Adding and subtracting numbers are easier when you look at their positions on the number line. You are going to want to draw a number line to get this. Make 0 the furthest point right. Make -18 the furthest point left. For now, it's fine for your number line to be marked in intervals of one unit. To begin with, the number -17 5/9 lies left of zero on the number line, very near but not quite at -18. Whenever we add an amount, like 14 4/5, we always move that amount to the right, in a positive direction. The resulting number will be getting larger. This problem is interesting because the resulting answer is still a negative number. That is because -17 5/9 is 17 5/9 units left of zero. Moving right on the number line 14 4/5 units does not make it back to zero. So the answer is still below zero, resulting in a negative number. But what number? Now that we understand the mechanics, let's do the arithmetic:
To subtract the fractions, the denominators have to be the same. We can adjust each fraction proportionately so that the denominators are the number 45 because 45 is the smallest number that is a factor of both 9 and 5.
-17 5/9 = -17 (5/9 X 5/5) = -17 25/45... and ...
+14 4/5 =+14 (4/5 X 9/9) = +14 36/45
But if you subtracted the numerators now you would get a negative number. Rather than deal with that twist, let's just borrow 1 in the form of 45/45 from 17... then the problem gets much easier. It looks like this:
-17 25/45 = -16 (25/45 + 45/45) which is -16 70/45
That's better. Now you can easily subtract the numerators:
Notice that -2 34/45 is actually a larger number than -17 25/45 because its position on the number line is closer to zero.
To summarize, I think that remembering addition always moves us right on the number line and remembering that subtraction always moves us left, in a negative direction, is the easiest way to deal with adding and subtracting negative and positive numbers.
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