Glories to Duranteshwar Mahadev and Aadyanagha Mahadevi 🙏!
Well! Many of you all must have had a phobia for mathematics (or some of you still have it). Right?
This blog aims at removing that phobia for mathematics. We have curated the content to simplify mathematical concepts so that you all do not have to spend extra money in coaching classes! That too free of cost!
This blog aims at focusing on :
1. Fractions
2. Decimals
3. Percentages
4. Averages
5. Ratios and Proportions.
Do refer to our introductory article on mathematical tricks for further assistance.
Mathematical Tricks by The Aadyanagha Foundation
Chapter 1 - Fractions
Think that it is your birthday and you have been gifted a chocolate cake. Of course you will not be able to eat the full cake as you will be sharing the cake among your friends, family and other guests in the birthday party.
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| A small slice cut from a whole cake |
So the cake is cut into 24 equal parts. Out of that, you eat 5 slices. So you ate only a portion of the cake.
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| Representation of a Fraction |
Fraction = A part of the whole
Numerator = Upper Part of the Fraction
Denominator = Lower Part of the Fraction; It is actually the total number of parts the whole is divided into.
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| Whole, Equal and Unequal Parts |
Now think of a situation. Can you say that the yellow circle is divided into equal parts? Can you say that the lower and upper parts of the circle are 1/2? No.
Hence, to determine a fraction correctly, the given whole must be divided into equal parts.
Proper, Improper and Mixed Fractions
i. Proper Fraction: When numerator is less than the denominator. Example - 2/3, 3/4, 4/5, 1/10, etc.
ii. Improper Fraction: When the numerator is more than the denominator. Example - 5/4, 4/3, 3/2, 23/12, 45/32, etc
iii. Mixed Fraction: Representation of an improper fraction into a whole number and proper fraction.
Examples –
5/4 is represented as (1 + 1/4 = 4/4 + 1/4).
3/2 is represented as (1 + 1/2 = 2/2 + 1/2).
Like and Unlike Fractions
Let us take a situation. There are three best friends. Mickey, Minnie and Jerry. They are very happy that they passed their examinations. They all go to a pizza shop and each buys an equal sized pizza for himself / herself.
Minnie's pizza is divided into 8 equal parts. Mickey's pizza is divided into 8 equal parts. Jerry's pizza is divided into 4 equal parts.
Minnie ate 3 slices from her pizza while Mickey ate 5 slices from his pizza. Since both the pizzas were cut into an equal number of slices, the slices were of the same size. Hence, it was easy to tell that Mickey is less hungry than Minnie now. The fractions here are like fractions as the denominators are same in both the fractions.
Now. Jerry too ate 3 slices from his pizza but he is less hungry than Minnie. How is that possible? Both ate 3 slices. Since both the pizzas were cut into an unequal number of slices, the slices were of a different size. Since Jerry's pizza was cut into lesser number of slices, his slices were of a larger size. The fractions here are unlike fractions as the denominators are different in both the fractions.
Like Fraction: When denominator of the set of fractions is same. Example - 2/5, 3/5 and 4/5.
When the denominator of given fractions is the same, then the fraction with the higher numerator has the higher value.
For example, 2/5 < 3/5 < 4/5.
Unlike Fraction: When denominator of the set of fractions is different. Example - 2/3 and 3/4. To compare them, we have to create equivalent fractions where the lowest common multiple of the denominators of both the fractions is the final denominator of both the fractions.
Equivalent and Simplified Fractions
Equivalent Fraction: When the numerator and denominator are multiplied by the same number, they cause such fractions. Example - Equivalent Fractions of 2/3 are 4/6, 6/9, 8/12, 10/15 and so on. Equivalent fractions of 3/4 are 6/8, 9/12, 12/16, 15/20 and so on.
