Category: Mathematics

Simple Interest is a quick and easy method for calculating the Interest Charged on a Loan or Principal Amount. The Concept of Simple Interest is quite famous and is used in many sectors such as finance, automobile, and banking. Go through the further modules to know about What is Simple Interest, Formula to Calculate Simple Interest, Solved Examples on How to Calculate the Simple Interest.

What is Simple Interest?

Simple Interest is the method for calculating the Interest Amount on a Certain Amount of Sum. SI is obtained by multiplying the interest rate, principal, time duration that elapses between payments. In general, it is calculated on a daily, monthly or annual basis.

In real times, money is not borrowed or lent for free. While repaying the amount you borrowed you need to pay a certain amount of interest along with the amount you have taken. In fact, the amount of interest you repay depends on the loan amount as well as the time for which you borrow and the rate of interest.

Simple Interest Formula

You need to be familiar with the Simple Interest Formula in order to understand the concept of Finances. The formula for Simple Interest helps you find the interest amount if principal, rate of interest, and time duration are given.

Formula to Calculate Simple Interest is SI = (P × R ×T) / 100

Where, P = Principal

R = Rate of Interest (in percentage)

T = Time Duration (in years)

However, the formula to find the Amount is given by Amount (A) = Principal (P) + Interest (I).

The amount is the total money you pay back at the end of the time you borrowed.

Simple Interest Formula for Months

The Formula to find Simple Interest for Months varies slightly compared to a yearly basis. Let us consider the Principal Amount be P and Rate of Interest per Annum be R and n be the time duration in months then the formula to Calculate SI is as such

Simple Interest for n months = (P × n × R)/ (12 ×100)

Solved Examples on Simple Interest

1. Raju takes a loan of Rs 20,000 from a bank for a period of 2 years. The rate of interest is 5% per annum. Find the interest and the amount he has to pay by the end of 2 years?

Solution:

Principal or Loan Sum = 20, 000

Time Duration T = 2 Yrs

Rate of Interest R = 5%

Formula to Calculate the Simple Interest = (P × R ×T) / 100

= (20,000*5*2)/100

= 200000/100

= 2000

Amount to be repeated by the end of the year = Principal + Interest

= 20, 000+2000

= 22, 000

Therefore, Raju needs to repay a total of Rs. 22, 000/- after the end of 2 years.

2. Mohan pays Rs 13000 as an amount on the sum of Rs 10000 that he had borrowed for 3 years. Find the rate of interest?

Solution:

Amount = 13, 000

Principal = 10,000

SI = Amount – Principal

= 13, 000 – 10, 000

= 3, 000

Time Duration = 3 yrs

Rate of Interest R = ?

SI = (P × R ×T) / 100

3, 000 =(10, 000*R*3)/100

R = (3000*100)/(10,000*3)

= 300000/30000

= 10%

Therefore, Rate of Interest is 10%.

3. Neela borrowed Rs 40,000 for 2 years at a rate of 4% per annum. Find the interest accumulated at the end of 2 years?

Solution:

P = Rs 40,000

R = 4%

T = 2 years

SI = (P × R ×T) / 100

= (40, 000*4*2)/100

= 3200

Interest accumulated at the end of 2 years is Rs. 3,200/-

A bar graph is a graph with rectangular bars. It is used to compare different categories. The most usual type of bar graph is vertical. One who is willing to learn more about the bar graph can refer to the following sections. On this page, we are providing the definition of a bar graph or column graph, solved example questions, and types of the bar graph.

Bar Graph Definition

A bar graph is a pictorial representation of grouped data in the form of horizontal or vertical rectangular bars, here the length of rectangular bars is equal to the measure of the data. The height of a rectangular bar totally depends on the numeric value it represents. The frequency distribution tables can be represented using bar charts which simplify the calculations and understanding of data.

Types of Bar Graphs or Column Graphs

In general, bar graphs can be horizontal or vertical rectangular bars. The main feature of a bar chart is its length or height. If the height of any bar graph is more, then the values are greater than the given data. In any bar graph, bars represent frequencies of distinctive values of a variable. The number of values on the x-axis of a bar graph and the y-axis of a column graph is known as the scale. The two types of bar graphs are the vertical bar graph and the horizontal bar graph. Apart from vertical, horizontal bar graphs, the other two types are grouped bar graphs, stacked bar graphs.

