Percentage to Fraction | Convert Percents to Fractions Calculator

During your math calculations, you might need quick conversions between Percentage to Fractions. Then take the help of the Procedure to Convert Percentage to Fraction over here and make your job easy. Know How to Convert a Percent into Fraction by going through the entire procedure listed. Refer to the Solved Examples and learn how to approach them while performing related calculations.

How to Convert a Percent to Fraction?

Follow the below-listed guidelines to change between Percentage to Fraction instantaneously. They are in the following fashion

  • Firstly obtain the given percentage and divide it by 100.
  • Use the number obtained as the numerator of the fraction and place 1 in the denominator of the fraction.
  • If the resultant obtained is a decimal value change it to a whole number. Count the number of places to the right of the decimal and multiply both numerator and denominator with 10x.
  • Reduce the Fraction to its Lowest Form by dividing both the numerator and denominator with their GCF.
  • Simplify the fraction obtained to a mixed number if required.

Solved Examples on Percent to Fraction Conversion

1. Express the following percentage into Fraction and reduce the fraction to its lowest terms?

(i) 14 % (ii) 125% (iii) 2%

Solution:

(i) 14 % = 14/100 = 7/50

(ii) 125% = 125/100 = 5/4

(iii) 2% = 2/100 = 1/50

2. Convert the following percentages to Fraction and reduce to the lowest forms?

(i) 62.5%

Solution:

62.5% = 62.5/100 = 0.625 = 0.625/1

Since the obtained value is a decimal check the number of decimal values to the right and multiply with the 10 raised to that power to make it a whole number.

In 0.625 no. of decimal places = 3

multiply both the numerator and denominator with 103

0.625/1 = (0.625*103)/(1*103)

= 625/1000

Reduce the obtained fraction to the lowest form by simply dividing both the numerator and denominator with the GCF.

625/1000 = (625÷125)/(1000÷125) = 5/8

3. Convert the following percentages to fraction and reduce to the lowest form?

(i) 25 %  (ii) 52 %  (iii) 40%

Solution:

(i) 25 % = 25/100 = 5/20

(ii) 52 % = 52/100 = 26/50

(iii) 40% = 40/100 = 4/10 = 2/5

4. Convert 13% to Fraction?

Solution:

Firstly place the percentage value over 100

= 13/100

Since the numerator is a whole number you can proceed further.

As the fraction can’t be reduced further and has no common divisors that itself is the required fraction value.

5. Convert 12.5% to fraction?

Solution:

Given Percentage = 12.5%

Firstly place the percentage value over 100 i.e. 12.5/100 = 0.125

Place the value obtained as a numerator with a denominator 1 in the fraction i.e. 0.125/1

Since the numerator is a decimal value change it to a whole number.

No. of decimal places to the right of the decimal value is 3. Multiply both the numerator and denominator with 103

= (0.125*103)/(1*103)

= 125/1000

Reduce the obtained fraction to the lowest form by simply dividing both the numerator and denominator with the GCF.

= (125÷125)/(1000÷125)

= 1/8

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Percentage (How to Calculate, Formula and Tricks)

In Maths, a Percentage is a Number or Ratio Expressed as a Fraction Over 100. In other words, percent means parts per hundred and is given by the symbol %. If we need to calculate the percent of a number you just need to divide the number by whole and multiply with 100. Percentages can be represented in any of the forms like decimal, fraction, etc.

You can get entire information regarding Percentage like Definition, Formula to Calculate Percentage, Conversions from Percentage to another form, and vice versa in the coming modules. Learn the Percentage Difference, Increase or Decrease Concepts too that you might need during your academics or in your day to day calculations.

List of Percentage Concepts

Access the concepts that you want to learn regarding the Percentage through the quick links available. Simply tap on the concept you wish to prepare and get the concerned information explained step by step. Clarify all your queries and be perfect in the corresponding topics.

Percentage Formula

To make your calculations quite simple we have provided the Percentage Formula here. Make use of it during your calculations and arrive at the solution easily.

Formula to Calculate Percentage is given by = (Value/Total Value) *100

How to Calculate Percentage of a Number?

To find the Percent of a Number check the following procedure

Consider the number to be X

P% of number = X

Removing the % sign we have the formula as under

P/100*Number = X

Percentage Change

% Change = ((New Value – Original Value)*100)/Original Value

There are two different types when it comes to Percent Change and they are given as under

  • Percentage Increase
  • Percentage Decrease

Percentage Increase

If the new value is greater than the original value that shows the percentage change in the value is increased from the original number. Percentage Increase is nothing but the subtraction of the original number from the new number divided by the original number.

% increase = (Increase in Value/Original Value) x 100

% increase = [(New Number – Original Number)/Original Number] x 100

Percentage Decrease

When the new value is less than the original value, that indicates the percentage change in the value shows the percent decrease in the original number. Percentage Decrease is nothing but the subtraction of new number from the original number.