Illustrations:
2/3 = (2 × 2)/(3 × 2) = 4/6
2/3 = (2 × 3)/(3 × 3) = 6/9
2/3 = (2 × 4)/(3 × 4) = 8/12
2/3 = (2 × 5)/(3 × 5) = 10/15
3/4 = (3 × 2)/(4 × 2) = 6/8
3/4 = (3 × 3)/(4 × 3) = 9/12
3/4 = (3 × 4)/(4 × 4) = 12/16
3/4 = (3 × 5)/(4 × 5) = 15/20
When the numerator of given fractions is same, then the fraction with the lower denominator has the higher value.
For example, there are two pizzas of equal sizes. One pizza is divided into 2 parts and the other pizza is divided into 4 parts.
The 1 slice of the first pizza would be bigger in size than the 1 slice of the other pizza. Hence, 1/2 is greater than 1/4.
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| Visual illustration of an equivalent fraction |
Simplified Fraction: When the numerator and denominator are divided by their highest common factor (greatest common divisor), they cause such fractions.
Illustrations :
1. 30/35 = (30 ÷ 5)/(35 ÷ 5) = 6/7
2. 42/48 = (42 ÷ 6)/(48 ÷ 6) = 7/8
3. 56/63 = (56 ÷ 7)/(63 ÷ 7) = 8/9
4. 50/55 = (50 ÷ 5)/(55 ÷ 5) = 10/11
Addition and Subtraction of Like Fractions
1. 4/7 + 2/7 = (4 + 2) / 7 = 6/7 (Denominator is the same)
2. 6/11 – 4/11 = (6 – 4)/11 = 2/11 (Denominator is same)
3. 7/9 + 2/9 = (7 + 2) / 9 = 9 / 9 = 1 (Since numerator is equal to denominator, the fraction becomes a complete whole; hence it becomes 1)
Addition and Subtraction of Unlike Fractions
1. 4/5 + 2/3
= (4 × 3) / (5 × 3) + (2 × 5) / (3 × 5)
= 12/15 + 10/15 = 22/15
= (15 + 7)/15
= 15/15 + 7/15
= 1 + 7/15
2. 4/5 - 2/3
= (4 × 3) / (5 × 3) - (2 × 5) / (3 × 5)
= 12/15 - 10/15
= 2/15
In the given numerical, we see that the denominators are different for both 4/5 and 2/3. So, we are using the concept of equivalent fractions to convert both fractions into like fractions by making the denominators same. The lowest common multiple would be the denominator of both fractions.
Step 1: We multiply 4 and 5 by 3 and get 12/15. Similarly, we multiply 2 and 3 by 5 and get 10/15.
Step 2: After adding 12/15 and 10/15, we see an improper fraction is formed (numerator > denominator). We convert it into a mixed fraction (representing an improper fraction as a whole number and proper fraction). Since the denominator is 15, we split 22 as 15 + 7. Then we show 22/15 as 15/15 + 7/15. In 15/15, 15 gets cancelled out and we get the number as 1. Hence 22/15 becomes 1 + 7/15.
Let’s solve 4/15 + 3/20
We first create an equivalent fraction for both to convert to them like fractions
We know that 60 is the lowest common multiple for both 15 and 20.
4/15 = (4 × 4)/(15 × 4) = 16/60
3/20 = (3 × 3)/(20 × 3) = 9/60
16/60 + 9/60 = 25/60 = (5 × 5)/(12 × 5) = 5/12
Multiplication of Fractions
1. 2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15
2. 2/5 × 7/8 = (2 × 7) / (5 × 8) = 7 / (5 × 4) = 7/20 (since 2 divides 8, so 2 becomes 1 and 8 becomes 4)
3. 4/15 × 9/10 = (4 × 9) / (15 × 10) = (2 × 3) / (5 × 5) = 6/25 (because 2 is a common factor of 4 and 10 ; while 3 is a common factor of 9 and 15)
Division of Fractions
1. 2/3 ÷ 4/5 = 2/3 × 5/4 = (2 × 5) / (3 × 4) = 5 / (3 × 2) = 5/6
2. 6/7 ÷ 8/9 = 6/7 × 9/8 = (6 × 9) / (7 × 8) = (3 × 9) / (7 × 4) = 27/28
Chapter 2 - Decimals
Let’s take a situation. We know 1 over = 6 balls.
