a)

A mixed number is a number consisting of an integer and a proper fraction, i.e. a number in the form abc where a is an integer and bc is a proper fraction: b is smaller than c.

An improper fraction is a fraction where the numerator is larger than the denominator, i.e. a number of the form de where the numerator, d, is greater than the denominator, e. 

To convert a mixed number into an improper fraction, multiply the integer part of the mixed number, a, by the denominator, c.

The numerator of the improper fraction will be equal to this added to what was already on the numerator of the proper fraction.

The denominator of the proper fraction will stay the same when it converts to an improper fraction to give a final answer of

(a×c)+bc.

i)

227=(2×7)+27=167.

ii)

2322=(2×22)+322=4722.

b)

To convert an improper fraction into a mixed number, you have to think about how many times the denominator can go into the numerator, and whatever is left over becomes the new numerator of the proper fraction. 

i) 296:  

29 can be divided by 6 fully 4 times, with 5 remaining.

4 becomes the integer part of the mixed number and 5 becomes the new numerator of the proper fraction. 

296=456.

It may be possible to simplify the proper fraction further by finding the highest common factor of the numerator and denominator. In this question, this is 1.

Therefore, it is not possible to simplify this further, and the final answer shall be

456.

ii) 228:

22 can be divided by 8 fully 2 times, with 6 left over.

2 becomes the integer part of the mixed number and 6 becomes the new numerator of the proper fraction. 

228=268.

It may be possible to simplify the proper fraction further by finding the highest common factor of the numerator and denominator. In this question, this is 2.

Simplifying by this value gives the final answer

234.