• Vertical Bar Graph: If the given data is represented vertically in a graph with the help of bars, where the bars denote the measure of data such graphs are called the vertical bar graphs.
• Horizontal Bar Graph: If the grouped data is represented horizontally in a graph with the help of bars, then those graphs are called the horizontal bar graphs.
• Grouped Bar Graph: It is used to represent the discrete value for more than one object. Here, the number of instances are combined together into a single bar. It is also known as the clustered graph.
• Stacked Bar Graph: It divides an aggregate into different parts. A stacked bar graph is also known as the composite bar graph.

How to draw Bar Graph or Column Graph?

• On a graph paper, draw two lines that represent x-axis (OX), y-axis (OY).
• Mark the points at equal intervals along the x-axis, also mention the names of data items whose values are plotted.
• Select a suitable scale. Find the heights of the bar for the given values.
• Mark off these heights or lengths parallel to the y-axis from the points taken in Step 2.
• Draw the bars of equal width for the heights marks in the above step along the x-axis. Those bars represent the numerical data.

Column Graph Solved example Questions

Example 1.

The annual earnings of a firm during the 8 consecutive years are given below:

Year	2010-2011	2011-2012	2012-2013	2013-2014	2014-2015	205-2016	2016-2017	2017-2018
Annual Earnings (in dollars)	150	120	170	180	160	190	200	205

Draw a bar graph representing the above data.

Solution:

Given that,

The annual earnings of a firm over the 8 consecutive years.

On a graph paper, draw a horizontal line OX and a vertical line OY, representing the x-axis and the y-axis respectively.

Take years along the x-axis at equal gaps.

Choose 1 small division = 10 dollars

Then, the heights of the bars are:

Annual income in 2010-11 = (1/10 * 150) = 15 small divisions

Annual income in 2011-12 = (1/10 * 120) = 12 small divisions

Annual income in 2012-13 = (1/10 * 170) = 17 small divisions

Annual income in 2013-14 = (1/10 * 180) = 18 small divisions

Annual income in 2014-15 = (1/10 * 160) = 16 small divisions

Annual income in 2015-16 = (1/10 * 190) = 19 small divisions

Annual income in 2016-17 = (1/10 * 200) = 20 small divisions

Annual income in 2017-18 = (1/10 * 205) = 20.5 small divisions

 

Bar graph showing firm annual income during eight consecutive years

At the points marked in Step 2, draw bars of equal width and of heights calculated.

Example 2.

In a firm of 500 employees, the percentage of monthly salary saved by each employee is given in the following table. Represent it using a bar graph.

Number of Employees	100	50	73	150	60	67
Savings in Percentages	20	30	50	60	70	80

Solution:

Given that,

The savings percentage of 500 employees in a firm.

Draw two lines OX, OY on graph paper.

Take savings along the x-axis, number of employees along the y-axis.

Choose 1 small division = 10 units on y-axis.

Then, the heights of the bars are:

The number of employees who are saving 20% of salary is (1/10 * 100) = 10 small divisions

The number of employees who are saving 30% of salary is (1/10 * 50) = 5 small divisions

The number of employees who are saving 50% of salary is (1/10 * 73) = 7.3 small divisions

The number of employees who are saving 60% of salary is (1/10 * 150) = 15 small divisions

The number of employees who are saving 70% of salary is (1/10 * 60) = 6 small divisions

The number of employees who are saving 80% of salary is (1/10 * 67) = 6.7 small divisions

Bar graph showing the percentage of monthly salary saved by each employee in a firm.

At the points marked in Step 2, draw bars of equal width and of heights calculated.

Example 3.

70 adults and 70 children were surveyed to find out how many servings of fruit and vegetables they eat per week. The results are shown in the bar graph below:

(i) How many adults and children combine eat between 6 and 10 servings of fruit and vegetables per week?

(ii) What is the mode for the number of servings of fruit and vegetables eaten per week by adults?

(iii) What is the difference between the number of children who eat over 2020 servings of fruit and vegetables per week and between 66 to 1010 servings of fruit and vegetables per week?

(iv) What basic trend do we notice from the data presented?

Solution:

Given graph shows how many fruits and vegetables adults and children eat per week.

(i) Here, we need to take readings of the two bars showing 6-10 servings per week category. The number of adults who are between 6 and 10 servings per week is a total of 12. The number of children who are between 6 and 10 servings per week is a total of 25.

Therefore, the number of adults and children combined who have between 6 and 10 servings per week is 12 + 25 = 37.

(ii) Simply, we need to locate the bar that represents children eating over 20 servings and the bar that represents children eating between 6 – 10 servings, find the difference between them.

