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26 Oct 2021, 11:43
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This is the \(4^{th}\) article in the Quant Fundamental Series where we have been discussing important properties from the topic of Numbers, a very important topic from the point of view of GMAT Quant.
The topics that we covered in the first 3 articles were on different aspects of Numbers – we did an article about special numbers like Zero, One and Two; then followed it up with an article on Perfect squares and cubes; in the third, we discussed the importance of Smart & quick calculations and also demonstrated some techniques for quick addition, subtraction, multiplication, squaring and cubing.
Those of you who have been following our articles till now have our gratitude. We hope that you have made good use of these articles and that you have been able to improve on your knowledge of Numbers.
And for all of you who have not had the chance to see our previous articles, do not worry; here are the links to these articles.
Zero One Two
Perfect Squares & Cubes
Smart calculations
In this article, we intend to cover important properties and strategies, which you need to know, to solve questions on Fractions and Decimals. These are two topics which contribute a fairly sizeable chunk of questions to the GMAT Quant question pool.
The fact that these numbers are not intuitive to think about, makes these doubly difficult to deal with, for a lot of test-takers. Our objective then, in this article, would be to review properties and discuss strategies and techniques that can help you overcome this hurdle, thereby affecting your performance positively.
The structure of this article is similar to the ones we have posted previously. In the first part i.e. today, the discussion will be about Fractions and all there is to them wrt GMAT Quant. In the second part, we shall discuss about Decimals in detail. In the third and the final part, we shall look at the conversions between the two and how this can be tested in questions. We shall also discuss some simple methods to estimate/approximate in questions on Fractions & Decimals
With this, it’s time to take a quick look at some basic aspects of Fractions.
Quick overview of the Basics
A fraction is any part of a whole. It is usually expressed by comparing this part with the whole. The number representing the whole is the denominator and the number representing the part is the numerator.
For example, a fraction like \(\frac{3}{5} \)represents 3 parts out of a whole of 5.
Sometimes, ratios are represented in the form of fractions. However, we need to remember that it is just one way of representing ratio; essentially, a ratio is a comparison of two quantities to establish which one is bigger and by how much. On the other hand, as we have seen by the definition of a fraction, a fraction is a part of a whole.
Types of Fractions
Based on the relative comparison between the numerator and the denominator, there are two types of Fractions:
Note that both of the above categories are not limited to positive values; there can be both positive and negative fractions of the two types listed above.
Proper Fractions: A proper fraction is a fraction in which the numerator is lesser than the denominator. Examples of proper fractions are \(\frac{1}{3}\), \(\frac{3}{5}\), \(\frac{5}{6}\) , -\(\frac{1}{2}\), -\(\frac{3}{4} \)and so on.
When plotted on the number line, all proper fractions lie in the interval from -1 to 1.
Improper Fractions: An improper fraction is a fraction in which the numerator is greater than the denominator. Examples of improper fractions are \(\frac{5}{3}, \frac{7}{5}, \frac{11}{9}, -\frac{3}{2}, -\frac{8}{3}\) and so on.
When plotted on the number line,
all positive improper fractions are greater than 1
all negative improper fractions are lesser than -1
What about Mixed Fractions then? Well, Mixed fractions are not a category by themselves; they are just another way of representing improper fractions.
Reducing a fraction to its lowest form
A very important property of fractions (and therefore, you can be sure that GMAT uses this to set up traps) is the property which says that a fraction should always be simplified to its simplest form.
This implies that you can see a fraction like \(\frac{30}{75}\) on a GMAT question; it is up to you to recognize that it is not in its simplest form and reduce it by removing the common factors.
Therefore, whenever you come across a question on the GMAT, involving fractions, always try to observe if the fractions have been given in the lowest form. If they are not, make sure that you reduce it to the simplest form before performing any other operation on it.
Comparison of Fractions
When fractions are to be compared, two very common methods that you may think of are to make the denominators same and compare the numerators OR vice versa. However, we are not going to cover those methods here since they are fairly simple and straightforward.
In this segment, we will cover two methods which are not frequently discussed when it comes to comparing fractions.