% Decrease = (Decrease in Value/Original Value) x 100

% Decrease = [( Original Number – New Number)/Original Number] x 100

Percentage Difference

If you need to find the Percentage Difference if two values are known then the formula to calculate Percentage Difference is given by

Percentage Difference = {|N1 – N2|/(N1+N2/2)}*100

Conversion of Fraction to Percentage

To convert fraction to percentage follow the below-listed guidelines.

  • Divide the numerator with the denominator.
  • Multiply the result with 100.
  • Simply place the % symbol after the result and that is the required percentage value.

Conversion of Decimal to Percentage

Follow the easy steps provided to change between Decimal to Percentage. They are as such

  • Obtain the decimal number.
  • Simply multiply the decimal value with 100 to get the percentage value.

Solved Examples on Percentage

1. What is 50% of 30?

Solution:

Given 50% of 30

= (50/100)*30

= 1500/100

= 15

Therefore, 50% of 30 is 15.

2. Find 20% of 40?

Solution:

Given 20% of 40

= (20/100)*40

= (20*40)/100

= 800/100

= 8

3. What is 15% of 60 equal to?

Solution:

= (15/100)*60

= (15*60)/100

= 900/100

= 9

4. There are 120 people present in an examination hall. The number of men is 50 and the number of women is 70 in the examination hall. Calculate the percentage of women present in the examination hall?

Solution:

Number of Women = 70

Percentage of Women = (70/100)*120

= (70*120)/100

= 8400/100

= 84%

The Percentage of Women in the Examination Hall is 84%.

5. What is the percentage change in the rent of the house if in the month of January it was Rs. 20,000 and in the month of March, it is Rs. 15,000?

Solution:

We can clearly say that there is a decrease in the rent

Decreased Value  = 20,000 – 15, 000

= 5, 000

Percent Change = (Decreased Value/Original Value)*100

= (5000/20,000)*100

= (1/4)*100

= 25%

Hence, there is a 25% decrease in the rent.

FAQs on Percentage

1. What is meant by Percentage?

A percentage is a Number or Ratio Expressed as a Fraction Over 100.

2. What is the Formula for Percentage?

The formula for Percentage is (Value/Total Value) *100

3. What is the Symbol of Percentage?

The percentage is denoted by the symbol %.

4. What is 10% of 45?

10% of 45 is given by 10/100*45 i.e. 4.5

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Compound Interest with Periodic Deductions | How to Calculate Periodic Compound Interest?

In this article, you will learn about How to Calculate Compound Interest with Periodic Deductions or Additions to the Amount. Practice the Problems in Periodic Compound Interest and learn how to solve the related problems. To help you understand the concept better we even listed the Formula along with Step by Step Solutions. Check out the Solved Examples for finding the Compound Interest with Periodic Deductions and learn the concept behind them.

Example Questions on Compound Interest with Periodic Deductions

1. Jasmine borrows $ 10,000 at a compound interest rate of 4% per annum. If she repays $ 2,000 at the end of each year, find the sum outstanding at the end of the second year?

Solution:

From the Given Data

Principal = $ 10, 000

Interest Rate = 4% Per Annum

Time = 1 Year

Interest = PTR/100

= 10,000*1*4/100

= $400

Amount of the loan after 1 year = Principal + Interest

= $10,000 + $400

= $10, 400

Jasmine Repays $2,000 after the first year

Therefore, New Principal for the 2nd year = $10, 400 – $2,000

= $8,400

Thus for 2nd Year Principal = $8, 400

Interest = 4%

Time = 1 Year

Interest = PTR/100

= 8,400*1*4/100

= $336

Amount after 2nd Year = Principal + Interest

= $8400+$336

= $8736

Therefore, Jasmine needs to pay an outstanding amount of $8736 dollars by the end of the 2nd Year.

2. David invests $ 10,000 at the beginning of every year in a bank and earns 5 % annual interest, compounded at the end of the year. What will be his balance in the bank at the end of two years?

Solution:

From the Given Data

Principal = $10, 000

Rate of Interest = 5%

Time = 1 Year

Interest = PTR/100

= 10,000*1*5/100

= $500

Therefore Amount after 1 Year = Principal + Interest

= $10,000+$500

= $10, 500

David deposits 10,000 at the beginning of the second year

New Principal becomes = $10, 500+ $10, 000

= $20, 500

Thus, for the 2nd Year

Principal = $20, 500

Interest Rate = 5%

Time = 1 Year

Interest = PTR/100

= $20, 500*1*5/100

= $1025

Amount after the 2nd Year = Principal + Interest

= $20, 500 + $1025

= $21, 525

Therefore, David will have $21, 525 in the bank after the end of 2 Years.

