Algebra - Solved Questions and Answers

Algebra is where we use letters (like x or y) to stand for unknown numbers, and then solve equations to find their values.

Algebra questions and answers are provided below for you to learn and practice.

Question 1: If there are 2 apples in a bag and x apples are added to it, making a total of 10 apples, how many apples were added?

Solution:

Let the number of apples added be x.

The total number of apples is 2 (initial) + x (added) = 10.

So, 2 + x = 10.

Subtract 2 from both sides to find x: x = 10 - 2.

Therefore, x = 8 apples were added.

Question 2: A pen costs x dollars. If 5 pens cost $15, what is the cost of one pen?

Solution:

Let the cost of one pen be x dollars.

The total cost for 5 pens is 5x = $15.

Divide both sides by 5 to find the cost per pen: x = 15 / 5.

Therefore, each pen costs $3.

Question 3: A train travels x miles in 2 hours. If it traveled 100 miles, how many miles does it travel in one hour?

Solution:

Let the distance traveled in one hour be x miles.

In 2 hours, the train travels 2x miles.

We know 2x = 100 miles.

Divide both sides by 2 to find x: x = 100 / 2.

Therefore, the train travels 50 miles in one hour.

Question 4: If x people can complete a task in 4 hours, and 8 people can complete the same task in 2 hours, how many people were originally there?

Solution:

Let the original number of people be x.

The work done is the same, so x people in 4 hours is equal to 8 people in 2 hours.

So, 4x = 2 * 8.

Simplify to find x: x = (2 * 8) / 4.

Therefore, x = 4 people were originally there.

Question 5: A rectangle's length is twice its width. If the width is x feet and the area is 50 square feet, what is the width of the rectangle?

Solution:

Let the width be x feet.

The length is 2x feet.

The area of the rectangle is length * width, so 2x * x = 50.

Simplify to find x: x² = 50 / 2.

Solving for x, we get x = sqrt(25) = 5 feet.

Question 6: Sarah has x dollars. After buying a book for $10, she has $30 left. How much money did Sarah have initially?

Solution:

Let Sarah's initial amount of money be x dollars.

After spending $10, she has x - 10 dollars left.

We know x - 10 = $30.

Add 10 to both sides to find x: x = 30 + 10.

Therefore, Sarah initially had $40.

Question 7: A school has x students in each class. If there are 5 classes and a total of 150 students, how many students are in each class?

Solution:

Let the number of students in each class be x.

Total students in 5 classes is 5x.

We know 5x = 150.

Divide both sides by 5 to find x: x = 150 / 5.

Therefore, there are 30 students in each class.

Question 8: A baker makes x cookies from a batch of dough. If he makes 120 cookies and divides them into 6 boxes, how many cookies are in each box?

Solution:

Let the number of cookies in each box be x.

The total number of cookies is 6x.

We know 6x = 120.

Divide both sides by 6 to find x: x = 120 / 6.

Therefore, each box contains 20 cookies.

Question 9: A baker used x cups of flour to make 3 cakes. If he used 15 cups in total, how many cups did he use for each cake?

Solution:

Let the cups of flour used for each cake be x.

The total flour used for 3 cakes is 3x cups.

So, 3x = 15.

Dividing both sides by 3, we get x = 15 / 3.

Therefore, x = 5 cups of flour per cake.

Question 10: In a garden, there are x roses and twice as many tulips. If there are 30 flowers in total, how many roses are there?

Solution:

Let the number of roses be x.

Then, the number of tulips is 2x.

The total number of flowers is x (roses) + 2x (tulips) = 30.

So, 3x = 30.

Dividing both sides by 3, x = 30 / 3.

Therefore, there are x = 10 roses.

Question 11: A car travels x kilometers in 1 hour. If it traveled 240 kilometers in 4 hours, what is its speed in kilometers per hour?

Solution:

Let the speed of the car be x kilometers per hour.

In 4 hours, the car travels 4x kilometers.

We know 4x = 240.

Dividing both sides by 4, x = 240 / 4.

So, the speed of the car is 60 kilometers per hour.

Question 12: A bottle contains x liters of water. After pouring out half of the water, 2 liters remain. How many liters were there initially?

Solution:

Let the initial amount of water be x liters.

Half of this amount is x / 2.

We know x / 2 = 2 liters.

Multiplying both sides by 2, x = 2 * 2.

Therefore, the bottle initially had 4 liters of water.

Question 13: In a class, there are x students. If 3 more students join the class, the total becomes 25. How many students were there initially?

Solution:

Let the initial number of students be x.

After 3 students join, the total is x + 3.

We know x + 3 = 25.

Subtracting 3 from both sides, x = 25 - 3.

So, there were initially 22 students in the class.

Question 14: A rectangle's length is twice its width w. If the rectangle's perimeter is 30 meters, what is its width?

Solution:

Let the width of the rectangle be w meters.

Its length is 2w meters.

The perimeter is 2(length + width) = 30.

Substituting length and width, we get 2(2w + w) = 30.

Simplifying, 6w = 30.

Dividing both sides by 6, w = 30 / 6.

So, the width of the rectangle is 5 meters.

Question 15: A pool is filled by x liters of water each minute. If it takes 10 minutes to fill 300 liters, how much water is filled each minute?

Solution:

Let the amount of water filled each minute be x liters.

In 10 minutes, the total filled is 10x liters.

We know 10x = 300.

Dividing both sides by 10, x = 300 / 10.

So, 30 liters of water is filled each minute.

