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GMAT Math: Terminating and Repeating Decimals

The topic of decimals, and patterns of decimals, seems to be of slightly greater interest to GMAC in the GMAT OG13e than in previous editions.  What decimals terminate?  What decimals repeat?  In this post, we’ll take a look at these questions.

 

Rational Numbers

Integers are positive and negative whole numbers, including zero.  Here are the integers:

{ … -3, -2, -1, 0, 1, 2, 3, …}

When we take a ratio of two integers, we get a rational number.  A rational number is any number of the form a/b, where a & b are integers, and b ≠ 0. Rational numbers are the set of all fractions made with integer ingredients.   Notice that all integers are included in the set of rational numbers, because, for example, 3/1 = 3.

 

Rational Numbers as Decimals

When we make a decimal out of a fraction, one of two things happens.  It either terminates (comes to an end) or repeats (goes on forever in a pattern).  Terminating rational numbers include:

1/2 = 0.5

1/8 = 0.125

3/20 = 0.15

9/160 = 0.05625

 

Repeating rational numbers include:

1/3 = 0.333333333333333333333333333333333333…

1/7 = 0.142857142857142857142857142857142857…

1/11 = 0.090909090909090909090909090909090909…

1/15 = 0.066666666666666666666666666666666666…

 

When Do Rational Number Terminate?

The GMAT won’t give you a complicated fraction like 9/160 and expect you to figure out what its decimal expression is.  BUT, the GMAT could give you a fraction like 9/160 and ask whether it terminates or not.  How do you know?

Well, first of all, any terminating decimal (like 0.0376) is, essentially, a fraction with a power of ten in the dominator; for example, 0.0376 = 376/10000 = 47/1250.  Notice we simplified this fraction, by cancelling a factor of 8 in the numerator.  Ten has factors of 2 and 5, so any power of ten will have powers of 2 and powers of 5, and some might be canceled by factors in the numerator , but no other factors will be introduced into the denominator.  Thus, if the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, then it can be written as something over a power of ten, which means its decimal expression will terminate.

If the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, the decimal expression terminates.  If there is any prime factor in the denominator other than 2 or 5, then the decimal expression repeats.  Thus,

 

1/24 repeats (there’s a factor of 3)

1/25 terminates (just powers of 5)

1/28 repeats (there’s a factor of 7)

1/32 terminates (just powers of 2)

1/40 terminates (just powers of 2 and 5)

 

Notice, as long as the fraction is in lowest terms, the numerator doesn’t matter at all. Since 1/40 terminates, then 7/40, 13/40, or any other integer over 40 also terminates. Since 1/28 repeats, then 5/28 and 15/28 and 25/28 all repeat; notice, though that 7/28 doesn’t repeat, because of the cancellation: 7/28 = 1/4 = 0.25.

 

Shortcut Decimals:

There are certain decimals that are good to know as shortcut, both for fraction-to-decimal conversions and for fraction-to-percent conversions.  These are

 

1/2 = 0.5

1/3 = 0.33333333333333333333333333…

2/3 = 0.66666666666666666666666666…

1/4 = 0.25

3/4 = 0.75

1/5 = 0.2 (and times 2, 3, and 4 for other easy decimals)

1/6 = 0.166666666666666666666666666….

5/6 = 0.833333333333333333333333333…

1/8 = 0.125

1/9 = 0.111111111111111111111111111… (and times other digits for other easy decimals)

1/11 = 0.09090909090909090909090909… (and times other digits for other easy decimals)

 

Irrationals

There’s another category of decimals that don’t terminate (they go on forever) and they have no repeating pattern.   These numbers, the non-terminating non-repeating decimals, are called the irrational numbers.  It is impossible to write any one of them as a ratio of two integers.  Mr. Pythagoras (c. 570 – c. 495 bce) was the first to prove a number irrational: he proved that the square-root of 2 — sqrt(2)  — is irrational.  We now know: all square-roots of integers that don’t come out evenly are irrational.  Another famous irrational number is pi, or pi, the ratio of a circle’s circumference to its diameter.  For example,

pi = 3.1415926535897932384626433832795028841971693993751058209749445923078164

0628620899862803482534211706798214808651328230664709384460955058223172535940812848111745

0284102701938521105559644622948954930381964428810975665933446128475648233786783165271201

909145648566923460348610454326648213393072602491412737…

 

That’s the first three hundred digits of pi, and the digits never repeat: they go on forever with no repeating pattern.  There are infinitely many irrational numbers: in fact, the infinity of irrational numbers is infinitely bigger than the infinity of the rational numbers, but that gets into some math (http://en.wikipedia.org/wiki/Aleph_number) that is much more advanced than the GMAT.

