An "improper" fraction is a fraction whose numerator is greater than the denominator, for example 5/2. Mixed numbers are mathematical expressions made up of an integer and a fractional part, for example 2+1/2. It is usually easier to imagine two and a half pizzas (2+1/2) rather than "five halves" of pizza. For this reason it is good to know how to transform a fraction into a mixed number and vice versa. Using the math operation of division is the fastest way to do this, but there is also an easier one if you have difficulty doing division.
Steps
Method 1 of 2: Using Division
Step 1. Start with an improper fraction
In our example we will consider the following fraction 15/4. This is unequivocally an improper fraction, since the numerator, 15, is greater than the denominator, 4.
If fractions or divisions worry you, you can use the second method of the article
Step 2. Rewrite the problem in the form of a division
In this case it is necessary to transform the fraction into a normal division and perform the calculations manually. The operation consists in dividing the numerator of the fraction by the denominator. In our example we will have to solve the following calculation 15 ÷ 4.
Step 3. Let's do the division
If you are not sure how to proceed, you can consult this article for more information on this. The execution of the example division will be much easier if you write down all the steps of the logical process to be carried out:
- Compare the divisor, 4, with the first digit of the dividend, ie 1. The number 4 is greater than 1, so we will need to include the next dividend digit as well.
- Compare the divisor, 4, with the first two digits of the dividend, ie 15. Now ask yourself "How many times is the number 4 in the number 15?" If you are unsure of the answer, try multiple times until you find the correct result using multiplication.
- The correct result is 3, so we return it to the line for the final result of the division.
Step 4. Let's calculate the remainder
Unless the numbers taken into consideration are multiples of each other, so they give an integer result, we will have a remainder. To calculate it, follow these simple instructions:
- Multiply the result by the divisor. In our example we will have to calculate 3 x 4.
- Write the product of the multiplication under the dividend. In our example we will have 3 x 4 = 12, so we bring the number 12 aligned below 15.
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Perform the subtraction of the result obtained from the dividend: 15 - 12 =
Step 3.. The latter is the rest of our first division.
Step 5. Now we express the result as a mixed number
Remember that a mixed number is made up of an integer and a fractional part. After performing the division represented by the improper fraction, we obtained all the information necessary to compose the resulting mixed number:
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The integer part is represented by the quotient of the division which in our case is
Step 3.;
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The numerator of the fractional part is represented by the rest of the fraction ie
Step 3.;
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The denominator of the fractional part remains that of the original improper fraction, therefore
Step 4..
- Now we write the final result in its correct form obtaining: 3+3/4.
Method 2 of 2: Alternative Method
Step 1. Make a note of the improper fraction to be processed
An improper fraction is defined as a fraction whose numerator is greater than the denominator. For instance 3/2 is an improper fraction because 3 is greater than 2.
- The number at the top of a fraction is called numerator while the one shown at the bottom denominator.
- The procedure described in this method is not ideal for very large fractions because it takes a long time to perform. If the numerator is much larger than the denominator, it is better to use the method that uses division because it is faster.
Step 2. Remember which fractions indicate unity
For example 2 ÷ 2 = 1 or 4 ÷ 4 = 1. This is true for any number divided by itself, since it will always result in one. In the case of fractions, the same result is obtained. For instance 2/2 = 1 as well as 4/4 = 1, so also 397/397 will be equal to 1.
Step 3. Divide the starting leg into two parts
This is a simple method of turning a fraction into an integer. Let's try to see if we can also apply it to a part of our improper starting fraction:
- In our example 3/2 the denominator (the number under the fraction sign) is 2.
- 2/2 it is a very simple fraction to simplify since the numerator and denominator are equal, so we can extract it from the original fraction and calculate the remainder.
- Reporting in written form the reasoning described in the previous step we will obtain: 3/2 = 2/2 + ?/2.
Step 4. Let's calculate the second part of the fraction
How do we identify the numerator of the second fraction into which we have divided the improper starting one? If you don't know how to add and subtract fractions, don't worry and read on. When the denominators of two fractions are equal we can ignore them and take into consideration only the relative numerators, thus transforming the problem into a simple addition between integers. Here are the steps related to our example 3/2 = 2/2 + ?/2:
- Only take into account the numerators (the numbers above the fraction line). In this case we have to solve this simple equation 3 = 2 + "?". What is the number that, substituted for the question mark, makes the equation true? In other words, what number added to 2 gives 3 as a result?
- The correct answer is 1 because 3 = 2 + 1.
- Now that we have found the solution to the problem, we can rewrite the equation by including the denominators: 3/2 = 2/2 + 1/2.
Step 5. Let's run the simplifications
We now know that our improper starting fraction can also be written as 2/2 + 1/2. We also learned that the fraction 2/2 = 1, just like in any other fraction in which the numerator and denominator are equal. This means we can simplify the fraction 2/2 replacing it with the number 1. At this point we will have 1 + 1/2, which represents exactly a mixed number! Our example problem has been solved.
- Once you have identified the correct solution, you will no longer need to add the "+" symbol, you can simply write 11/2.
- Remember that a mixed number is made up of an integer part and a proper fraction.
Step 6. Repeat the above steps if the remaining fraction is still improper
In some cases the fractional part of the mixed number obtained by the method described is still an improper fraction (where the numerator is even greater than the denominator). When this happens, the procedure must be repeated, transforming the fraction obtained into a second mixed number. When finished, don't forget to add the integer part obtained from the first simplification process to the one you will get now (in our example it was "1"). For example, let's try to transform the improper fraction 7/3 in a mixed number:
- 7/3 = 3/3 + ?/3;
- 7 = 3 + ?;
- 7 = 3 + 4;
- 7/3 = 3/3 + 4/3;
- 7/3 = 1 + 4/3.
- As you can see, the fractional part of the mixed number obtained in this example is still an improper fraction, so for the moment set aside the whole part (i.e. 1) and repeat the decomposition process starting from the new fraction: 4/3 = 3/3 + ?/3;
- 4 = 3 + ?;
- 4 = 3 + 1;
- 4/3 = 3/3 + 1/3;
- 4/3 = 1 + 1/3;
- The fraction obtained is a proper fraction, so the work is done. Remember to add the whole part of the first mixed number obtained i.e. 1: 1 + 1 + 1/3 = 2+1/3.