In the first over, India scores 12 runs.
In the second over, India scores 4 runs.
In the third over, India scores 14 runs.
In the fourth over, India scores 8 runs.
In 4 overs = (12 + 4 + 14 + 8) = 38 runs
Run Rate = Runs in a given amount of overs / Number of balls = 38/4 = 9.5
We cannot represent the run rate in the form of a fraction on television. So, we convert it to a decimal. In short, Decimal Point is the tiny dot. Everything left to it is a whole number and to its right is a portion.
Some standard Fractions converted to Decimal equivalents:
1/10 = 0.1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
Let’s understand decimals with respect to place value.
Let us write this in expanded form.
987.642 = (9 × 100) + (8 × 10) + 7 + (1/10 × 6) + (1/100 × 4) + (1/1000 × 2) = 900 + 80 + 7 + 0.6 + 0.04 + 0.002 = 987 + 0.642
Let’s understand how to derive decimals from division process.
Let us take an example where we divide 29 by 4.
We all know 29 is not a multiple for 4.
i. The multiple of 4 closest to 29 is 28.
ii. So, the part of the whole number left to the decimal point is 7 (a whole number).
iii. Now we get a remainder 1 (29 – 28 = 1). We need to divide it by 4.
iv. We can also show 29 as 29.0 and 29.00.
v. We bring down an invisible 0 to divide the remainder 1. As a result, it becomes 10.
vi. The multiple of 4 closest to 10 is 8. We put a 2 after the decimal point.
vii. We again get 2 as a remainder (10 – 8 = 2). We again bring down another 0 as a result of which 2 becomes 20.
viii. We know 20 is a multiple of 4. So, we put 5 after 2.
ix. We get the final answer as 7.25 when we divide 29.00 by 4.
When we represent 29 / 4 as a mixed fraction, we get a mixed fraction as 7 + 1/4.
In the same way when we represent it as a decimal, we get it as 7.25.
Types of Decimals
i. Terminating Decimals: These have a clear ending after a point.
Examples -
1/2 = 0.5; 1/4 = 0.25; 1/8 = 0.125
ii. Recurring Decimals: No matter how many zeroes you bring down, they do not end.
Examples -
1/3 = 0.333333…. ; 1/6 = 0.166666666…. ; 1/9 = 0.111111….
Mathematical Operations on Decimals
1. Let us take a situation where we purchased cashews worth ₹ 24.35, almonds worth ₹ 26.55 and peanuts worth ₹ 31.21. What is the total bill?
Let us use the Left to Right Addition Method.
Cost of cashews = ₹ (20 + 4 + 0.3 + 0.05)
Cost of almonds = ₹ (20 + 6 + 0.5 + 0.05)
Cost of peanuts = ₹ (30 + 1 + 0.2 + 0.01)
Total Bill = ₹ (20 + 4 + 0.3 + 0.05) + ₹ (20 + 6 + 0.5 + 0.05) + ₹ (30 + 1 + 0.2 + 0.01)
= ₹ [(20 + 20 + 30) + (4 + 6 + 1) + (0.3 + 0.5 + 0.2) + (0.05 + 0.05 + 0.01)]
= ₹ (70 + 11 + 1.0 + 0.11) = ₹ 82.11
2. The selling price of a perfume is ₹ 123.45, its discount is ₹5.6 and its cost price is ₹ 98.76. Is it a profit or a loss?
Let us use Zigzag Subtraction Method to calculate the final selling price.
Final selling price = ₹ (123.45 - 5.6) = ₹ (123.45 - 5 - 0.6) = ₹ (118.45 - 0.6) = ₹ 117.85
Since the selling price > cost price, it is a profit.
