Problem 1
1/2 = 2/4
The mixed number 4 & 1/2 is the same as 4 & 2/4
Then convert that mixed number to an improper fraction using the formula below
a & b/c = (a*c+b)/c
4 & 2/4 = (4*4+2)/4
4 & 2/4 = 18/4
We could reduce this, but I'll keep it unreduced so that we keep the denominator of 4.
Now think of 3/4 as "3 fourths" and 18/4 as "18 fourths".
We could easily replace "fourths" with "unit" to say 3 units and 18 units respectively.
The problem can be rephrased into this question: "how many groups of 3 units are in 18 units?"
At this point, it's probably clear that there are 6 groupings since 18/3 = 6.
It might help to set up a drawing where there are 6 circles, and inside each circle would be 3 dots. So there are 3*6 = 18 dots total.
Answer: 6
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Problem 2
3/4 = 9/12
2/3 = 8/12
2 2/3 = 2 8/12
2 & 8/12 = (2*12+8)/12 ... using the formula from earlier
2 & 8/12 = 32/12
Think of "twelfths" as a unit
So asking "how many groups of 9/12 are in 32/12" is the same as asking "how many groups of 9 units are in 32 units".
Divide the two numbers to find out
32/9 = (27+5)/9
32/9 = 27/9 + 5/9
32/9 = 3 + 5/9
32/9 = 3 & 5/9
Notice that if you were to convert the mixed number 3 & 5/9 to an improper fraction (using that formula mentioned), then you should get 32/9 back again.
Answer: 3 & 5/9
