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To convert a mixed fraction to an improper fraction, multiply the whole number by the denominator, add the numerator, then write the result over the original denominator.
For 4 2/3 that gives (4 x 3) + 2 = 14, so 4 2/3 as an improper fraction is 14/3. The value hasn’t changed, only the way it’s written.
Mixed fractions and improper fractions: the difference
A fraction tells us how many equal parts of a whole we have. The numerator is the number above the dividing line and the denominator is the number below it.
In a proper fraction, the numerator is smaller than the denominator, e.g. 1/4.
A mixed fraction, also known as a mixed number, is made up of a whole number followed by a proper fraction, e.g. 2 3/4, which reads as two and three quarters.
An improper fraction is one where the numerator is equal to or greater than the denominator, e.g. 11/5. It holds the same value as the equivalent mixed fraction, but the whole number isn’t shown separately.
For every mixed fraction there is an equivalent improper fraction, and vice versa. Mixed fractions are easier to picture in daily life, but improper fractions are far easier to handle in calculations, which is why questions usually ask you to convert before you multiply, divide or add.
The conversion itself takes three steps.
Step 1: Multiply the whole number by the denominator
Say you’re asked to express 4 2/3 as an improper fraction. First take the whole number, in this case 4, and multiply it by the denominator of the fractional part, in this case 3:
4 x 3 = 12
The number 12 is what we take as our resulting product.
Step 2: Add the product to the numerator
Next, take the resulting product of 12 and add it to the numerator of the fractional part, which in our example of 4 2/3 is 2:
12 + 2 = 14
The result here becomes the numerator of the improper fraction.
Step 3: Write the new numerator over the original denominator
The final step is to place the new numerator over the original denominator from the mixed fraction.
In our example, this gives us 14/3.
So, 4 2/3 expressed as an improper fraction is 14/3, each representation holding the same value.
Example questions
Here are some example questions for you to try out yourself. Although we’ve explained the process in three steps, finding the numerator for an improper fraction can be done in one simple equation:
(whole number x denominator) + numerator
It’s then just a case of placing the resulting figure over the original denominator.
Example question 1
Express the mixed fraction 3 2/3 as an improper fraction.
Multiply the whole number by the denominator of the fractional part, then add the numerator of the fractional part:
(3 x 3) + 2 = 11
Place that figure over the original denominator to get the improper fraction: 11/3.
So, 3 2/3 expressed as an improper fraction is 11/3.
Example question 2
Express the mixed fraction 24 5/6 as an improper fraction.
Using one single equation, multiply the whole number by the denominator of the fractional part and add the numerator of the fractional part:
(24 x 6) + 5 = 149
Now write the resulting figure over the original denominator of the mixed fraction to find the equivalent improper fraction: 149/6.
So, 24 5/6 expressed as an improper fraction is 149/6.
Example question 3
Express the mixed fraction 106 7/8 as an improper fraction.
Again, using one equation, multiply the whole number by the denominator and then add on the numerator:
(106 x 8) + 7 = 855
Now take the resulting figure, placing it over the original denominator to find the improper fraction of 855/8.
106 7/8 expressed as an improper fraction is 855/8.
Converting back: improper fractions to mixed numbers
Questions often run in the other direction and ask you to express an improper fraction as a mixed number. Divide the numerator by the denominator. The number of whole times it goes in becomes the whole number, and whatever is left over becomes the new numerator, sitting over the same denominator.
Take 14/3. Three goes into 14 four times, which accounts for 12 and leaves a remainder of 2. So 14/3 is 4 2/3, which is exactly where we started.
Take 149/6. Six goes into 149 twenty-four times, which accounts for 144 and leaves a remainder of 5. So 149/6 is 24 5/6.
Take 855/8. Eight goes into 855 one hundred and six times, which accounts for 848 and leaves a remainder of 7. So 855/8 is 106 7/8.
Converting back is a quick way to check your working: if the mixed number you started with reappears, your improper fraction is right.
Why the conversion is worth doing
Improper fractions are much easier to work with in a calculation.
To multiply 4 2/3 by 2, convert first: 14/3 x 2 = 28/3. Three goes into 28 nine times with 1 left over, so converting back gives 9 1/3. Trying to handle the whole number and the fractional part separately is where mistakes tend to creep in.
Addition works the same way. To add 1 1/2 and 2 1/3, convert to 3/2 and 7/3, put both over a common denominator of 6 to get 9/6 and 14/6, then add them for 23/6. Six goes into 23 three times with 5 left over, so the answer is 3 5/6.
Practice fraction questions
Fractions of this kind appear throughout numerical reasoning tests, usually inside a longer question and usually without a calculator. If you would rather drill the arithmetic on its own first, basic numeracy tests are the closer match.
Timed practice is what makes the conversion automatic, so try a set of free aptitude tests to see where you currently stand.