Let us use Left to Right Subtraction Method.
Amount of profit = ₹ 117.85 - ₹ 98.76 = ₹ (100 + 10 + 7 + 0.8 + 0.05) - ₹ (90 + 8 + 0.7 + 0.06)
= ₹ [100 + (10 - 90) + (7 - 8) + (0.8 - 0.7) + (0.05 - 0.06)] = ₹ [(100 - 80 - 1) + (0.1 - 0.01)] = ₹ [19 + 0.09] = ₹ 19.09
3. The speed of a car is 15.4 km/hr. If the time taken is 2.5 hours, what is the distance traveled by the car?
Since Distance = Speed × Time ;
Distance traveled by the car = (15.4 × 2.5) km = [(154/10) × (25/10)] km = [3850/100] km = 38.5 km
4. The selling price of each share is ₹ 2.25. The earnings from all shares is ₹ 69.75. How many shares were sold?
Number of shares sold = 69.75 ÷ 2.25 = (69.75 × 100) ÷ (2.25 × 100) = 6975 ÷ 225 = 31
Chapter 3 - Percentages
You all must have heard of statements like :
i. "I am 100% sure this has happened."
ii. "Wow you got 100% marks."
iii. "Cent per cent."
What does the word percent mean? The word 'cent' means 100. So per cent means per 100. Hence, a percentage (symbolized by '%' symbol) is a fraction where the denominator is 100. The use of percentages increased because keeping 100 as a base was easy for calculations.
Percentages are commonly used in report cards, profit and loss rates, discount rates, interest rates, investment rates, inflation rates, etc.
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| Illustration for a percentage |
In the given picture, we see 20 small squares out of 100 small squares colored in purple in the first large square ; while in the second large square, we see 50 small squares out of 100 small squared colored in purple.
Relationship between Fractions, Decimals and Percentages
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| Fractions, Decimals and Percentages converting to each other |
To convert a fraction or a decimal into percentage, multiply it by 100 and add a % symbol after it.
Why we do that? Because by multiplying by 100% (i.e. 100/100), we are making it an equivalent fraction where the denominator is a 100.
Illustration :
1. 1/2 = (1 × 50) / (2 × 50) = 50/100 = 50%
2. 1/4 = (1 × 25) / (4 × 25) = 25/100 = 25%
3. 0.54 = 54/100 × 100/100 = 54/100 × 100% = 54%
4. 0.36 = 36/100 × 100/100 = 36/100 × 100% = 36%
Now let us understand the concept of opposites.
Suppose we take a situation where 94% students passed in the mathematics test. How many failed?
We all know 100% means complete!
So the number of students who failed = 100% - 94% = 6%
Let us understand this from an example.
There are 200 students in the class. 188 students passed.
So logically, the number of students who failed = 200 - 188 = 12
% of students who passed = 188/200 × 100% = 94%
% of students who failed = 12/200 × 100% = 6%
% of all students = 94% + 6% = 100%
Let us try some numerical problems.
1. The speed of a bike is 40 km/hr. The total distance is 200 km. The distance traveled by now is 60 km. The speed increases by 25% and the remaining distance is covered. What is the total time taken?
Increase in the speed = (25/100 × 40) km/hr = 10 km/hr
Remaining distance = (200 - 60) km = 140 km
Speed for remaining distance = (40 + 10) km/hr = 50 km/hr
Time taken to cover 60 km = (60 ÷ 40) hours = 1.5 hours
Time taken to cover 140 km = (140 ÷ 50) hours = 2.8 hours
Total time taken to travel 200 km = (1.5 + 2.8) hours = 4.3 hours
2. A customer buys three items. A shampoo worth ₹ 180 ; a body soap worth ₹ 40 ; a hair oil worth ₹ 120 ; a face cream worth ₹ 60. The discount is of 10%. The tax on the bill is 5%. What is the total value of the bill?
