Your sample problem: (32 1/2) / (4 1/4)
My Way (with all the gory details): we are going to operate on the numerator and the denominator separately, then do the division.
Step 1: 32 1/2 = 32*(2/2) + 1/2 = (32*2)/2 + 1/2 = 64/2 + 1/2 = (64 + 1) / 2 = 65/2
a) Convert number to a fraction.
b) Any number divided by itself equals 1. In this case (2/2) = 1
c) Any number can be multiplied by 1 without affecting its value. In this case 32*(2/2) = 32
Step 2: 65/2 = (65/2)*(2/2) = (65 *2) / (2*2) = 130/4
a) Recognizing that 4 is the smallest common denominator, so 65/2 was converted to 130/4 using same rules stated above.
b) Any number divided by itself equals 1. In this case (2/2) = 1
c) Any number can be multiplied by 1 without affecting its value. In this case (2/2) = 1
Step 3: 4 1/4 = 4*(4/4) + 1/4 = (4*4)/4 + 1/4 = 16/4 +1/4 = (16+1)/4 = 17/4
a) Convert number to a fraction.
b) Any number divided by itself equals 1. In this case (4/4) = 1
c) Any number can be multiplied by 1 without affecting its value. In this case 4*(4/4) = 16
Step 4: (130/4) = 130/4 * (1/1) = (130*1)/(1*4) = 130 * (1/4*1) = 130 * (1/4)
a) Factor out the common denominator, (1/4).
b) Any number divided by itself equals 1. In this case (1/1) = 1
c) Any number can be multiplied by 1 without affecting its value. In this case 130/4 *(1/1) = 130/4
Step 5: (17/4) = 17/4 * (1/1) = (17*1)/(1*4) = 17 * (1/(4*1)) = 17 *(1/4)
a) Factor out the common denominator, (1/4).
b) Any number divided by itself equals 1. In this case (1/1) = 1
c) Any number can be multiplied by 1 without affecting its value. In this case 17/4 *(1/1) = 17/4
Step 6: (1/4)/(1/4) = 1
a) Recognize there is a factor of (1/4) in both the numerator and the denominator.
b) Any number divided by itself equals 1. In this case (1/4)/(1/4) = 1
Step 7: 130 * (1/4) / 17 *(1/4) = (130/17) * (1/4)/(1/4) = (130/17)*1 = 130/17
a) The (1/4) fraction in the numerator cancels the (1/4) in the denominator.
b) Any number divided by itself equals 1. In this case (1/4)/(1/4) = 1
Step 8: (130/17) = ~7.65
a) The result, the answer, equals 130/17
b) Restated the answer as a decimal fraction.
Wow!, that was a lot of typing.