3. John lent $ 5,000 at a compound interest rate of 10% per annum. If he repays $ 500 at the end of the first year and $ 1,000 at the end of the second year, find his outstanding loan at the beginning of the third year?

Solution:

From the Given Data

Principal = $ 5,000

Interest rate = 10%

Time = 1 Year

Interest = PTR/100

= 5000*1*10/100

= $500

Amount after 1 year = Principal + Interest

= $5000+$500

= $5500

John repays $500 at the end of first year thus new principal = $5500 – $500

= $5000

Thus, For 2nd Year

New Principal = $5000

Time = 1 Year

Interest Rate = 10%

Interest = PTR/100

= $5000*1*10/100

= $500

Amount after 2nd Year = Principal + Interest

= $5,000+$500

= $5500

John repays $1000 after the end of 2nd year

Thus for third year New Principal = $5500 – $1000

= $4500

Therefore the outstanding loan at the beginning of the third year is $4500.

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Compound Interest by Using Formula | How to Calculate Compound Interest using Formula?

Do you feel any difficulty in calculating the Compound Interest? Not anymore after going through this article. Finding Compound Interest using the Formula is quite simple and you don’t have to do hectic calculations, unlike the manual methods. You just need to substitute the inputs and perform basic maths to obtain the Calculate Compound Interest instantly.

For the sake of your convenience, we have listed the Formulas for finding Compound Interest Annually, Half-Yearly, Quarterly along with Solved Examples. Refer to the Step by Step Solutions provided and understand the concept easily.

Compound Interest Formula in Different Cases

In general, Compound Interest is the Interest calculated on the Principal and the Interest accumulated over the previous period. We have listed the Compound Interest Formulas in various cases like When Interest Rate is Compounded Annually, Half-Yearly, Quarterly in the coming modules by taking enough examples.

  • Compound Interest Formula when Rate is Compounded Annually
  • Compound Interest when Rate is Compounded Half-Yearly
  • Compound Interest Quarterly Formula
  • When the Interest is Compounded Annually but rates are different for different years
  • Interest is compounded annually but time is a fraction

Annually Compound Interest Formula

We know the Formula to Calculate the Amount is A = P(1+r/n)nt

Where A= Amount

P= Principal

R= Rate of Interest

n= Number of times interest is compounded per year

If the Interest Rate is Compounded Annually we have the Formula as A = P(1+R/100)t

CI = A – P

Examples

1. Find the amount of $4000 for 2 years, compounded annually at 5% per annum. Also, find the compound interest?

Solution:

We know the formula to calculate Amount is A = P(1+R/100)n

P = $4000, R = 5%, n = 2 years

Substitute the input data we get the equation as such

A = 4000(1+5/100)2

= 4000(105/100)2

= $4410

CI = A – P

= $4410- $4000

= $410

Therefore, Compound Interest is $410.

2. Calculate the compound interest (CI) on Rs. 3000 for 1 year at 10% per annum compounded annually?

Solution:

The Formula to Calculate the Amount is A = P(1+R/100)n

P = Rs. 3000, n = 1 year R = 10%

Substituting the input data in the formula and we get

A = 3000(1+10/100)1

= 3000(1.1)

= Rs. 3300

CI = A – P

= 3300 – 3000

= Rs. 300

Half-Yearly Compound Interest Formula

Let us calculate the Compound Interest on a Principal P kept for 1 Year and at an Interest Rate R% compounded half-yearly

As the Interest is compounded half-yearly Interest Amount will vary after the first 6 months. The Interest for the next 6 months will be calculated on the remaining amount after the first 6 months.

Principal = P, Rate = R/2 %, Time = 2n

A = P(1+R/100)n

Substitute R/2 and 2n in terms of Rate and Time in the above formula

A = P(1+R/2*100)2n

CI = A – P

Example

Calculate the compound interest to be paid on a loan of Rs. 20,000 for 3/2 years at 10% per annum compounded half-yearly?

Solution:

From the given data

Principal = 20,000

R = 10%

n = 3/2

A= P(1+R/2*100)2n

Substitute the input values in the formula and we have

A = 20,000(1+10/200)2*3/2

= 20, 000(1+10/200)3

= 20,000(1.157)

= Rs. 23152

CI = A – P

= 23,152 – 20,000

= Rs. 3152

Compound Interest Quarterly Formula

Let us find the Compound Interest Kept on a Principal for 1 year and a Rate of R% compounded quarterly. Since CI is compounded quarterly principal amount will change after the first 3 months. The next 3 months(second quarter) interest will be calculated on the amount left after the first 3 months. Third-quarter Interest will be calculated on the amount left after the first 6 months. Last quarter will be found on the amount left after the first 9 months.