Cost of all products = ₹ (180 + 40 + 120 + 60) = ₹ 400
Discount Amount = ₹ (10% × 400) = ₹ 40
Tax Amount = ₹ (5% × (400 + 40)) = ₹ (5% × 440) = ₹ 22
Final Value of the Bill = ₹ (440 + 22) = ₹ 462
Chapter 4 - Averages
There are four friends. Donald, Daisy, Daffy and Melissa. They see some chocolate bars. Donald takes 8 bars, Daisy takes 6 bars, Daffy takes 9 bars and Melissa takes 5 bars. Don't you think there is an inequality? Yes!
The Chocolate Angel steps in and asks them to give her back all the chocolate bars so that all the bars are distributed equally.
Total number of chocolate bars = 8 + 6 + 9 + 5 = 28
How many bars will each friend get?
Since there are 4 friends, we divide the number of chocolate bars by the number of friends.
Number of chocolate bars each friend will get = 28 ÷ 4 = 7
Now each friend gets a 'fair share'. This 'fair share' is known as an average.
You all must have heard of statements like :
1. "The average Indian spends 10 hours at work."
2. "The average crime rates in India are high and unreported, etc."
Remember the first example we said with respect to the run rate (in the Chapter 2 (Decimals) in this blog)? The run rate is nothing but the average runs scored in an over.
Now lets take a situation.
In the first over, India scores 12 runs.
In the second over, India scores 4 runs.
In the third over, India scores 14 runs.
In the fourth over, India scores 8 runs.
In the fifth over, India scores a certain number of runs because of which the run rate becomes 11. How many runs did India score in the fifth over?
Total runs in 5 overs = Number of overs × run rate = (5 × 11) runs = 55 runs
Runs in the fifth over = 55 - (12 + 4 + 14 + 8) = 55 - 38 = 17 runs
Chapter 5 - Ratios and Proportions
Now take a situation where you make a chocolate - date - walnut cake. It involves the following ingredients :
5 cups of flour ; 4 cups of milk ; 3 cups of chocolate powder ; 2 cups of date paste ; 1 cup of walnuts.
How much will 2 chocolate - date - walnut cakes require?
10 cups of flour ; 8 cups of milk ; 6 cups of chocolate powder ; 4 cups of date paste ; 2 cups of walnuts.
How much will 6 chocolate - date - walnut cakes require?
30 cups of flour ; 24 cups of milk ; 18 cups of chocolate powder ; 12 cups of date paste ; 6 cups of walnuts.
Just imagine if for 2 cakes we only added more flour and kept other quantities intact, would the cake be tasty? No. Without the addition of the milk, the cakes will not be soft. Without the increase in chocolate powder, the cakes would not be chocolaty. Without the addition of the date paste the cakes will not be sufficiently sweet. Without the addition of the walnuts, the crunchiness in the cakes would be less.
Here the relationship between the ingredients is called a ratio (symbolized by ':').
Ratio of flour to milk to chocolate powder to date paste to walnuts = 5 : 4 : 3 : 2 : 1
When you maintain the ratio with the making of every cake, you ensure the taste is good. Keeping that rule the same is called proportion.
Let us take some more examples :
1. There are 6 tigers and 7 lions needed for a circus show.
Ratio of tigers to lions = 6 : 7
Now imagine if we would write the ratio of lions to tigers, would we still write 6 : 7? No. We would write it as 7 : 6. The order of the numbers must match the order of the words!
2. There are 2 puppies, 3 rabbits and 4 hamsters in a family.
Ratio of hamsters to rabbits to puppies = 4 : 3 : 2
Simplified Ratios
Think of a situation where you are making a pudding that requires 24 almonds and 27 cashews, what would be the ratio of almonds to cashews?