 

Practice Question

1) {(0.16666...)/(0.44444...)} =

(A) 2/27

(B) 3/2

(C) 3/4

(D) 3/8

(E) 9/16

 

Practice Question Explanation

1) From our shortcuts, we know 0.166666666666… = 1/6, and 0.444444444444… = 4/9.  Therefore (1/6)*(9/4) = 3/8.  Answer = D

 

 

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49 Responses to GMAT Math: Terminating and Repeating Decimals

  1. debra knauf August 23, 2016 at 4:48 pm #

    in the decimal 0.15.why is there a line over the number 15

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert August 24, 2016 at 10:32 am #

      Hi Debra,

      Happy to help! 🙂 The line means that the numbers occurring under the line repeat infinitely. I hope that clarifies!

  2. ta'jhai August 16, 2016 at 4:45 pm #

    is 1.765 a rational or irrational number and where would it fall o a number line????

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert August 18, 2016 at 9:02 am #

      Hi Ta’jhai!

      In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. A good example of an irrational number is pi.

      The decimal 1.765 that you mention can be written as many fractions, including 353/200. Therefore, it is a rational number. It would fall between 1 and 2 on a number line of only whole numbers. It would be possible to give an infinite number of places it would fall depending on the specificity of your number line units. For example, it could also be between 1.764 and 1.766 if you had decimals that specific.

      I hope this helps! 🙂

  3. Emma Immel August 14, 2016 at 12:30 pm #

    How do you decide by which power
    of 10 to multiply an equation when writing a
    decimal with repeating digits as a fraction?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert August 15, 2016 at 4:35 am #

      Hi Emma 🙂

      Usually, we use this method only for terminating decimals. To determine which power of ten to use, you need to count the number of decimal places from the decimal point to the end of the terminating number. That will give you the power of 10 by which to multiply. For example, let’s say we have the decimal

      0.004137

      To see what power of 10 we should multiply by to write this decimal as a fraction, we start at the decimal point and count the places until reaching the final digit, 7. There are 6 digits, so we will multiply this decimal by 10^6:

      0.004137*10^6 = 4137

      And to write the decimal as a fraction, we need to divide this integer 4137 by 10^6:

      0.004137 = 4137/1,000,000

      For repeating decimals, there’s no power of 10 by which we can multiply to be able to rewrite the decimal as a fraction. Rather, dealing with repeating rational numbers is a matter of recognizing familiar patterns and guess and check to determine the equivalent fraction.

      I hope this helps, at least a little 🙂

  4. Yvez July 18, 2016 at 5:51 am #

    Can I know where will I put the terminating and repeating decimals if I make a map about Real Numbers ?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert July 23, 2016 at 7:49 pm #

      If I understand correctly, you’re asking where repeating decimals would fit in a concept map of real numbers, such as this one: http://edtech2.boisestate.edu/dougwolfe/images/real_numbers2.jpg . In any concept map of different categories of real numbers, numbers with repeating decimals would fall under the category of rational numbers— numbers that can be expressed in fraction form. All repeating decimal numbers have a fraction expression. 1/3 is 0.333333333 repeating, 2/7 is 0.285714 repeating, and so on. Rational numbers also includes terminating decimals, such as 2.25 (9/4 in fraction form).