The formula of Compound Interest Compounded Quarterly is given as

A = P(1+R/4/100)4n

CI = A – P

Example

Find the compound interest on $12,000 if Nick took a loan from a bank for 6 months at 8 % per annum, compounded quarterly?

Solution:

From the given data P = $12, 000

R = 8% per Annum, (8/4)% per quarter = 2% per quarter

T = 6 months = 2 Quarters

A = P(1+R/100)n

= 12,000(1+2/100)2

= 12,000(1+0.02)2

= 12,000(1.02)2

= Rs. 12,484

CI = A – P

= 12,484 – 12,000

= Rs. 484

When the Interest is Compounded Annually but rates are different for different years

Suppose the Interest Compounded Annually be different in different years. In the first year if the Interest Rate is p % per annum and for the second year if it is q % then

Amount Formula is given by = P*(1+p/100)*(1+q/100)

This formula can be extended for any number of years.

To get the Compound Interest Subtract Principal from Amount i.e. CI = A – P

Example

Find the amount of $10, 000 after 2 years, compounded annually, if the rate of interest being 3 % p.a. during the first year and 4 % p.a. during the second year. Also, find the compound interest?

Solution:

Formula to Calculate the Amount is A = P*(1+p/100)*(1+q/100)

From the given data P = $10, 000, p = 3%, q = 4%

Substitute the inputs in the formula to calculate the Amount and the equation is as under

A = 10,000(1+3/100)(1+4/100)

= 10, 000(1.03)(1.04)

= $10712

CI = A – P

= $10712 – $10, 000

= $712

Interest is Compounded Annually but time is a fraction

For instance, if time is 5 3/4 years then Amount is given as under

A = P * (1 + R/100)5 * [1 + (3/4 × R)/100]

Example

Find the compound interest on $ 30,000 at 6 % per annum for 3 years. Solution Amount after  3 3/4 years?

Solution:

Amount after 3 3/4 years is given by A = $ [30,000 × (1 + 6/100)3 × (1 + (3/4 × 8)/100)]

= $[30,000 * (1 + 0.06)3 * (1 + 6/100)]

=$[30,000*(1.06)3*(1.06)]

= $37874

CI = A – P

= $37874 – $30, 000

= $874

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Decimal to Percentage | Decimal to Percent Conversion Calculator

You will no longer feel the Decimal to Percentage Conversion difficult anymore with our article. Get the detailed procedure to convert percent to a decimal and perform required conversions easily. Furthermore, have a look at the example problems provided and clarify all your concerns while solving related problems.

How to Write a Percent as a Decimal?

Follow the below-provided steps and convert percent to decimal easily. They are as under

  • Obtain the number in decimal form
  • Convert from decimal to a percentage by simply multiplying with 100 and remove the % sign.

Another method for converting percentage to decimal is to simply move the decimal point 2 places to the right and add a percent(%) sign.

Solved Examples on Conversion of Decimal to Percent

1. Express Decimal 0.13 as Percentage?

Solution:

Given Decimal 0.13

Multiply the decimal value with 100 and place the % symbol after the result

0.13*100

= 13%

Therefore, 0.13 in Percentage is 13%.

2. Convert the following decimal values to Percent?

(i) 0.073 (ii) 0.002 (iii) 1.012

Solution:

(i) 0.073

Multiply the decimal value with 100 and place the % symbol after the result

0.073*100 = 7.3%

(ii) 0.002

Multiply the decimal value with 100 and place the % symbol after the result

0.002*100 = 0.2%

(iii) 1.012

Multiply the decimal value with 100 and place the % symbol after the result

1.012*100 = 101.2%

3. Convert the following decimals to Percentage?

(i) 0.4 (ii) 0.54

Solution:

(i) 0.4

Simply move the decimal point 2 places to the right and add a percent(%) sign.

0.4 → 40

Therefore, 0.4 as a Percentage is 40%.

(ii) 0.54

Simply move the decimal point 2 places to the right and add a percent(%) sign.

0.54 → 54

Therefore, 0.54 as a Percentage is 54%.

4. Convert 0.935 to percent?

Solution:

Simply move the decimal point 2 places to the right and add a percent(%) sign.

0.935 → 93.5%

Therefore, 0.935 as a Percentage is 93.5%.

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Difference of Compound Interest and Simple Interest | How to Calculate Difference Between SI and CI?

The Major Difference Between Simple Interest and Compound Interest is just that Simple Interest is calculated on the Principal whereas Compound Interest is calculated on the Principal Amount along with the Interest accumulated for a certain period of time. Both Simple and Compound Interest are widely used concepts in the majority of financial services. Check out Solved Example Problems for finding the difference between CI and SI in the later sections. Get to know about the concept Difference of Compound Interest and Simple Interest in detail by going through the entire article.

How to find the Difference Between Simple Interest and Compound Interest?