Comparing fractions using cross multiplication
This method is very useful when it comes to comparing two fractions, which is also its limitation. In this method, to compare fractions, we first write the fractions side by side, as shown in the example below:
Compare \(\frac{7}{9}\) and \(\frac{5}{8}\)
Multiply the numerator of the first fraction with the denominator of the second and write it on the LHS; multiply the numerator of the second fraction with the denominator of the first and write it on the RHS.
The side that has the bigger product will have the bigger fraction.
Multiplying 7 with 8 we have 56 on the LHS; multiplying 5 with 9, we have 45 on the RHS. Since 56 is greater than 45, \(\frac{7}{9}\) is greater than \(\frac{5}{8}\).
A quick way to verify your answer is to pick up a calculator and check. \(\frac{7}{9}\) = 0.777777.. whereas \(\frac{5}{8}\) = 0.625. Clearly, this method works.
Let’s take another example to consolidate our learnings.
Compare \(\frac{11}{17} \)with\(\frac{ 9}{14}\).
Multiplying 11 with 14, we have 154; multiplying 9 with 17, we have 153. Therefore, \(\frac{11}{17} is bigger than \frac{9}{14}\).
With these examples, we hope we have made this method clear. Let’s now look at another useful method to compare fractions.
Comparing fractions with common differences
This method can be used to compare fractions in which there is a constant difference between the numerator and the denominator, in all the fractions that are being compared.
If all fractions being compared are Proper fractions, the size of the numerator will dictate the order; in the case of improper fractions, the order will depend on the size of the denominator.
If,
The fraction with the biggest numerator will be the biggest and the one with the smallest numerator will be the smallest.
Let’s take a very simple example to compare 4 fractions, viz., \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4} and \frac{4}{5}\). These are very easy fractions for you to verify your final answer since most of you will be able to calculate the values.
We see that in these fractions, there is a common difference of 1 between each of the numerators and the denominators; we also see that all the fractions are proper fractions.
Therefore, the fraction with the biggest numerator i.e. \(\frac{4}{5}\) is the biggest. It can be easily verified that this is true.
You may now try your hand at this sample problem:
Compare \(\frac{11}{17}, \frac{13}{19}, \frac{17}{23} and \frac{19}{25}\).
Did you answer that \(\frac{19}{25}\) is the biggest? Well, that’s the right answer.
If,
The fraction with the smallest denominator will be the biggest and the one with the biggest denominator will be the smallest.
Like the previous technique, let’s take a couple of examples to understand. Lets start by comparing the fractions \(\frac{6}{5}, \frac{5}{4}, \frac{4}{3} and \frac{3}{2}\).
The fraction with the smallest denominator is \(\frac{3}{2}\) which is why it is clearly the biggest fraction (you can go ahead and verify by finding out the actual values of the fractions)
For the second example, let’s just flip the fractions that we considered for the first case. Let’s compare \(\frac{17}{11}, \frac{19}{13}, \frac{23}{17} and \frac{25}{19}\).
Since 11 is the smallest denominator, \(\frac{17}{11}\) is the biggest fraction.
We hope that these examples have served to clarify any doubts you may have had as you were reading through the description of the technique.
These two techniques can come in handy in questions on estimation/approximation, percentages and ratios, where two or more fractions have to be compared to arrive at the answer.
With this, we come to the end of the first part of this article on Fractions and Decimals. The important learnings from this part were:
1) Proper fractions lie between -1 and 1 on the Number Line
2) Improper fractions are either greater than 1 or less than -1
3) It is very important to work with fractions in their simplest form
4) Method of cross multiplication is a great method to compare 2 fractions
5) When there is a common difference between the numerator and the denominator of all given proper fractions, the biggest fraction will have the greatest numerator and vice versa.
6) When there is a common difference between the numerator and the denominator of all given improper fractions, the biggest fraction will have the smallest denominator and vice versa.
In the second part of this article, tomorrow, we shall cover some important properties and problem solving techniques related to Decimals.
Goodbye!