The number of children who eat over 20 servings of fruit and vegetables is 4. The number of children who eat between 6 and 10 servings of fruit and vegetables is 25. Therefore, the difference between the two categories is 25 – 4 = 21

(iii) The mode is the most frequently occurring value. Since this question is about the adults, we do not need to consider the bars for children. All we need to do is see which bar is highest for the adults.

The 16 – 20 bar has the highest length i.e 22. Therefore the mode is 16 – 20 servings.

(iv) The most basic trend we can obtain from the given information is that adults eat more portions of fruit and vegetables per week than children.

FAQs on Bar Graph or Column Graph

1. How do you define a bar graph?

The bar graph represents the categorical data using rectangular bars. It shows a comparison between discrete categories.

2. What are the different types of bar graphs?

The different types of bar graphs are vertical bar graphs, grouped bar graphs, horizontal bar graphs, and stacked bar graphs.

3. Is a Histogram the same as a Bar Graph?

Bar graphs are used when your data is in categories. But histogram is used when the data is continuous.

4. When a bar graph is used?

The bar graph is used to compare the items between different groups over time. These are used to measure the changes over a period of time. When the changes are larger, a bar graph is the best option to represent the data.

The pie chart represents data in a circular graph. The pieces of the chart or graph show the data size. The various components are represented by the pieces of a circle and the whole circle is the sum of the values of all components. One can get the pie chart definition, formula, use, solved examples, and steps to create a pie chart in the below sections.

Pie Chart Definition

A pie chart is also a graph and is a type of pictorial representation of data. It divides the circle into various sectors in order to explain the numeric values. Each section is a proportionate part of the whole circle. We use a pie chart to find the composition of something. The pie chart is also known as the circle chart.

A pie chart is used for data representation. The total of all data in a pie is equal to 360° and the total value of a pie is always 100%.

The central angle for a component formula is given as

Central angle for a component = (Value of the component / Sum of the values of all components) * 360°

Steps to Make a Pie Chart

Follow the below-mentioned steps to create a pie chart/ circle chart for the given data.

• Enter the given data into a table to make the process easy for you.
• Find the sum of values in the table.
• Divide each value in the table by sum and multiply the result with 100 to get the percent.
• To get the degrees of each value, substitute the values in the formula i.e (Value of each component/sum) * 360A pie chart is used for the data representation. The total of all data in a pie is equal to 360° and the total value of a pie is always 100%.
• The central angle for a component formula is given as Central angle for a component = (Value of the component / Sum of the values of all components) * 360°.
• Draw a circle and use a protractor to measure the degree of each sector.

Advantages and Disadvantages of Pie Chart

Advantages

• It is simple and easy to understand
• Data can be represented visually as a fractional part of the whole.
• It helps in providing an effective communication tool for the uninformed audience.

Disadvantages

• It can represent only one set of data. So, you need a series to compare multiple sets.
• If you have too many pieces of data, and even you add labels and numbers may not help here, they themselves may become crowded and hard to read.

Example Questions & Answers

Example 1.

The following table shows the expenditure in percentage incurred on the construction of a house in a city:

Item	Brick	Cement	Steel	Labour	Miscellaneous
Expenditure (in percentage)	18%	30%	10%	12%	30%

Draw a pie chart for the above data.

Solution:

Total percentage = 100

Central angle for a component = (Value of the component / Sum of the values of all components) * 360°

Calculation of central angles:

Item	Expenditure (in percentage)	Central Angle
Brick	18%	((18 / 100) * 360°) = 64.8°
Cement	30%	((30 / 100) * 360°) = 108°
Steel	10%	((10 / 100) * 360°) = 36°
Labour	12%	((12 / 100) * 360° = 43.2°
Miscellaneous	30%	((30 / 100) * 360°) = 108°

Construction for creating a pie chart

Steps of construction:

1. Draw a circle of any convenient radius.

2. Draw a horizontal radius of the circle.

3. Draw sectors starting from the horizontal radius with central angles of 64.8°, 108°, 36°, 43.2°, and 108° respectively.

4. Shade the sectors differently using different colors and label them.

Thus, we obtain the required pie chart, shown in the above figure.

Example 2.

The marks scored by a student in his examination are shown below:

Subject	English	Hindi	Maths	Science	Social
Marks	70	85	76	88	92

Draw a pie chart for the above data.

Solution:

The given table contains the marks of a student in an exam.