Let us discuss in detail how to find the Difference between Simple Interest and Compound Interest. They are along the lines

Consider the Rate of Interest is the same for both Compound Interest and Simple Interest

Difference = Compound Interest for 2 years – Simple Interest for 2 Years

= P{(1+r/100)2-1} – P*r*2/100

= P*r/100*r/100

=((P*r/100)*r*1)/100

Solved Examples on Difference of Compound Interest and Simple Interest

1. Find the difference of the compound interest and simple interest on $ 10,000 at the same interest rate of 10 % per annum for 2 years?

Solution:

Simple Interest = PTR/100

= 10,000*10*2/100

= $2000

To find the Compound Interest firstly calculate the Amount

Amount A = P(1+R/100)n

A = 10,000(1+10/100)2

= 10,000(110/100)2

= 10,000(1.21)

= $12,100

Compound Interest = Amount – Principal

= 12,100 – 10,000

= $2,100

Difference between CI and SI = CI for 2 Years – SI for 2 Years

= $2100- $2000

= $100

2. What is the sum of money on which the difference between simple and compound interest in 2 years is $ 100 at the interest rate of 5% per annum?

Solution:

Simple Interest = PTR/100

Principal = P

T = 2 years

Substituting the given data in the formula for the simple interest we have

SI = (P*2*5)/100

To find the Compound Interest firstly, find out the Amount

Amount A = P(1+R/100)n

= P(1+5/100)2

CI = Amount – Principal

= P(1+5/100)2 – P

= P((1+5/100)2 -1)

Given the difference between CI and SI = $100

P((1+5/100)2 -1) – (P*2*5)/100 = $100

P((105/100)2 -1)-10P/100 = $100

P(1.1025-1)-10P/100 = $100

100P(0.1025)-10P =$10000

110.25P-10P = $10000

100.25P = $10000

P = $10000/100.25

= $99.75

Therefore, the Sum of Money is $99.75

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Uniform Rate of Growth | How to Calculate Uniform Growth Rate?

In our Day-Day Lives, we see entities such as Population of a City, Value of a Property, Weight of a Child, Height of a Tree that grow over a period of time. We call the Increase in Quantity as Growth and the Growth Per Unit Time is known as the Rate of Growth. If the Growth Rate occurs at the Same Rate then we call it Uniform Increase or Uniform Growth Rate. Check the Formulas for Uniform Rate of Growth and the Solved Problems for finding the Principal of Compound Interest in Uniform Rate of Growth explained step by step.

How to find the Uniform Rate of Growth?

Learn how to calculate the uniform growth rate by referring to the below modules.

If the Present Value P of a quantity increases at the rate of r % per unit of time then the value Q of the quantity after n units of time is as such

Q = P(1+r/100)n

Growth is obtained by subtracting the Increased Value from the Actual Value

Growth = Q – P

= P(1+r/100)n – P

= P{(1+r/100)n -1}

Solved Examples on Principal of Compound Interest in Uniform Rate of Growth

1. The population of the town increases by 8% every year. If the present population is 7000, what will be the population of the town after 2 years?

Solution:

Given Current Population = 7000

n = 2 Years

Rate of Interest = 8%

Q = P(1+r/100)n

Substitute the input values in the above formula

= 7000(1+8/100)2

= 7000(1+0.08)2

= 7000(1.08)2

= 7000(1.1664)

= 8164

Population of a Town after 2 years is 8164.

2. John buys a plot of land for $ 20,000. If the value of the land appreciates by 10% every year then find the profit that John will make by selling the plot after 3 years?

Solution:

P = $20,000

interest rate = 10%

n = 3 years

Q = P(1+r/100)n

= 20,000(1+10/100)3

= 20,000(1+0.1)3

= 20,000(1.1)3

= $26620

Profit made by John = $26620 – $20,000

= $6,620

3. Mike purchased a bike for Rs. 45,000. If the cost of his bike is appreciated at a rate of 5% per annum, then find the cost of the bike after 3 years?

Solution:

Initial Price = Rs. 45, 000

Rate of Appreciation = 5 % Per Annum

n = 3 years

Q = P(1+r/100)n

= 45,000(1+5/100)3

= 45,000(1+0.05)3

= 45,000(1.05)3

= Rs. 52093

Therefore, the Cost of the Bike after 3 years is Rs. 52093.

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Compound Interest when Interest is Compounded Yearly| How to Calculate Compound Interest Annually?

Let us learn How to Calculate Compound Interest when Interest Rate is Compounded Yearly in the coming modules. Check out Formula, Solved Examples on finding the Compound Interest Annually. We tried explaining each and every step on how to find Compound Interest. Computing the Compound Interest using Growing Principal can be difficult if the time period is long.

How to find Compound Interest when Interest is Compounded Annually?

