ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

How to Solve Terminating Fractions and Repeating Decimals Involving 9 in the Denominator

Updated on April 3, 2017
1701TheOriginal profile image

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly improve it.

Changing the Fraction into a Decimal

Oftentimes in mathematics, we will have a fraction that when converted into a decimal will go on and on forever. Because of this never-ending nature to the number, we call them repeating decimals. An example is 0.1666666666… The fractional form of this, 1/6, can be thought of as terminating because the fraction does not require us to go on indefinitely and will be a simple ratio of two numbers, ending our search for the precise value and hence the use of the word terminates. We can just write the decimal as a ratio of two numbers and stop there.

When you are presented with a problem that will involve these conversions, it is nice to have a short-cut, and math is full of them. They are our friend so long as we understand why we are using it. And of course the easier the short-cut the better. Fortunately, when you have a fraction with a nine in the denominator, such an easy route exists.

Let us take a look at 1/9. Normally, when we convert a fraction into a decimal, we count how many times the denominator goes into the numerator. This is simply a division problem. But right away we know that 9 does not go into 1, so we must write that it goes into it 0 times. Now, we add a decimal behind the 0 and add a 0 to the 1, so now it is 10. 9 certainly goes into 10, and when you find the difference, it is 1 again. Applying this trick over and over again, we find that 1/9 = 0.1111111… or written in correct form 0.̅1. Now try 5/99. You will find that after division, the answer is 0.̅0̅5. So how do I go about this so that it is easier than division?

Simply follow this formula: 1. The numerator of the fraction will be the number that is in the decimal. 2. The number of 9’s in the denominator is the number of decimal places we will need. 3. Put a bar over the number that is behind the decimal place. In the case of 1/9, 1 is in the decimal and we have only one 9, so we move the decimal over by one. Now we have 0.1, and once we place a bar over it, becomes 0.̅1. For 5/99, 5 will be in the decimal, we have two 9’s so we move the decimal over two spots to make it 0.05, and finally throwing a bar over it makes it 0.̅0̅5.

Source

Changing the Decimal into a Fraction

What if we have a repeating decimal and I want to rewrite it as a terminating fraction? We could simply work backwards from what we did above. For 0.̅1̅9, we have a 19 after the decimal point, so it will go in the numerator and sincewe have 2 decimal places, so that means we need two 9’s in the denominator, for a final answer of 19/99.

We can also use another trick to figure out the answer. Let’s make x the decimal we are trying to make into a fraction. If 0.̅7 is the decimal, then 0.̅7 = x. Multiply both sides by 10 gives us 7.̅7 = 10x. If I take away x from both sides, that gives me 7 = 9x, or x = 7/9. The easy trick to remember here is that the number of decimal places is the number of 10’s I need. Since there was only one in the above example, I only need to multiply both sides by one 10. For the one we did before, 0.̅1̅9, we need to multiply both sides by two 10’s, or 100, to solve the problem with this technique.

Now, try it out for yourself. Give the problems below a shot and see if you can get the right answer.

© 2013 Leonard Kelley

Comments

    0 of 8192 characters used
    Post Comment

    No comments yet.

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: "https://hubpages.com/privacy-policy#gdpr"

    Show Details
    Necessary
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Marketing
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Statistics
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)