The total marks scored by a student = 411

Central angle for a component = (Each subject marks / total marks) * 360°

Calculation of central angles:

Subject	Marks	Central Angle
English	70	(70 / 411) * 360° = 61.31°
Hindi	85	(85 / 411) * 360° = 74.45°
Maths	76	(76 / 411) * 360° = 66.56°
Science	88	(88 / 411) * 360° = 77.08°
Social	92	(92 / 411) * 360° = 80.58°

Construction for creating a pie chart

Steps of construction:

1. Draw a circle.

2. Draw sectors starting from the horizontal radius with central angles of 61.31°, 74.45°, 66.56°, 77.08°, 80.58° respectively.

3. Shade the sectors differently using different colors and label them.

Thus, we obtain the required pie chart, shown in the above figure.

Example 3.

The data on the mode of transport used by 1000 students are given below:

Mode of Transport	Bus	Cycle	Train	Car	Scooter
Number of Students	150	50	200	100	500

Represent the data on a pie chart.

Solution:

The given table shows the mode of transport used by 1000 students to reach the school.

Total Number of Students = 1000

Central angle for a component = (Value of the component / Sum of the values of all components) * 360°

Calculation of central angles:

Mode of Transport	Number of Students	Central Angle
Bus	150	(150 / 1000) * 360 = 54°
Cycle	50	(50 / 1000) * 360 = 18°
Train	200	(200 / 1000) * 360 = 72°
Car	100	(100 / 1000) * 360 = 36°
Scooter	500	(500 / 1000) * 360 = 180°

Construction for creating a pie chart

1. Draw a circle of any convenient radius.

2. Draw a horizontal radius of the circle.

3. Draw sectors starting from the horizontal radius with central angles of 54°, 18°, 72°, 36°, and 180°.

4. Shade the sectors differently using different colors and label them.

Thus, we obtain the required pie chart, shown in the above figure.

FAQs on Pie Chart

1. What are the uses of a pie chart?

A pie chart is used to represent categorical data, to compare areas of growth in a business like turnover profit and exposure.

2. What are the pie chart examples?

The real-life examples of the pie charts are a representation of kinds of cars sold in a month, types of food items liked by people in a room, marks scored by students in a class, etc

3. What is the formula to calculate the percentage of each component of a pie chart?

The pie chart formula to calculate the central angle for a component is as follows:

Central angle for a component = (Value of the component / Sum of the values of all components) * 360°

Subtraction of algebraic expressions is subtracting one expression from another expression. A detailed explanation is given about the Subtraction of algebraic expressions in this article. Go through each step of solving problems and understand how simply a Subtraction of algebraic expressions problem can be solved. Students can get complete knowledge of Algebraic Expressions by referring to this page.

How to Find Subtraction of Algebraic Expressions?

Follow the below-listed steps to Subtract Algebraic Expressions and arrive at the solution easily. They are along the lines

• Write the given expressions in standard form.
• After that arrange one expression under another expression with the like terms come in the same column.
• The main part of the subtraction of algebraic expressions is changing the sign of every individual term of the second expression to get the inverse of the expression.
• Lastly, add the like terms and get the final expression.

Subtraction of Algebraic Expressions Solved Examples

1. Subtract 3a + 4b – 2c from 5a – 2b + 2c

Solution:
Note down both given expressions and rearrange them if required.
3a + 4b – 2c = 3a + 4b – 2c
5a – 2b + 2c = 5a – 2b + 2c
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
5a – 2b + 2c
3a + 4b – 2c
Change the signs of each term available in the second row and fins the inverse of the second expression.
3a + 4b – 2c = -(3a + 4b – 2c) = – 3a – 4b + 2c
Add the like terms by arranging two expressions in columns to get the final expression.
5a – 2b + 2c
– 3a – 4b + 2c
—————————-
2a – 6b + 4c

The required expression is 2a – 6b + 4c

2. Subtract 4x² – 7x – 5 from 6 + 2x – 3x².

Solution:
Note down both given expressions and rearrange them if required.
6 + 2x – 3x² = – 3x² + 2x + 6
4x² – 7x – 5 = 4x² – 7x – 5
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
– 3x² + 2x + 6
4x² – 7x – 5
Change the signs of each term available in the second row and fins the inverse of the second expression.
4x² – 7x – 5 = -(4x² – 7x – 5) = – 4x² + 7x + 5
Add the like terms by arranging two expressions in columns to get the final expression.
– 3x² + 2x + 6
– 4x² + 7x + 5
—————————-
– 7x² + 9x + 11

The required expression is – 7x² + 9x + 11

3. Subtract 4x + 2y – 4z from 10x – 6y + 2z

Solution:
Note down both given expressions and rearrange them if required.
10x – 6y + 2z = 10x – 6y + 2z
4x + 2y – 4z = 4x + 2y – 4z
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
10x – 6y + 2z
4x + 2y – 4z
Change the signs of each term available in the second row and fins the inverse of the second expression.
4x + 2y – 4z = -(4x + 2y – 4z) = – 4x – 2y + 4z
Add the like terms by arranging two expressions in columns to get the final expression.
10x – 6y + 2z
– 4x – 2y + 4z
—————————-
6x – 8y + 6z