If the Interest is Compounded Annually then Formula to Calculate the Compound Interest is given by

A = P(1+r/100)n

Where A is the Amount

P = Principal

r = rate of interest per unit time

n = Time Duration

CI can be obtained by subtracting the Principal from the Amount

CI = A – P

= P(1+r/100)n – P

= P{1+r/100)n – 1}

Solved Examples on Compound Interest when Interest is Compounded Annually

1. Find the amount and the compound interest on $8, 000 in 2 years and at 5% compounded yearly?

Solution:

Principal = $8, 000

r = 5%

n = 2

A = P(1+r/100)n

Substitute the Input Values in the formula of Amount

A = 8,000(1+5/100)2

= 8000(1+0.05)2

= 8000(1.05)2

= $8820

CI = A – P

= $8820 – $8000

= $820.

2. Find the amount of $12,000 for 2 years compounded annually, the rate of the interest being 5 % for the 1st year and 6 % for the second year?

Solution:

A = P*(1+p/100)*(1+q/100)

= 12,000(1+5/100)(1+6/100)

= 12,000(1.05)(1.06)

= $13356

Amount after 2 years is $13356.

3. Calculate the compound interest (CI) on Rs. 10, 000 for 3 years at 8% per annum compounded annually?

Solution:

Principal = Rs. 10,000

n = 3

r = 8%

A = P(1+r/100)n

= 10,000(1+8/100)3

= 10,000(1.08)3

= Rs. 12,597

CI = A – P

= 12, 597 – 10, 000

= Rs. 2, 597

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Speed of Train | How to Calculate the Motion or Speed of a Train?

Learn What is meant by Speed and How to Calculate the Motion or Speed by going through the entire article. Check out Formula to find the Speed of a Train when Train Passes through a Stationary Object and Train Passing through a Stationary Object of Certain Length. We have mentioned Solved Examples explaining step by step for a better understanding of the concept. Learn about Time and Distance Concepts too from here and get a good hold of the concept by practicing the sample problems available on a regular basis.

How to find the Speed of a Train?

There are two different scenarios to calculate the Speed of a Train. We explained each scenario in detail by taking a few examples. Understand the concept behind them easily. They are as under

  • Train Passes through a Stationary Object
  • Train Passes through a Stationary Object having Some Length

When a Train Passes through a Stationary Object

Consider the length of the train along with the engine be x. When the end of the train passes the object then the engine of the train travels the distance equal to the train length.

Time taken by train to pass the stationary object = length of the train/speed of the train

When a Train Passes through a Stationary Object having Some Length

When the end of the train passes the stationary object having a certain length, then the engine of the train moves a distance equal to the sum of both the length of the train and the stationary object.

Time taken by the train to pass stationary object = (length of the train+length of the stationary object)/speed of the train

Solved Examples on Speed of a Train

1. Find the time taken by a train 125 m long, running at a speed of 108 km/hr in crossing the pole?

Solution:

Length of the Train = 125m

Speed of Train = 108 km/hr

= 108*5/18

= 30 m/sec

Time taken by train to cross the pole = 125m/30m/sec

= 4.1 sec

2. A train 320 m long is running at a speed of 60 km/hr. what time will it take to cross a 160 m long tunnel?

Solution:

Length of the train = 320m

Length of the tunnel = 160m

Length of Train+Length of Tunnel = (320+160) = 480m

Speed of train = 60 km/hr

= 60*5/18

= 16.66 m/sec

Time taken by train to cross the tunnel = 480m/16.66m/sec = 28.8 sec

3. A train is running at a speed of 75 km/hr. if it crosses a pole in just 15 second, what is the length of the train?

Solution:

Speed of the train = 75 km/hr

Speed of the train = 75 × 5/18 m/sec = 20.83 m/sec

Time taken by the train to cross the pole = 15 seconds

Therefore, length of the train = 20.83 m/sec × 15 sec = 312.5 m

4. A train 250 m long crosses the bridge 120 m in 20 seconds. Find the speed of the train in km/hr?

Solution:

Length of the train = 250m

Length of the Bridge = 120 m

Total Length = Length of Train + Lenth of Bridge

= 250+120

= 370 m

Time taken to cross the bridge = 20 seconds

Speed of the Train = Length/Time = 370m/20 sec = 18.5 m/sec

To convert m/sec to km/hr multiply with 18/5

Speed of the Train in km/hr = 18.5*18/5

= 66.6 km/hr

Therefore, the speed of the train is 66.6 km/hr.

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Domain and Range of a Relation

Domain and Range of a Relation

Before learning about Domain and Range of a Relation firstly know what Relations are. A relation is a rule that relates an element from one set to the other set. Consider two non-empty sets A and B then the relation is a subset of Cartesian Product AxB.

