A step-by-step guide: converting Fractions to Decimals
Published July 20th 2024
Last edited July 22nd 2024
Fractions to decimals conversions are popular interview questions at major trading firms like Optiver, Flow Traders, Akuna Capital, and Group One. This article provides you with a step-by-step guide on how to mentally solve these questions. We also include four examples to get you up to speed quickly with hands-on practice.
Step-by-Step Guide
Integer Division
Before diving into the steps of converting a fraction to a decimal, it's important to understand the concept of integer division. Normally, when dividing a number, you calculate both the quotient and the decimal part. For example, 10 divided by 4 is 2.5. With integer division, you only calculate how many times the whole number of 4 fits into 10, along with the remainder. So, 10 divided by 4 results in 2 (2 times 4) with a remainder of 2. This type of division will be used in our method to solve fractions to decimals.
Step-by-Step Guide
You can convert fractions to decimals mentally using the following steps:
- Divide the numerator by the denominator using integer division. Determine how often the denominator fits into the numerator. This gives us the number left from the decimal separator. Keep the remainder in mind as it will be needed for further calculations. In computing, the remainder is also known as the modulo.
- Take the remainder and multiply it by 10. Divide that number by the denominator. Write down the result and keep track of the new remainder. This gives the second digit of the decimal.
- Repeat step 2 until you reach one of the following scenarios:
- The remainder is 0: there are no more digits, so the calculation is complete. This is called a terminating or non-repeating decimal. However, not all fractions resolve to a terminating decimal.
- A repeating pattern of digits appears. For example, 0.3333333 or 0.12456456456. Since the remainder is divided by the same denominator each time, the sequence will recur indefinitely. The number of decimals after which a repeating pattern occurs can be very large.
- You have reached your desired level of accuracy. If you only need five digits of the decimal, calculate until the sixth digit and use that digit to round the fifth digit.
Example 1: Converting 5/8 to a Decimal
- Start with dividing 5 by 8 using integer division. We find the first digit 0 with a remainder of 5.
- Multiply the remainder by 10. 50 divided by 8 equals 6 with a remainder of 2.
- Continue the process: 20 divided by 8 equals 2 with a remainder of 4.
- Continue the process: 40 divided by 8 equals 5 with a remainder of 0.
No further division is needed as the remainder is 0. Our final result is 0.625. Finding the remainder of 0, indicates we have completely solved the fraction and there are no more decimals to be calculated. These decimals are called non-repeating or terminating decimals.
Example 2: Converting 7/12 to a Decimal
- Start with dividing 7 by 12 using integer division. We find the first digit to be 0 with a remainder of 7.
- Multiply the remainder by 10. 70 divided by 12 equals 5 with a remainder of 10.
- Continue the process: 100 divided by 12 equals 8 with a remainder of 4.
- Continue the process: 40 divided by 12 equals 3 with a remainder of 4.
- Continue the process: 40 divided by 12 equals 3 with a remainder of 4.
We find the result to be 0.583333... . In this example, we found a repeating remainder, indicating repeating decimal. The repeating digit sequence is called the reptend or repetend.
Example 3: Converting 1/7 to a Decimal
- We start with dividing 1 by 7 using integer division. We find the first digit to be 0 with a remainder of 1.
- Multiply the remainder by 10. 10 divided by 7 equals 1 with a remainder of 3.
- Continue the process: 30 divided by 7 equals 4 with a remainder of 2.
- Continue the process: 20 divided by 7 equals 2 with a remainder of 6.
- Continue the process: 60 divided by 7 equals 8 with a remainder of 4.
- Continue the process: 40 divided by 7 equals 5 with a remainder of 5.
- Continue the process: 50 divided by 7 equals 7 with a remainder of 1.
- Since we have returned to the initial remainder of 1, we know the pattern 142857 will repeat.
We now know the first twelve digits of the decimal notation: 0.142857142857... .
Example 4: Converting 13/17 to a Decimal
- Start with the integer division of 13 by 17. We find the result to be 0 with a remainder of 13.
- Multiply the remainder by 10: 130 divided by 17 equals 7 with a remainder of 11.
- Continue the process: 110 divided by 17 equals 6 with a remainder of 8.
- Continue the process: 80 divided by 17 equals 4 with a remainder of 12.
- Continue the process: 120 divided by 17 equals 7 with a remainder of 1.
- Continue the process: 10 divided by 17 equals 0 with a remainder of 10.
- Continue the process: 100 divided by 17 equals 5 with a remainder of 15.
- Continue the process: 150 divided by 17 equals 8 with a remainder of 14.
- Continue the process: 140 divided by 17 equals 8 with a remainder of 4.
- Continue the process: 40 divided by 17 equals 2 with a remainder of 6.
- Continue the process: 60 divided by 17 equals 3 with a remainder of 9.
- Continue the process: 90 divided by 17 equals 5 with a remainder of 5.
- Continue the process: 50 divided by 17 equals 2 with a remainder of 16.
- Continue the process: 160 divided by 17 equals 9 with a remainder of 7.
- Continue the process: 70 divided by 17 equals 4 with a remainder of 2.
- Continue the process: 20 divided by 17 equals 1 with a remainder of 3.
- Continue the process: 30 divided by 17 equals 1 with a remainder of 13. Notice that the remainder of 13 is equal to our initial division, meaning the decimals will keep repeating from here on.
We find 0.7647058823529411 and we know that the next group of decimals is 7647058823529411, since the remainder is the same as the initial division.
Closing Remarks
By following these steps, you can convert fractions to decimals mentally. Practice with our Fractions to Decimals practice tool to build your confidence and improve your speed in performing these calculations.