The required expression is 6x – 8y + 6z

4. Subtract – 5ab + 2a² from 3a² + 9ab.

Solution:
Note down both given expressions and rearrange them if required.
– 5ab + 2a² = 2a² – 5ab
3a² + 9ab = 3a² + 9ab
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
3a² + 9ab
2a² – 5ab
Change the signs of each term available in the second row and fins the inverse of the second expression.
2a² – 5ab = -(2a² – 5ab) = – 2a² + 5ab
Add the like terms by arranging two expressions in columns to get the final expression.
3a² + 9ab
– 2a² + 5ab
—————————-
a² + 14ab

The required expression is a² + 14ab

5. Subtract 2x² – 5xy + 9y² – 4 from 7xy – 2x² – 4y² + 7.

Solution:
Note down both given expressions and rearrange them if required.
7xy – 2x² – 4y² + 7 = – 2x² + 7xy – 4y² + 7
2x² – 5xy + 9y² – 4 = 2x² – 5xy + 9y² – 4
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
– 2x² + 7xy – 4y² + 7
2x² – 5xy + 9y² – 4
Change the signs of each term available in the second row and fins the inverse of the second expression.
2x² – 5xy + 9y² – 4 = -(2x² – 5xy + 9y² – 4) = – 2x² + 5xy – 9y² + 4
Add the like terms by arranging two expressions in columns to get the final expression.
– 2x² + 7xy – 4y² + 7
– 2x² + 5xy – 9y² + 4
—————————-
– 4x² + 12xy – 13y² + 11

The required expression is – 4x² + 12xy – 13y² + 11

6. What should be subtracted from 3a³ – 5a² + 7a – 8 to obtain 2a² – 4a + 3 ?

Solution:
Let ‘S’ denote the required expression.
Given that S subtracted from 3a³ – 5a² + 7a – 8 to get 2a² – 4a + 3.
(3a³ – 5a² + 7a – 8) – S = 2a² – 4a + 3
S = (3a³ – 5a² + 7a – 8) – (2a² – 4a + 3)
Note down both given expressions and rearrange them if required.
3a³ – 5a² + 7a – 8 = 3a³ – 5a² + 7a – 8
2a² – 4a + 3 = 2a² – 4a + 3
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
3a³ – 5a² + 7a – 8
0   + 2a² – 4a + 3
Change the signs of each term available in the second row and fins the inverse of the second expression.
0   + 2a² – 4a + 3 = -(0   + 2a² – 4a + 3) = – 0  – 2a² + 4a – 3
Add the like terms by arranging two expressions in columns to get the final expression.
3a³ – 5a² + 7a – 8
– 0  – 2a² + 4a – 3
—————————-
3a³ – 7a² + 11a – 11

The required expression is 3a³ – 7a² + 11a – 11

7. Subtract 5a³ – 6a² + 2a – 10 from the sum of 5a³ + 6a² + 9 and 8a² – 4?

Solution:
Given that Subtract 5a³ – 6a² + 2a – 10 from the sum of 5a³ + 6a² + 9 and 8a² – 4.
Therefore, firstly, find the sum of 5a³ + 6a² + 9 and 8a² – 4.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
5a³ + 6a² + 9 + 8a² – 4
Arrange the like terms together.
5a³ + 6a² + 8a² + 9 – 4
Now, find the sum of the numerical coefficients of all terms.
5a³ + 14a² + 5
The required expression is 5a³ + 14a² + 5.
Now, subtract 5a³ – 6a² + 2a – 10 from 5a³ + 14a² + 5.
Note down both given expressions and rearrange them if required.
5a³ + 14a² + 5 = 5a³ + 14a² + 5
5a³ – 6a² + 2a – 10 = 5a³ – 6a² + 2a – 10
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
5a³ + 14a² + 0.a + 5
5a³ – 6a² + 2a – 10
Change the signs of each term available in the second row and fins the inverse of the second expression.
5a³ – 6a² + 2a – 10 = -(5a³ – 6a² + 2a – 10) = – 5a³ + 6a² – 2a + 10
Add the like terms by arranging two expressions in columns to get the final expression.
5a³ + 14a² + 0.a + 5
– 5a³ + 6a² – 2a + 10
—————————-
0 + 20a² – 2a + 15