The domain is the set of all first elements of the ordered pairs. The range on the other hand is the set of all second elements of the ordered pairs. However, Range includes only the elements used by the function. There lies in a trick in the range i.e. Set B can be equal to the range of relation or bigger than that. This is because there can be elements in Set B that aren’t related to Set A.

Domain and Range Definition

If R be a relation from Set A to Set B then the Set of first elements belonging to the ordered pair is called the Domain of R. We can represent the Domain as such

Dom(R) = {a ∈ A: (a, b) ∈ R for some b ∈ B}

The Set of Second Components belonging to the ordered pair is called the Range of R. It can be denoted as follows

R = {b ∈ B: (a, b) ∈R for some a ∈ A}

Thus, Domain and Range are given by Domain (R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}.

Solved Examples on Domain and Range of a Relation

1. State the domain and range of the following relation: (eye color, student’s name).
A = {(blue, John), (green, William), (brown,Wilson), (blue, Moy), (brown, Abraham), (green, Dutt)}. State whether the relation is a function?

Solution:

Domain: {blue, green, brown} Range: {John, William, Wilson, Moy, Abraham, Dutt}

No, the relation is not a function since the eye colors are repeated.

2. State the domain and range of the following relation: {(4,3), (-1,7), (2,-3), (7,5), (6,-2)}?

Solution:

The domain is the first component of the ordered pairs. Whereas, Range is the Second Component of the ordered pairs. Remove the duplicates if any are present.

Domain = {4, -1, 2, 7, 6} Range = {3, 7, -3, 5, -2}

3. From the following Arrow Diagram find the Domain and Range and depict the relation between them?

Solution:

Arrow Diagram

Domain = {3, 4, 5}

Range = {3, 4, 5, 6}

R = {(3, 4), (4, 6), (5, 3), (5, 5)}

4. Determine the domain and range of the relation R defined by

R = {x – 2, 2x + 3} : x ∈ {0, 1, 2, 3, 4, 5}

Solution:

Given x = {0, 1, 2, 3, 4, 5}

x = 0 ⇒ x – 2 = 0 – 2 = -2 and 2x + 3 = 2*0 + 3  = 3

x = 1 ⇒ x-2 = 1-2 = -1 and 2x+3 = 2*1+3 = 5

x = 2 ⇒ x-2 = 2-2 = 0 and 2x+3 = 2*2+3 = 7

x = 3 ⇒ x-2 = 3-2 = 1 ad 2x+3 = 2*3+3 = 9

x = 4 ⇒ x-2 = 4-2 = 2 and 2x+3 = 2*4+3 = 11

x = 5 ⇒ x-2 = 5-2 =3 and 2x+3 = 2*5+3 = 13

Hence R = {-2, 3), (-1, 5), (0, 7), (1, 9), (2, 11), (3, 13)

Domain of R = {-2, -1, 0, 1, 2, 3}

Range of R = {3, 5, 7, 9, 11, 13}

5. The below figure shows a relation between Set x and Set y. Write the same in Roster Form, Set Builder Form, and find the domain and Range?

Solution:

In the Set Builder Form R = {(x, y): x is the square of y, x ∈ X, y ∈ Y}

In Roster Form R = {(2, 1)(4, 2)}

Domain = {2, 4}

Range = {1, 2}

6.  The Arrow Diagram Shows the Relation R from Set C to Set D. Write the relation R in Roster Form?

Solution:

We observe the relation R using the Arrow Diagram Above

From that Relation R in Roster Form = {(2,20) ; (2, 40) ; (4, 40) ; (3, 30)}

Rational Numbers

Rational Numbers | Definition, Types, Properties, Standard Form of Rational Numbers

In Maths, Rational Numbers sound similar to Fractions and they are expressed in the form of p/q where q is not equal to zero. Any fraction that has non zero denominators is called a Rational Number. Thus, we can say 0 also a rational number as we can express it in the form of 0/1, 0/2 0/3, etc. However, 1/0, 2/0 aren’t rational numbers as they give infinite values.

Continue reading further modules to learn completely about Rational Numbers. Get to know about Types of Rational Numbers, Difference Between Rational and Irrational Numbers, Solved Examples, and learn how to Identify Rational Numbers, etc. In order to represent Rational Numbers on a Number Line firstly change them into decimal values.

Definition of Rational Number

Rational Number in Mathematics is defined as any number that can be represented in the form of p/q where q ≠ 0. On the other hand, we can also say that any fraction fits into the category of Rational Numbers if bot p, q are integers and the denominator is not equal to zero.

How to Identify Rational Numbers?

You need to check the following conditions to know whether a number is rational or not. They are as follows

  • It should be represented in the form of p/q, where q ≠ 0.
  • Ratio p/q can be further simplified and expressed in the form of a decimal value.

The set of Rational Numerals include positive, negative numbers, and zero. It can be expressed as a Fraction.