Ratio of almonds to cashews = 24 : 27 = (24 ÷ 3) : (27 ÷ 3) = 8 : 9
Why did we do this? In real life situations, such big numbers will only make calculations harder. Hence, we divide the terms of the ratio by their highest common factor to make the ratio look simpler.
Equivalent Ratios
Think of the example of the tigers and lions. Suppose there are three circus shows, each require these number of tigers and lions, what would be the ratio?
Ratio of tigers to lions = (6 × 3) : (7 × 3) = 18 : 21
This kind of a ratio is known as an equivalent ratio where all the terms of the ratio are multiplied by the same number.
These are the secrets behind proportions.
Relationship between Ratios, Fractions and Percentages
Suppose there are 6 roses and 4 lotuses in a pot.
Total number of flowers in the pot = (6 + 4) flowers = 10 flowers
Portion of Roses in the pot = 6/10 = (3 × 2) / (5 × 2) = 3/5
Portion of Lotuses in the pot = 4/10 = (2 × 2) / (5 × 2) = 2/5
% of Roses in the pot = 3/5 × 100% = 60%
% of Lotuses in the pot = 2/5 × 100% = 40%
Ratio of Roses to Lotuses in the Pot = 6 : 4 = (3 × 2) : (2 × 2) = 3 : 2
Understanding the concept of equivalent ratios, it can also be written as 60 : 40.
Further, it can also be written as 3/5 : 2/5.
If we notice carefully, the numerators in both the fractions are the terms of the ratio. On the other hand, the denominator of the fractions is the sum of the terms of the ratio (3 + 2 = 5).
Product of Means and Extremes in Proportions
As mentioned before, equivalent ratios and simplified ratios are the secret behind proportions.
1. Let us first take the example of almonds and cashews
8 : 9 :: 24 : 27
Here 8 and 27 are the extremes while 9 and 24 are means.
Product of Means = 24 × 9 = 216
Product of Extremes = 27 × 8 = 216
2. Now let us take the example of tigers and lions
6 : 7 :: 18 : 21
Product of Means = 18 × 7 = 126
Product of Extremes = 21 × 6 = 126
From these examples, we understand Product of Means = Product of Extremes.
Let us solve some examples.
1. The ratio of mangoes to papayas is 3 : 4. If we have 9 mangoes, how many papayas will we have?
Let the number of papayas be 'x'.
The proportion is = 3 : 4 :: 9 : x
This means 3/4 = 9/x
Product of Extremes = Product of Means
3x = (9 × 4) = 36
x = 36 ÷ 3 = 12
So, we have 12 papayas.
2. The ratio of marks in English to Mathematics is 8 : 7. If the score in Mathematics is 42, what is the score in English? Also give us the percentage in each subject and average marks scored, considering the full marks in each paper is 50.
Let the marks scored in English be 'x'.
The proportion is = 8 : 7 :: x : 42
This means 8/7 = x/42
Product of Means = Product of Extremes
7x = 42 × 8
x = (42 × 8) ÷ 7 = 6 × 8 = 48
Hence, the marks in English is 48.
% marks in English = 48/50 × 100% = 96%
% marks in Mathematics = 42/50 × 100% = 84%
Average marks scored = (48 + 42) ÷ 2 = 90 ÷ 2 = 45
Conclusion
And that’s it! We have now covered some of the most important everyday mathematical concepts — Fractions, Decimals, Percentages, Averages, and Ratios & Proportions — in a simple and practical way.
Mathematics is not about memorizing complicated rules or spending huge amounts on coaching classes. It is about understanding the logic behind numbers and using simple tricks to make calculations easier. Whether you are sharing a cake, calculating run rates in cricket, preparing a perfect recipe, or checking your exam marks, these concepts are everywhere in our daily lives.
We hope this blog has helped reduce your fear of mathematics and given you useful tools to solve problems confidently. Remember — practice is the key. Try solving a few problems from your textbook or daily life using the methods we discussed. You will surely see the difference!
Thanks and Regards,
The Aadyanagha Foundation.