The required expression is 20a² – 2a + 15

8. What should be subtracted from 10a³ – 15a² + 9a – 21 to obtain 6a² – 14a + 36 ?
Solution:
Let ‘S’ denote the required expression.
Given that S subtracted from 10a³ – 15a² + 9a – 21 to get 6a² – 14a + 36.
(10a³ – 15a² + 9a – 21) – S = 6a² – 14a + 36
S = (10a³ – 15a² + 9a – 21) – (6a² – 14a + 36)
Note down both given expressions and rearrange them if required.
10a³ – 15a² + 9a – 21 = 10a³ – 15a² + 9a – 21
6a² – 14a + 36 = 6a² – 14a + 36
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
10a³ – 15a² + 9a – 21
0   + 6a² – 14a + 36
Change the signs of each term available in the second row and fins the inverse of the second expression.
0 + 6a² – 14a + 36 = -(0 + 6a² – 14a + 36) = – 0  – 6a² + 14a – 36
Add the like terms by arranging two expressions in columns to get the final expression.
10a³ – 15a² + 9a – 21
– 0  – 6a² + 14a – 36
—————————-
10a³ – 21a² + 23a – 57

The required expression is 10a³ – 21a² + 23a – 57

9. Subtract 21a³ – 3a² + 4a – 6 from the sum of 6a³ + 7a² + 15 and 7a² – 16?

Solution:
Given that Subtract Subtract 21a³ – 3a² + 4a – 6 from the sum of 6a³ + 7a² + 15 and 7a² – 16.
Therefore, firstly, find the sum of 6a³ + 7a² + 15 and 7a² – 16.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
6a³ + 7a² + 15 + 7a² – 16
Arrange the like terms together.
6a³ + 7a² + 7a² + 15 – 16
Now, find the sum of the numerical coefficients of all terms.
6a³ + 14a² – 1
The required expression is 6a³ + 14a² – 1.
Now, subtract 21a³ – 3a² + 4a – 6 from 6a³ + 14a² – 1.
Note down both given expressions and rearrange them if required.
6a³ + 14a² – 1 = 6a³ + 14a² – 1
21a³ – 3a² + 4a – 6 = 21a³ – 3a² + 4a – 6
Write both expressions one below another such that the expressions will subtract each other with the like terms come in the same column.
6a³ + 14a² + 0.a – 1
21a³ – 3a² + 4a – 6
Change the signs of each term available in the second row and fins the inverse of the second expression.
21a³ – 3a² + 4a – 6 = -(21a³ – 3a² + 4a – 6) = – 21a³ + 3a² – 4a + 6
Add the like terms by arranging two expressions in columns to get the final expression.
6a³ + 14a² + 0.a – 1
– 21a³ + 3a² – 4a + 6
—————————-
– 15a³ + 17a² – 4a + 5

The required expression is – 15a³ + 17a² – 4a + 5

Addition of Algebraic Expressions is the process of collecting like terms and adding them. Identify and add the coefficients of like terms and sum them to find the final expression of given problems. We have explained clear ways to find the Addition of Algebraic Expressions. We even provided different examples for different methods to solve the addition of algebraic expressions. Therefore, check practice questions, answers, and explanations for every problem and get the best knowledge to solve algebraic expressions.

Methods to Solve an Addition of Algebraic Expressions

Students can perform the addition of algebraic expressions using two methods. Use the best and easy method for you and find the final expression. The two methods to find the addition of algebraic expressions are given below.
1. Horizontal Method
2. Column Method

How to find the addition of algebraic expressions using the Horizontal Method?

The Horizontal Method is the simplest way to find the addition of algebraic expressions. Just by following the below step by step procedure, students can estimate the addition of algebraic expressions.

• Firstly, note down the given expressions.
• Place the given expressions in a row and separate them with the addition symbol in between them.
• Re-arrange the given terms by grouping or placing like terms together.
• Simplify the coefficients of like terms.
• Finally, write the resultant expression in standard form.

How to find the addition of algebraic expressions using the Column Method?

The Column method is also called a Vertical Method. The Addition of Algebraic Expressions can be estimated using the vertical method by writing expressions in separate rows. Before you write separate rows, you need to arrange the expressions with like terms in the correct order. Have a look at the below procedure to exactly know what is the process to find the addition of algebraic expressions using the Column Method.

• Write the given expressions.
• Place one expression below the other expression with the like terms come in the same column.
• Next, add the like terms column-wise with their coefficients.
• Finally, fins the resultant expression in the standard form.

