In the first two instalments of this article, we did a comprehensive discussion on Fractions and Decimals, including some strategies and techniques that can help you solve questions coming on the Quant section of the GMAT.
If you have not read any of these parts, here’s a link for you to do so, so that you can go through the basic concepts and make better use of this third part of the article on Fractions & Decimals:
Fractions & Decimals - Part 1Fractions & Decimals - Part 2 In the third and last part of this article, today, we shall see some techniques to convert between fractions and decimals. Questions related to these techniques have shown up occasionally on the GMAT, hence, it’s important that you learn them. They are not difficult to learn either.
So, let’s get started.
First up, we look at identifying fractions that yield terminating decimals.
Fractions yielding Terminating Decimals Not all fractions convert to terminating decimals. So, is there a way to quickly identify if a fraction would yield a terminating decimal without actually dividing? The answer is YES.
When a fraction is expressed in its simplest form, if the denominator is of the form \(2^x * 5^y\) where x and y are non-negative integers, then the fraction will yield a terminating decimal. For example, \(\frac{73}{100}\) would yield a terminating decimal since the denominator can be expressed only in terms of powers of 2s and 5s.
Note that \(\frac{73}{100}\) is already in its lowest form.
Let’s take another example, say \(\frac{144}{375}\). Now, \(\frac{144}{375}\) is not in it’s simplest form and can be further simplified to \(\frac{48}{125}\). The denominator is \(5^3\) and hence this fraction will also yield a terminating decimal.
Note that, in both the examples we took, the numerator did not have any say in whether the fraction would yield a terminating decimal or not. It was only the denominator that dictated all the terms.Therefore, to check if the fraction would give you a terminating decimal,
Reduce the fraction to its lowest form
Check if the denominator can be written in the form of \(2^x * 5^y\)
Corollary – If the denominator has any other prime number, it will NOT yield a terminating decimal. Sometimes, it’s easier to figure out that the denominator has powers of primes other than 2 or 5; in such cases, go ahead and conclude that the decimal will be recurring, you won’t be wrong.
From finding out which fractions yield terminating decimals, let’s now look at how to convert terminating decimals to fractions.
Converting Terminating decimals to fractions A terminating decimal, as the name implies, stops after a certain number of digits. As discussed in the second part of this article,
terminating decimals and recurring decimals both represent rational numbers. Therefore, it is possible to convert both types to the p/q form.
In this segment, we look at how to convert terminating decimals to the p/q form.
Let’s convert 0.264 to the \(\frac{p}{q}\) form
1)
Consider all the significant digits in the decimals and write them down as a number; this will represent the numerator of the fraction The significant digits are 264, therefore, the numerator is 264.
2)
The number of significant digits will represent the power of 10 that will represent the denominator of the fraction. There are three significant digits, therefore, the denominator is \(10^3\) i.e. 1000.
Hence, the corresponding fraction is\(\frac{ 264 }{ 1000 }\)
3)
Do not forget to simplify to the lowest form. \(\frac{264 }{ 1000}\) can be simplified to \(\frac{33}{125}\).
Convert 0.7793 to the \(\frac{p}{q}\) form
The significant digits are 7793, therefore, the numerator is 7793.
There are three significant digits, therefore, the denominator is \(10^4\) i.e. 10000.
Hence, the corresponding fraction is \(\frac{7793 }{ 10000}\) which cannot be simplified further since the numerator and denominator have no common factor.
Converting Pure recurring decimals to fractions As discussed in the second part of this article, a pure recurring decimal is one in which all the digits occurring in the decimal recur.
Let us recall that all recurring decimals have a period of recursion. This represents the number of different digits that recur. For example, in the decimal 0.72727272…. the period is 2 since there are 2 digits that recur.
Being able to recognize the period of the decimal is important when it comes to converting recurring decimals to fractions.
Let’s convert 0.2727227……. to the \(\frac{p}{q}\) form
1)
Consider all the significant digits in the decimal and write them as a number; this number represents the numerator. The significant digits are 27, therefore, the numerator is 27.
2)
Consider the period of the recurring decimal, form a number with as many 9s as the period; this number represents the denominator. The period of the decimal is 2, therefore, the denominator is 99
Hence, the corresponding fraction is\( \frac{27 }{ 99} \)
3)
Do not forget to simplify to the lowest form\(\frac{27}{99}\) can be simplified to \(3 / 11\).
Convert 0.123123123….. to the\(\frac{ p}{q}\) form
The significant digits are 123, therefore, the numerator is 123.
The period of the decimal is 3, therefore, the denominator is 999
Hence, the corresponding fraction is \(\frac{123 }{ 999}\) which can be simplified to \(\frac{41}{333}\) and no further.