  5. Mohammad Alim July 12, 2016 at 9:23 am #

    can you help me
    why 0 is included in even numbers

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert July 13, 2016 at 12:07 am #

      Hi Mohammad,

      This is just a quirk of how numbers are classified. While 0 is neither negative nor positive, it is considered an even number. 🙂

  6. tigoy July 2, 2016 at 12:36 am #

    sir,

    how to convert 0.07 into fraction and 7 is repeating decimal…
    please sir help me

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert July 5, 2016 at 9:53 am #

      Hi there 🙂

      Happy to help! Firstly, let’s look at 0.07 or seven hundredths. As its name suggest, seven hundredths is 7 parts of 100. So, we can rewrite 0.07 as

      0.07 = 7/100.

      In terms of 0.777…, here’s a nice trick. For repeating decimals that have the same repeating digit, from 0.111.. to 0.888…, as fractions, they are that digit over 9:

      0.111… = 1/9
      0.888… = 8/9
      etc.

      So, 0.777… repeating is equivalent to 7/9.

      Hope this helps! 🙂

  7. Autumn May 13, 2016 at 7:57 am #

    Hey can u help me what’s the decimal expansion of the rational number 1/9 repeats.

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert May 18, 2016 at 11:02 am #

      Hi Autumn,

      1/9 = 0.111… where 1 repeats indefinitely in the decimal part of the number.

  8. tanu May 12, 2016 at 7:17 pm #

    sir,
    please help me in mine situation as my exams are nearer.
    which least number must be divided
    to 2/5×7 to make it terminating decimal expansion
    plz sir reply me fast
    I m waiting for your reply

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert May 18, 2016 at 11:21 am #

      Hi Tanu 🙂

      Let’s look at the two factors in the denominator:

      5: 1/5 = 0.2 and is therefore a terminating decimal
      7: 1/7 = 0.142857… The decimal does not terminate but rather repeats as 1428567.

      So, if we want to create a terminating decimal from 2/(5*7), we must multiply by 7. By multiplying by 7, we can cancel the term of 7 in the denominator, leaving us with a factor of 5, which will give us a terminating decimal:

      2/(5*7)*7 = 2/5 = 0.4

      I hope that helps!

  9. Rohan May 10, 2016 at 12:28 am #

    Pi is an irrational number. Pi is ratio of circumference to its diameter. But if we make a circle with rational diameter say x then we will get a circle with circumference of rational value i guess. Now if we take the ratio it should be rational. Then why pi is irrational

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert May 18, 2016 at 10:17 am #

      Hi Rohan 🙂

      Pi is an irrational number because when written as a decimal number it does not terminate or repeat. The circumference of a circle, C, is defined as 2(pi)r, where r is the radius of the circle. Since 2r = d, then C = d(pi) and C/d = pi. With that in mind, you’ll find that we cannot write pi as a ratio of two rational numbers. Typically in GMAT questions, you’ll be given either the radius or diameter as a rational number. That means that the circumference will be pi multiplied by that rational number. Since pi is irrational, the circumference in that case will also be an irrational number.

      That said, physically measuring the exact circumference of a circle with a rational radius is not possible and requires estimation, which can result in a very good approximation of the circumference. The precision of that approximation can be tested by comparing the experimental C/r ratio with pi.

      And if you’re interested, you can read up about the proof that pi is an irrational number on Wikipedia.

      Hope this helps 🙂

  10. samridhi May 7, 2016 at 8:13 am #

    hey can u please help me with a question
    whether pie{22/7} is a rational number or not?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert May 7, 2016 at 1:27 pm #

      Hi Samridhi,

      Good question! 🙂 The popular approximation of 22/7 = 3.1428571428571… is close but not quite accurate for pi. In either case, pi is a famous irrational number! I hope that helps. 🙂

  11. Zsombi April 29, 2016 at 11:36 am #

    Why do calculators put a 7 as last decimal digit when dividing numbers like 2, 5, 20, 40, 80, 110, 140 and so on by 3? Calculating by hand the result has a never ending 6 as decimal. How do they get the 7?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert May 2, 2016 at 3:39 am #

      Hi Zsombi,

      Good question! 🙂

      The calculator is forced to round up by the end of its display, and a repeating 6 decimal would warrant rounding up to 7. But you are right, it is actually a never-ending 6. I hope that makes sense!