Addition of Algebraic Expressions Solved Examples

1. Add 5a + 7b – 6c, b + 2c – 3a and 2a – 5b – 3c

Solution:Given expressions are 5a + 7b – 6c, b + 2c – 3a and 2a – 5b – 3c.

Horizontal Method:
There are three variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
5a + 7b – 6c + b + 2c – 3a + 2a – 5b – 3c.
Arrange the like terms together.
5a – 3a + 2a + 7b + b – 5b – 6c + 2c – 3c.
Now, find the sum of the numerical coefficients of all terms.
4a + 3b – 7c.

The required expression is 4a + 3b – 7c.

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
5a + 7b – 6c, – 3a + b + 2c, 2a – 5b – 3c
Note down the like terms below each other and add them in column-wise.
+ 5a + 7b – 6c
– 3a +  b + 2c
+ 2a – 5b – 3c
—————————-
4a + 3b – 7c

The required expression is 4a + 3b – 7c.

2. Add 7x² + 8y – 9, 3y + 2 – 3x² and 3 – y + 3x².

Solution:
Given expressions are 7x² + 8y – 9, 3y + 2 – 3x² and 3 – y + 3x².

Horizontal Method:
There are three variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
7x² + 8y – 9 + 3y + 2 – 3x² + 3 – y + 3x².
Arrange the like terms together.
7x² – 3x² + 3x² + 8y + 3y – y – 9 + 2 + 3.
Now, find the sum of the numerical coefficients of all terms.
7x² + 10y – 4.

The required expression is 7x² + 10y – 4.

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
7x² + 8y – 9,  – 3x² + 3y + 2 and + 3x² – y + 3
Note down the like terms below each other and add them in column-wise.
+ 7x² + 8y – 9
– 3x²  + 3y + 2
+ 3x² – y  + 3
—————————-
7x² + 10y – 4

The required expression is 7x² + 10y – 4.

3. Add 4x² – 2xy + 4y², 3xy – 5y² + 9x² and 2y² + xy – 7x².

Solution:
Given expressions are 4x² – 2xy + 4y², 3xy – 5y² + 9x² and 2y² + xy – 7x².

Horizontal Method:
There are three variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
4x² – 2xy + 4y² + 3xy – 5y² + 9x² + 2y² + xy – 7x².
Arrange the like terms together.
4x² + 9x² – 7x² – 2xy + 3xy + xy + 4y² – 5y² + 2y².
Now, find the sum of the numerical coefficients of all terms.
6x² + 2xy + y².

The required expression is 6x² + 2xy + y².

4. Add 10a² + 3b² – c², 2b² + 6c² – 7a² and 3a² – 8b² – 6c².

Solution:
Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
10a² + 3b² – c², 2b² + 6c² – 7a² and 3a² – 8b² – 6c².
Note down the like terms below each other and add them in column-wise.
+ 10a² + 3b² – c²
– 7a² + 2b² + 6c²
+ 3a² – 8b² – 6c²
—————————-
6a² – 3b² – c²

The required expression is 6a² – 3b² – c².

5. Add the 4x + 8y and 2x + 3y.
Solution:
Given expressions are 4x + 8y and 2x + 3y.

Horizontal Method:
There are two variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
4x + 8y + 2x + 3y
Arrange the like terms together.
4x + 2x + 8y + 3y
Now, find the sum of the numerical coefficients of all terms.
6x + 11y

The required expression is 6x + 11y.

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
4x + 8y and 2x + 3y
Note down the like terms below each other and add them in column-wise.
4x + 8y
2x + 3y
—————————-
6x + 11y

The required expression is 6x + 11y.

6. Add 3x + 9y + 5 and 4x + 3y + 2

Solution:
Given expressions are 3x + 9y + 5 and 4x + 3y + 2.

Horizontal Method:
There are two variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
3x + 9y + 5 + 4x + 3y + 2
Arrange the like terms together.
3x + 4x + 9y + 3y + 5 + 2
Now, find the sum of the numerical coefficients of all terms.
7x + 12y + 7

The required expression is 7x + 12y + 7.

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
3x + 9y + 5 and 4x + 3y + 2
Note down the like terms below each other and add them in column-wise.
3x + 9y + 5
4x + 3y + 2
—————————-
7x + 12y + 7

The required expression is 7x + 12y + 7.

7. Add 12x + 4y + 21z and 32x – 2y – 16z

Solution:
Given expressions are 12x + 4y + 21z and 32x – 2y – 16z.