Question 16: A store sells x apples in a day. If it sold 120 apples in 4 days, how many apples does it sell each day?

Solution:

Let the number of apples sold each day be x.

In 4 days, the total sold is 4x apples.

We know 4x = 120.

Dividing both sides by 4, x = 120 / 4.

Therefore, the store sells 30 apples each day.

Question 17: A train is x meters long and passes a pole in 3 seconds. If it passes the pole at a speed of 10 meters per second, how long is the train?

Solution:

Let the length of the train be x meters.

Speed = Distance / Time, so 10 = x / 3.

Multiplying both sides by 3, x = 10 * 3.

Therefore, the train is 30 meters long.

Question 18: A farmer has x chickens and 4 times as many cows. If he has a total of 100 animals, how many chickens does he have?

Solution:

Let the number of chickens be x.

The number of cows is 4x.

The total number of animals is x (chickens) + 4x (cows) = 100.

So, 5x = 100.

Dividing both sides by 5, x = 100 / 5.

Thus, there are 20 chickens.

Question 19: In the expression 7x³ - 4x² + 9x - 5, what is the coefficient of x²?

Solution:

Identify the term that contains x². The term is -4x².

The coefficient of x² is the number in front of the x² term, which is -4.

Question 20: Find the coefficient of x in the polynomial 3x⁴ - 2x³ + 5x - 7.

Solution:

Look for the term with x (to the first power). The term is 5x.

The coefficient of x is the number in front of the x term, which is 5.

Question 21: What is the coefficient of the constant term in the expression 8x² - 6x + 4?

Solution:

The constant term is the term without any variable, which is 4 in this case.

The coefficient of the constant term is the number itself, so it is 4.

Question 22: Simplify 3x + 5x.

Solution:

Combine the like terms by adding the coefficients.

3x + 5x becomes (3 + 5)x.

Therefore, the simplified expression is 8x.

Question 23: Find the result of 7y - 3y.

Solution:

Subtract the coefficients of the like terms.

7y - 3y becomes (7 - 3)y.

Thus, the result is 4y.

Question 24: Simplify 4a + 7b - 3a.

Solution:

Combine the like terms 4a and -3a.

4a - 3a becomes (4 - 3)a.

The expression simplifies to a + 7b.

Question 25: Calculate the sum of 5x + 8y and 3x - 2y.

Solution:

Add the like terms separately.

Combine 5x and 3x to get 8x.

Combine 8y and -2y to get 6y.

The result is 8x + 6y.

Question 26: What is the result of subtracting 2x - 3y from 6x + 4y?

Solution:

Subtract each term separately.

6x - 2x results in 4x.

4y - (-3y) becomes 4y + 3y, resulting in 7y.

The answer is 4x + 7y.

Question 27: Simplify the expression 9m - 5n + 2m + 3n.

Solution:

Combine like terms 9m and 2m to get 11m.

Combine -5n and 3n to get -2n.

The simplified expression is 11m - 2n.

Question 28: Find the sum of 3p + 4q and -2p - 6q.

Solution:

Add the like terms separately.

3p + (-2p) results in p.

4q + (-6q) results in -2q.

The final expression is p - 2q.

Question 29: Simplify the expression 4x² + 5x - 3 + 2x² - 7x + 6.

Solution:

Combine like terms.

Add 4x² and 2x² to get 6x².

Combine 5x and -7x to get -2x.

Add -3 and 6 to get 3.

The simplified expression is 6x² - 2x + 3.

Question 30: Find the result of subtracting 3y² - 4y + 7 from 5y² + 2y - 1.

Solution:

Subtract each term separately.

5y² - 3y² results in 2y².

2y - (-4y) becomes 2y + 4y, resulting in 6y.

-1 - 7 becomes -8.

The answer is 2y² + 6y - 8.

Question 31: Simplify the sum of 2x³ - 3x² + x - 5 and -x³ + 4x² - 2x + 6.

Solution:

Combine like terms.

Add 2x³ and -x³ to get x³.

Combine -3x² and 4x² to get x².

Add x and -2x to get -x.

Combine -5 and 6 to get 1.

The final expression is x³ + x² - x + 1.

Question 32: Give an expression for the following case:
The total cost (C) when buying x number of notebooks, each costing $3, and y number of pens, each costing $1.50.

Solution:

The cost for notebooks is $3 per notebook, so the cost for x notebooks is 3x.

The cost for pens is $1.50 per pen, so the cost for y pens is 1.50y.

The total cost, C, is the sum of these costs: C = 3x + 1.50y.

Question 33: Give an expression for the following case:

The perimeter (P) of a rectangle with a length of (2x + 5) meters and a width of (x + 3) meters.

Solution:

The perimeter of a rectangle is given by P = 2(length + width).

Substituting the given lengths, P = 2((2x + 5) + (x + 3)).

Simplifying, P = 2(3x + 8).

Question 34: Give an expression for the following case:
The area (A) of a triangle with a base of (x + 4) meters and a height of (2x - 3) meters.

Solution:

The area of a triangle is given by A = 1/2(base * height).

Substituting the given measurements, A = 1/2((x + 4) * (2x - 3)).

Simplifying, A = (x + 4)(x - 1.5).

Question 35: Give an expression for the following case:
The total distance (D) traveled when driving x kilometers at a speed of 60 km/hr and then an additional y kilometers at 80 km/hr.

Solution:

The distance does not depend on the speed, so the total distance is simply the sum of the two individual distances.

Therefore, D = x + y.

Related Links:

• Algebraic Formulas
• Algebraic Expressions
• Polynomials