  12. Gauri April 24, 2016 at 5:52 am #

    Hii
    Can u plz help me
    I have a question
    Is root 6 + root 9
    Rational??

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert April 25, 2016 at 11:56 am #

      Hi Gauri,

      Happy to help! The sum of a irrational number and rational number is irrational. In this case, we have

      sqrt(6) + sqrt(9) = sqrt(6) + 3

      Sqrt(6) is an irrational number and is 2.4494… w

      So, the sum is

      2.4494… + 3 = 5.4494…

      The three dots represent the irrational decimal part of sqrt(6). Since the decimal does not terminate, the decimal in the sum will not terminate either. Therefore sqrt(6) + sqrt(9) results in a irrational number.

      Hope this helps 🙂

  13. jason January 25, 2016 at 9:35 am #

    What if the number is 3.067 but only the 7 has a repeating line over the number. Is that rational or irrational?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert February 7, 2016 at 8:40 am #

      Hi Jason,

      A number can be a rational number and not have the first decimal place repeating. Consider some common repeating decimals:

      1/6 = 0.1666…
      5/6 = 0.8333…
      1/12 = 0.08333…
      5/12 = 0.41666…

      I do not know off hand what the fraction representation of your suggested 3.0677… would be, but presumably we could find a ratio of two integers that would create that number, making it a rational number. 🙂

  14. Ty October 12, 2015 at 4:21 pm #

    No. On a number line 1 is farther away from 0 than .75

  15. Camaya October 7, 2015 at 2:44 pm #

    Would .75 be bigger than 1?

    • Braun October 30, 2015 at 12:51 am #

      No it would be smaller because 1 is a whole number and .75 is 3/4 of a whole so 1 is bigger

  16. irfan khan June 27, 2015 at 12:16 am #

    Thanks

  17. Hamza March 2, 2015 at 4:42 am #

    Great article Mike!!!

    • Mike MᶜGarry
      Mike March 2, 2015 at 11:46 am #

      Dear Hamza,
      I’m glad you found this helpful! Best of luck to you!
      Mike 🙂

      • Matt Hall April 21, 2015 at 11:59 pm #

        Hi Mike definitely great explanation about rationals and irrationals.

        I have a question.
        Is there some sort of formula to determine when a decimal number is Irrational? so it can be implemented by code.

        Thank you.

        Matt Hall,

        • Mike MᶜGarry
          Mike April 23, 2015 at 11:06 am #

          Matt,
          I’m happy to respond. 🙂 Everything about the relationship of rationals and irrationals defies all attempts to encapsulate it in a formula. For simple GMAT purposes, the GMAT will give you a decimal with, say, 10 or 12 places after the decimal showing. Either there will be a simple repeating pattern or not. If there’s a repeating pattern, the number is rational. For GMAT purposes, if the decimals shown contain no repeating pattern, then we can assume the decimal is irrational. Technically, the GMAT never asks about rational or irrational anyway: that’s already between the GMAT.
          Now, in the bigger picture, it’s certainly true that there are decimals that have repeating patterns that consist, say, of a string of some large number of decimals. For example, the decimal pattern of 1/29 repeats a pattern that is 28-decimal-places long:
          1/29 =
          0.03448275862068965517241379310344827586206896551724137931
          03448275862068965517241379310344827586206896551724137931
          03448275862068965517241379310344827586206896551724137931 . . .
          (courtesy of Wolfram Alpha)
          As I am sure you appreciate, if you have only, say, the first 12 decimals places, there are an infinity of possible rational numbers and another, larger infinity of possible irrational numbers that start with those initial 12 decimals. Having 12 decimal places, while extremely precise, is known mathematically as a decimal approximation, a sharply curtailed approximation of the infinite decimal. From a mathematical point of view, if we have 12 decimal places, we have essentially nothing. Mathematician regularly examine decimals to millions and billions of decimal places. For example, in 2014, pi was calculated to 13,300,000,000,000 decimals. If we are looking at any number that can be printed on a single sheet of paper, on a single line of text, then, from a purely mathematical point of view, that’s kiddie pool stuff.
          Basically, there’s no code as efficient as the human mind on this. However many decimals we have, the question is simple: do we see a repeating pattern or not? The human brain is a better pattern-detecting and pattern-matching machine than any computer.
          Does all this answer your question?
          Mike 🙂

  18. John Doe July 29, 2014 at 11:52 pm #

    Example of 9/160 at top of the page is wrong. You forgot to add 0 after decimal point.