Horizontal Method:
There are three variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
12x + 4y + 21z + 32x – 2y – 16z
Arrange the like terms together.
12x + 32x + 4y – 2y + 21z – 16z
Now, find the sum of the numerical coefficients of all terms.
44x + 2y -5z

The required expression is 44x + 2y -5z.

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
12x + 4y + 21z and 32x – 2y – 16z
Note down the like terms below each other and add them in column-wise.
12x + 4y + 21z
32x – 2y – 16z
—————————-
44x + 2y -5z

The required expression is 44x + 2y -5z.

8. Add 6x³ – 4y³ and 9x³ – 5y³
Solution:
Given expressions are 6x³ – 4y³ and 9x³ – 5y³.

Horizontal Method:
There are two variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
6x³ – 4y³ + 9x³ – 5y³
Arrange the like terms together.
6x³ + 9x³ – 4y³ – 5y³
Now, find the sum of the numerical coefficients of all terms.
15x³ – 9y³

The required expression is 15x³ – 9y³.

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
6x³ – 4y³ and 9x³ – 5y³.
Note down the like terms below each other and add them in column-wise.
9x³ – 5y³
6x³ – 4y³
—————————-
15x³ – 9y³

The required expression is 15x³ – 9y³.

9. Add 3a² + 5b² + 7c² – 9abc and 2a² – 4b² + 6c² + 8abc
Solution:
Given expressions are 3a² + 5b² + 7c² – 9abc and 2a² – 4b² + 6c² + 8abc.

Horizontal Method:
There are three variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
3a² + 5b² + 7c² – 9abc + 2a² – 4b² + 6c² + 8abc
Arrange the like terms together.
3a² + 2a² + 5b² – 4b² + 7c² + 6c² – 9abc + 8abc
Now, find the sum of the numerical coefficients of all terms.
5a² + b² + 13c² – abc

The required expression is 5a² + b² + 13c² – abc.

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
3a² + 5b² + 7c² – 9abc and 2a² – 4b² + 6c² + 8abc
Note down the like terms below each other and add them in column-wise.
3a² + 5b² + 7c² – 9abc
2a² – 4b² + 6c² + 8abc
—————————-
5a² + b² + 13c² – abc

The required expression is 5a² + b² + 13c² – abc.

10. Add 2xy² + 6x²y – 9x²y – 4xy² + 5 and 3x²y + 2xy²

Solution:
Given expressions are 2xy² + 6x²y – 9x²y – 4xy² + 5 and 3x²y + 2xy².

Horizontal Method:
There are three variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
2xy² + 6x²y – 9x²y – 4xy² + 5 + 3x²y + 2xy²
Arrange the like terms together.
2xy² + 2xy² + 6x²y + 3x²y – 9x²y – 4xy² + 5
Now, find the sum of the numerical coefficients of all terms.
4xy² + 9x²y – 9x²y – 4xy² + 5

The required expression is 4xy² + 9x²y – 9x²y – 4xy² + 5.

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
2xy² + 6x²y – 9x²y – 4xy² + 5 and 3x²y + 2xy²
Note down the like terms below each other and add them in column-wise.
2xy² + 6x²y – 9x²y – 4xy² + 5
2xy² + 3x²y +0    + 0    + 0
—————————-
4xy² + 9x²y – 9x²y – 4xy² + 5

The required expression is 4xy² + 9x²y – 9x²y – 4xy² + 5.

11. Add 2x² + 3y – 4z², 5y + 3x², 4x² + 9z² – 8y and 3y – 3x².

Solution:
Given expressions are 2x² + 3y – 4z², 5y + 3x², 4x² + 9z² – 8y and 3y – 3x².

Horizontal Method:
There are three variables available.
Note down the like terms and then find the sum of the numerical coefficients of all terms.
2x² + 3y – 4z² + 5y + 3x² + 4x² + 9z² – 8y + 3y – 3x²
Arrange the like terms together.
2x² + 3x² + 4x² – 3x² + 3y + 5y – 8y + 3y – 4z² + 9z²
Now, find the sum of the numerical coefficients of all terms.
6x² + 3y + 5z²

The required expression is 6x² + 3y + 5z².

Column Method:
Arrange the given expressions in the same order and write them in rows. Note down the like terms below each other and add them in column-wise.
Rearrange the given expressions.
2x² + 3y – 4z², 3x² + 5y, 4x² – 8y + 9z² and – 3x² + 3y
Note down the like terms below each other and add them in column-wise.
2x² + 3y – 4z²
3x² + 5y + 0
4x² – 8y + 9z²
– 3x² + 3y + 0
—————————-
6x² + 3y + 5z²

The required expression is 6x² + 3y + 5z².