If we start with the mixed number three and five-eighths, it can be read as shorthand notation for the sum of three and five/eighths. This represents long division as follows. The whole number 3 is the quotient, the numerator 5 is the remainder, and the denominator 8 is the divisor.
The mixed number can be converted to an improper fraction by multiplying quotient and divisor, adding the remainder, then dividing the quantity by the divisor. This can be represented vertically as a fraction, or horizontally as a standard order of operations. The result is 29/8, or 29 divided by 8. If long division is carried out next, the answer can be expressed in three different ways.
If expressed as a decimal, the decimal point can be moved two places to the right, implicitly multiplying by 100, to get the equivalent percentage. Hence, the quantity can be expressed equivalently as a mixed number, improper fraction, decimal, or percentage.
The next image shows various ways for the number to be expressed using pie and bar diagrams.
Three and five-eighths can be viewed as three whole pies or rectangles, with a fourth pie or rectangle divided into eight pieces with three pieces that are empty.
As the above photo shows, the four rectangles can be stacked end to end.
The pies or rectangles can be equivalently divided into 16 congruent pieces, with the fourth unit having six empty pieces. Hence, three and ten-sixteenths is equivalent to three and five-eighths.
The final image shows two of the whole units being stuck together, then converted to a sum of three numbers: two plus one plus five-eighths. Using the associative property of addition, or the order of operations, the next step shows three plus five-eights. In the final step, we return to the mixed number that we started with in this demonstration.
Mr. Ken Marciel