was orchestrated under the Jefferson Administration and nearly doubled the size of the United States adding on almost 828,000 sq miles.

Choice B is incorrect, the Treaty of Guadalupe Hidalgo added 525,000 sq miles to the United States and Mexico lost nearly half of their land. This was still a smaller land acquisition than the Louisiana Purchase.

Choice C is incorrect, the Texas Annexation incorporated the Republic of Texas into the United States and brought in about 390,000 sq miles of land to the Union. This was also less than the Louisiana Purchase.

Choice D is incorrect, the Gadsden Purchase was the last time that the United States acquired land from Mexico with this $10 million purchase which added territory onto New Mexico and Arizona. This acquisition was for about 30,000 sqare miles which was less than the Louisiana Purchase.

12. D
Texas was an independent nation from about 1836 to 1845 with its own government and president. Mexico did not recognize this independence, but they effectively were independent until annexation by the United States in 1845.

Choice A is incorrect, Delaware was the first state in the Union in 1787, it was not created during westward expansion. Before being a state it was a colony of Great Britain, not a sovereign nation.

Choice B is incorrect, California claimed to be an independent nation for less than a year, but never installed a civil administration let alone a president.

Choice C is incorrect, Alaska was a territory of Russia before becoming part of the Union, and never was a sovereign nation.

13. C

The Transcontinental Railroad greatly reduced the amount of time it took to get across the United States from almost half a year to about a week.

Choice A is incorrect, the telegraph was the fastest way for information to travel across the Continental United States. People could not travel by telegraph however.

Choice B is incorrect, the Clearmont steamboat marked the start of westward expansion, but it did not provide the shortest trip across the United States.

Choice D is incorrect, the Northwest Passage is a sea route that runs through the Arctic Ocean from the tips of Russia and Alaska above Canada and out into the Atlantic Ocean. It was first discovered in 1903, but remained closed for most of each year until about 2010. This passage was dangerous and never was significantly used for the purposes of western expansion.

14. A
With Arizona’s statehood in 1912, all the west had been conquered and incorporated into the United States. There was nowhere else to expand in Western North America.

Choice B is incorrect, the end of the Mexican American War in 1848 was more than half a century before the end of western expansion.

Choice C is incorrect, the Completion Speech does not exist.

Choice D is incorrect, President William Mckinley’s Assassination in 1901 did not have an effect on westward expansion.

15. C
Many Native Americans were forcefully relocated out of the Deep South if they failed to assimilate. This occurred during much of the 1830s. Most deaths occurred near the end when the army detained many Native Americans who had not left and forced them west, many died due to disease, exposure and starvation.

Choice A is incorrect, the Trail of Tears occurred before the United States military and the various Sioux tribes were at war.

Choice B is incorrect, Irish immigrants were never forcefully removed en mass from New York

Choice D is incorrect, while the Transcontinental Railroad claimed many lives in its construction no part of it was ever called the Trail of Tears.

Written by:
Modified: September 29th, 2017
Published: August 22nd, 2016