The topics that we covered in the first 3 articles were on different aspects of Numbers – we did an article about special numbers like Zero, One and Two; then followed it up with an article on Perfect squares and cubes; in the third, we discussed the importance of Smart & quick calculations and also demonstrated some techniques for quick addition, subtraction, multiplication, squaring and cubing.
Those of you who have been following our articles till now have our gratitude. We hope that you have made good use of these articles and that you have been able to improve on your knowledge of Numbers.
And for all of you who have not had the chance to see our previous articles, do not worry; here are the links to these articles.
Zero One Two
Perfect Squares & Cubes
Smart calculations
In this article, we intend to cover important properties and strategies, which you need to know, to solve questions on Fractions and Decimals. These are two topics which contribute a fairly sizeable chunk of questions to the GMAT Quant question pool.
The fact that these numbers are not intuitive to think about, makes these doubly difficult to deal with, for a lot of test-takers. Our objective then, in this article, would be to review properties and discuss strategies and techniques that can help you overcome this hurdle, thereby affecting your performance positively.
The structure of this article is similar to the ones we have posted previously. In the first part i.e. today, the discussion will be about Fractions and all there is to them wrt GMAT Quant. In the second part, we shall discuss about Decimals in detail. In the third and the final part, we shall look at the conversions between the two and how this can be tested in questions. We shall also discuss some simple methods to estimate/approximate in questions on Fractions & Decimals
With this, it’s time to take a quick look at some basic aspects of Fractions.
Quick overview of the Basics
A fraction is any part of a whole. It is usually expressed by comparing this part with the whole. The number representing the whole is the denominator and the number representing the part is the numerator.
For example, a fraction like \(\frac{3}{5} \)represents 3 parts out of a whole of 5.
Sometimes, ratios are represented in the form of fractions. However, we need to remember that it is just one way of representing ratio; essentially, a ratio is a comparison of two quantities to establish which one is bigger and by how much. On the other hand, as we have seen by the definition of a fraction, a fraction is a part of a whole.
Types of Fractions
Based on the relative comparison between the numerator and the denominator, there are two types of Fractions:
- 1) Proper Fractions
2) Improper Fractions
Note that both of the above categories are not limited to positive values; there can be both positive and negative fractions of the two types listed above.
Proper Fractions: A proper fraction is a fraction in which the numerator is lesser than the denominator. Examples of proper fractions are \(\frac{1}{3}\), \(\frac{3}{5}\), \(\frac{5}{6}\) , -\(\frac{1}{2}\), -\(\frac{3}{4} \)and so on.
When plotted on the number line, all proper fractions lie in the interval from -1 to 1.
Improper Fractions: An improper fraction is a fraction in which the numerator is greater than the denominator. Examples of improper fractions are \(\frac{5}{3}, \frac{7}{5}, \frac{11}{9}, -\frac{3}{2}, -\frac{8}{3}\) and so on.
When plotted on the number line,
all positive improper fractions are greater than 1
all negative improper fractions are lesser than -1
What about Mixed Fractions then? Well, Mixed fractions are not a category by themselves; they are just another way of representing improper fractions.
Reducing a fraction to its lowest form
A very important property of fractions (and therefore, you can be sure that GMAT uses this to set up traps) is the property which says that a fraction should always be simplified to its simplest form.
This implies that you can see a fraction like \(\frac{30}{75}\) on a GMAT question; it is up to you to recognize that it is not in its simplest form and reduce it by removing the common factors.
Therefore, whenever you come across a question on the GMAT, involving fractions, always try to observe if the fractions have been given in the lowest form. If they are not, make sure that you reduce it to the simplest form before performing any other operation on it.
Comparison of Fractions
When fractions are to be compared, two very common methods that you may think of are to make the denominators same and compare the numerators OR vice versa. However, we are not going to cover those methods here since they are fairly simple and straightforward.
In this segment, we will cover two methods which are not frequently discussed when it comes to comparing fractions.