Examples of Rational Numbers

p q p/q Rational
20 4 20/4 =5 Rational
2 2000 2/2000 = 0.001 Rational
100 10 100/10 = 10 Rational

Types of Rational Numbers

You can better understand the concept of sets by having a glance at the below diagram.

Rational Numbers

  • Real numbers (R) include All the rational numbers (Q).
  • Real numbers include the Integers (Z).
  • Integers involve Natural Numbers(N).
  • Every whole number is a rational number as every whole number can be expressed in terms of a fraction.

Standard Form of Rational Numbers

A Rational Number is said to be in its standard form if the common factors between divisor and dividend is only one and therefore the divisor is positive.

For Example, 12/24 is a rational number. It can be simplified further into 1/2. As the Common Factors between divisor and dividend is one the rational number 1/2 is said to be in its standard form.

Positive and Negative Rational Numbers

Positive Rational Numbers Negative Rational Numbers
If both the numerator and denominator are of the same signs. If numerator and denominator are of opposite signs.
All are greater than 0 All are less than 0
Example: 12/7, 9/10, and 3/4 are positive rational numbers Example: -2/13, 7/-11, and -1/4 are negative rational numbers

Arithmetic Operations on Rational Numbers

Let us discuss how to perform basic operations i.e. Arithmetic Operations on Rational Numbers. Consider p/q, s/t as two rational numbers.

Addition: Whenever we add two rational numbers p/q, s/t we need to make the denominator the same. Thus, we get (pt+qs)/qt.

Ex: 1/3+3/4 = (4+3)/12 = 7/12

Subtraction: When it comes to subtraction between rational numbers p/q, s/t we need to make the denominator the same and then subtract.

Ex: 1/2-4/3 = (3-8)/6 = -5/6

Multiplication: While Multiplying Rational Numbers p/q, s/t simply multiply the numerators and the denominators of the rational numbers respectively. On multiplying p/q with s/t then we get (p*s)/(q*t)

Ex: 1/3*4/2=4/6

Division: Division of p/q & s/t is represented as (p/q)÷(s/t) = pt/qs

Ex: 1/4÷4/3 =1*3/4*4 = 3/16

Properties of Rational Numbers

  • If we add a zero to a Rational Number you will get the Rational Number Itself.
  • Addition, Subtraction, Multiplication of a Rational Number yields in a Rational Number.
  • Rational Number remains the same on multiplying or dividing both the numerator and denominator with the same factor.

There are few other properties of rational numbers and they are given as under

  • Closure Property
  • Commutative Property
  • Associative Property
  • Distributive Property
  • Identity Property
  • Inverse Property

Representation of Rational Numbers on a Number Line

Number Line is a straight line diagram on which each and every point corresponds to a real number. As Rational Numbers are Real Numbers they have a specific location on the number line.

Rational Numbers Vs Irrational Numbers

There is a difference between Rational Numbers and Irrational Numbers. Fractions with non zero denominators are called Rational Numbers. All the numbers that are not Rational are Called Irrational Numbers. Rational Numbers can be Positive, Negative, or Zero. To specify a negative Rational Number negative sign is placed in front of the numerator.

When it comes to Irrational Numbers you can’t write them as simple fractions but can represent them with a decimal. You will endless non-repeating digits after the decimal point.

Pi (π) = 3.142857…

√2 = 1.414213…

Solved Examples

Example 1.

Identify whether Mixed Fraction 1 3/4 is a Rational Number or Not?

Solution: The Simplest Form of Mixed Number 1 3/4 is 7/4

Numerator = 7 which is an integer

Denominator = 4 which is an integer and not equal to 0.

Thus, 7/4 is a Rational Number.

Example 2.

Determine whether the given numbers are rational or irrational?

(a) 1.45 (b) 0.001 (c) 0.15 (d) 0.9 (d) √3

Solution:

Given Numbers are in Decimal Format and to find out whether they are rational or not we need to change them into fraction format i.e. p/q. If the denominator is non zero then the number is rational or else irrational.

Decimal Number Fraction Rational Number
1.45 29/20 Yes
0.001 1/1000 Yes
0.15 3/20 Yes
0.9 9/10 Yes
√ 3 1.732… No

FAQs on Rational Numbers

1. How to Identify a Rational Number?

If the Number is expressed in the form of p/q where p, q are integers and q is non zero then it called a Rational Number.

2. Is 5 a Rational Number?

Yes, 5 is a Rational Number as it can be expressed in the form of 5/1.

3. What do we get on adding zero to a Rational Number?

On Adding Zero to a Rational Number, you will get the Same Rational Number.

4. What is the difference between Rational and Irrational Numbers?

Rational Numbers are terminating decimals whereas Irrational Numbers are Non-Terminating Decimals.