    • Mike MᶜGarry
      Mike July 30, 2014 at 10:13 am #

      Dear “John”,
      Very perceptive! Thank you very much for pointing out that typo. I just corrected it. Best of luck to you, my friend.
      Mike 🙂

      • Braun October 30, 2015 at 12:56 am #

        Can you please give an example of the division problem of 1/8 converted into a decimal and can you tell me if the answer is terminating or if it’s repeating?

  19. Anusha July 14, 2014 at 9:43 pm #

    If a denominator have prime factors of only 2 or only or both then the number terminates ( because any denominator will be in the form ten’s , so factors of 10 is 2* 5 )

    If the denominator factors are other than 2 and 5 then the decimal repeats
    But taking an ex 1/14 ( factors are 2 and 7 ) it has a repeating decimal.

    It’s very useful,thank you.Is my understanding is Right?

    • Mike MᶜGarry
      Mike July 15, 2014 at 2:46 pm #

      Dear Anusha,
      Your understanding is correct. For example, 1/35 has 5 & 7 in the denominator, so this would be a repeating non-terminating decimals.
      1/35 = 0.0285714285714285714 …
      But, notice, if we put something divisible by 7 in the numerator, then the sevens would cancel, and the fraction would terminate:
      21/35 = 3/5 = 0.6
      Does all this make sense?
      Mike 🙂

  20. anonmoyous May 22, 2014 at 10:25 pm #

    If the number is itself 2 & 5. And if multiplies it with 5/2 and gives answers in which digit in ones place is 0 then that number will terminates always for example
    3/4 it will terminates
    4*5/2=10 ad we got 0 at ones digit we can easily identify it.

    • Mike MᶜGarry
      Mike May 23, 2014 at 10:37 am #

      Dear “anonmoyous” or “anonymous”,
      Yes, I believe if I understand you correctly, we are saying very much the same thing. You seem to understand this issue well, which is great. Best of luck to you.
      Mike 🙂

      • Fariha siddiqui February 4, 2016 at 9:46 am #

        Give me some reasons why 1/6 is irrational number?

        • Magoosh Test Prep Expert
          Magoosh Test Prep Expert February 5, 2016 at 8:22 am #

          Hi Fariha,

          Happy to help! 🙂 Can you tell me why you think 1/6 is an irrational number? 1/6 is actually not an irrational number because we can express it as a ratio of integers or a fraction! I hope that helps. 🙂

  21. Nishant Sondhi December 21, 2013 at 10:25 am #

    Hi Mike…

    Thanks a ton for this explanation..I just gave a Practice Test from GMAC ( the new Exam pack) and got two questions to do with terminating decimals…Have my GMAT in 36 hours from now..I have absolutely loved the Magoosh product…Hope to get a good score on Monday and write a good testimonial for Magoosh 🙂

    • Mike MᶜGarry
      Mike December 21, 2013 at 5:56 pm #

      Dear Nishant,
      You are quite welcome. I am very glad you found this blog article helpful, and I’m glad you like the Magoosh product. Best of luck to you my friend!
      Mike 🙂

  22. Saketh July 13, 2013 at 11:17 pm #

    Found what i was looking for!
    Thanks this really helped! 🙂

    • Mike MᶜGarry
      Mike July 14, 2013 at 12:00 pm #

      Dear Saketh,
      I’m glad it helped you. Best of luck to you!
      Mike 🙂

  23. Sara May 7, 2013 at 5:05 pm #

    I didn’t find what I was looking for, but it is very useful! 🙂
    Thank You for spreading your knowledge with others.
    Sincerely,
    Sara

    • Mike MᶜGarry
      Mike May 8, 2013 at 12:48 pm #

      Well, thank you. I hope you find that for which you seek. Best of luck to you.
      Mike 🙂


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