Comparing fractions using cross multiplication
This method is very useful when it comes to comparing two fractions, which is also its limitation. In this method, to compare fractions, we first write the fractions side by side, as shown in the example below:
Compare \(\frac{7}{9}\) and \(\frac{5}{8}\)
Multiply the numerator of the first fraction with the denominator of the second and write it on the LHS; multiply the numerator of the second fraction with the denominator of the first and write it on the RHS.
The side that has the bigger product will have the bigger fraction.
Multiplying 7 with 8 we have 56 on the LHS; multiplying 5 with 9, we have 45 on the RHS. Since 56 is greater than 45, \(\frac{7}{9}\) is greater than \(\frac{5}{8}\).
A quick way to verify your answer is to pick up a calculator and check. \(\frac{7}{9}\) = 0.777777.. whereas \(\frac{5}{8}\) = 0.625. Clearly, this method works.
Let’s take another example to consolidate our learnings.
Compare \(\frac{11}{17} \)with\(\frac{ 9}{14}\).
Multiplying 11 with 14, we have 154; multiplying 9 with 17, we have 153. Therefore, \(\frac{11}{17} is bigger than \frac{9}{14}\).
With these examples, we hope we have made this method clear. Let’s now look at another useful method to compare fractions.
Comparing fractions with common differences
This method can be used to compare fractions in which there is a constant difference between the numerator and the denominator, in all the fractions that are being compared.
If all fractions being compared are Proper fractions, the size of the numerator will dictate the order; in the case of improper fractions, the order will depend on the size of the denominator.
If,
- all fractions are proper fractions
there is a common difference between the Numerator and Denominator of all fractions
The fraction with the biggest numerator will be the biggest and the one with the smallest numerator will be the smallest.
Let’s take a very simple example to compare 4 fractions, viz., \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4} and \frac{4}{5}\). These are very easy fractions for you to verify your final answer since most of you will be able to calculate the values.
We see that in these fractions, there is a common difference of 1 between each of the numerators and the denominators; we also see that all the fractions are proper fractions.
Therefore, the fraction with the biggest numerator i.e. \(\frac{4}{5}\) is the biggest. It can be easily verified that this is true.
You may now try your hand at this sample problem:
Compare \(\frac{11}{17}, \frac{13}{19}, \frac{17}{23} and \frac{19}{25}\).
Did you answer that \(\frac{19}{25}\) is the biggest? Well, that’s the right answer.
If,
- all fractions are improper fractions
there is a common difference between the Numerator and Denominator of all fractions
The fraction with the smallest denominator will be the biggest and the one with the biggest denominator will be the smallest.
Like the previous technique, let’s take a couple of examples to understand. Lets start by comparing the fractions \(\frac{6}{5}, \frac{5}{4}, \frac{4}{3} and \frac{3}{2}\).
The fraction with the smallest denominator is \(\frac{3}{2}\) which is why it is clearly the biggest fraction (you can go ahead and verify by finding out the actual values of the fractions)
For the second example, let’s just flip the fractions that we considered for the first case. Let’s compare \(\frac{17}{11}, \frac{19}{13}, \frac{23}{17} and \frac{25}{19}\).
Since 11 is the smallest denominator, \(\frac{17}{11}\) is the biggest fraction.
We hope that these examples have served to clarify any doubts you may have had as you were reading through the description of the technique.
These two techniques can come in handy in questions on estimation/approximation, percentages and ratios, where two or more fractions have to be compared to arrive at the answer.
With this, we come to the end of the first part of this article on Fractions and Decimals. The important learnings from this part were:
1) Proper fractions lie between -1 and 1 on the Number Line
2) Improper fractions are either greater than 1 or less than -1
3) It is very important to work with fractions in their simplest form
4) Method of cross multiplication is a great method to compare 2 fractions
5) When there is a common difference between the numerator and the denominator of all given proper fractions, the biggest fraction will have the greatest numerator and vice versa.
6) When there is a common difference between the numerator and the denominator of all given improper fractions, the biggest fraction will have the smallest denominator and vice versa.
In the second part of this article, tomorrow, we shall cover some important properties and problem solving techniques related to Decimals.
Goodbye!
27 Oct 2021, 19:14
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25 Oct 2024, 01:03
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