Converting a Mixed Recurring decimal to fractions As compared to pure recurring decimals, converting mixed recurring decimals comes with a few additional steps. However, they will not cause you a lot of distress.
Let’s 0.16666666……. to the \(\frac{p}{q}\) form
1)
Consider all the significant digits in the decimal and write them as a number. The significant digits form the number 16.
2)
Consider all the non-recurring significant digits in the decimal and write them as a number. The non-recurring digits form the number 1.
3)
Now, subtract the number formed in Step 2 from the number formed in Step 1; the result will represent the numerator. Subtracting 1 from 16, we get 15 which represents the numerator
4)
The denominator of the fraction will have as many 9s as the period of the decimal followed by as many 0s as the number of non-recurring digits. The period of the decimal is 1 and the number of non-recurring digits is 1, therefore, the denominator is 90
Hence, the corresponding fraction is 15 / 90
5)
Do not forget to simplify to the lowest form. 15/90 can be simplified to 1 / 6.
Convert 0.012353535….. to the \(\frac{p}{q}\) form
The significant digits 01235 form the number 1235. The non-recurring digits 012 form the number 12. Subtracting 12 from 1235, we get 1223 which represents the numerator.
The period of the decimal is 2 and the number of non-recurring digits is 3, therefore, the denominator is 99000
Hence, the corresponding fraction is \(\frac{1223 }{ 99000}\) which cannot be simplified further since 1223 is a prime number.
We hope that these methods of converting the different types of decimals to fractions is fairly clear by now. Some of the decimals that we considered for demonstrating the procedure might have looked intimidating because of the calculations they entailed. Do not worry, that’s good because if you were able to grasp the techniques from these examples, the decimals you will see on the actual GMAT will look easier because they won’t be as complicated as some of these.
In the final segment of today’s article, we shall discuss approximation/estimation methods that can be employed in questions on Fractions and Decimals.
Estimation Techniques Talking of questions on estimation/approximation, it’s important to know that you can get better at estimating, with the right kind and amount of practice. At the outset, it might seem like it’s an art (well, it is!), but there’s definitely science working behind it.
Estimation/approximation-based problems can be seen on both Fractions and Decimals.
In case of fractions, typically, a fraction that is close to a commonly occurring fraction will be provided. Therefore, a great way to estimate it is to approximate it to be the commonly occurring fraction. Note that, for you to be able to do this, you will have to know the exact values of common fractions like \(\frac{1}{2},\frac{ 1}{4}, \frac{1}{5}, \frac{3}{4}, \frac{2}{3}, \frac{4}{5}\) and so on.
It is also important to note that the fraction to be estimated should first be reduced to the lowest form – this is something we have highlighted in the first part of this article as well.
For example, in a question on percentage change, let us say that we were able to simplify the expression and obtain a fraction like \(\frac{9}{23}\) * 100; close observation tells us that the numerator is close to 8 and the denominator is close to 24. Therefore, can we approximate \(\frac{9}{23}\) to \(\frac{1}{3}\) and say it’s approximate value is 0.33?
Well, there’s one step that one should not forget. By reducing the numerator and increasing the denominator, we have actually reduced the overall value of the original fraction. As such, the actual value of the fraction should be slightly more than 0.33.
But, how high can I go? Should I take 0.35 or 0.40? This is the next question that arises. For this, we can again use the approximation method to find an upper limit. Observation also tells us that numerator can be made 10 and denominator 25, which reduces to \(\frac{2}{5}\) or 0.40.
Now, we can fairly estimate that \(\frac{9}{23}\) should be in between 0.33 and 0.40. This will not only help you eliminate the percentages in this range but will also help you pick the percentage which IS in this range (knowing the GMAT, it is very probable that you will most likely have one option like this in a question of this level).
Of course, you need to put in a lot of practice to be able to do all of this (almost at a sub-conscious level) in about 30 to 40 seconds. Considering there aren’t too many direct questions on estimation/approximation of fractions and decimals, we recommend that you start estimating/approximating whenever the opportunity presents itself. Develop a mindset to never go till the very end for the final answer. This way, you will automatically force yourself to estimate and slowly but steadily, you will become a natural at it.
Now, when it comes to estimating decimals, the techniques that we discussed to round off decimals can come in handy.
Depending on the accuracy required by the question/observed in the answer options, you can decide to round it off to the appropriate digit.
For example, 15.76 can be rounded off to 15.8 or 16 depending on the requirements of the question. However, approximating 15.76 to 20 would be too far-fetched and incorrect. Try to avoid such estimations.
This segment on Estimation can come in very handy in a lot of questions on GMAT Quant, so please make sure that you have understood it correctly and put in enough practice, as described above.
With this, we come to the end of the third and final part of this article on Fractions and Decimals. We hope that this will prove a very useful article for all of y’all regardless of which stage of your preparation you are in, for the GMAT.
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