# Music and Math Project

## Subtraction

Also when starting to subtract numbers in the Land of Twelve, it is good to first see what happens if you subtract a small number (smaller than twelve), for instance $$4\mathrm{X}_{12} - 6_{12}$$. Because the last digit in the first number, $$\mathrm{X}_{12}$$, is bigger than $$6_{12}$$, this calculation is pretty easy: $4 \mathrm{X}_{12} - 6_{12} = 44_{12}.$ The calculation of $$55_{12} - \mathrm{Y}_{12}$$ is a bit harder, but again you can first subtract $$5_{12}$$ to get $$50_{12}$$, and then subtract whatever is left from $$\mathrm{Y}_{12}$$, namely $$6_{12}$$: $55_{12} - \mathrm{Y}_{12} = 55_{12} - 5_{12} - 6_{12} = 50_{12} - 6_{12} = 46_{12}.$ To subtract bigger numbers, you can use column subtraction. Remember how in the Land of Ten you would borrow (or trade) numbers if you were subtracting a larger digit from a smaller? That still works the same. Let's look at this example: $\begin{array}{cccc} & & & \\ 7 & \mathrm{X} & 3 &\\ 2 & 5 & \mathrm{Y} & - \\ \hline & & ? \end{array}$ Because the $$\mathrm{Y}$$ is bigger than the $$3$$, we need to borrow a 'twelve' from the $$\mathrm{X}$$. The $$\mathrm{X}$$ stands on the place for the multiples of twelve, so it stand for $$\mathrm{X}0$$. Subtracting $$10$$, we get $$90$$. So the $$\mathrm{X}$$ becomes a $$9$$ and if we add the borrowed $$12$$ to the $$3$$ we get $$13$$. The result is $\begin{array}{cccc} 7 & 9 & 13 &\\ 2 & 5 & \mathrm{Y} & - \\ \hline & ? & 4 \end{array}$ We can now subtract the $$5$$ from the $$9$$ $\begin{array}{cccc} 7 & 9 & 13 &\\ 2 & 5 & \mathrm{Y} & - \\ \hline & 4 & 4 \end{array}$ And finally, we subtract $$2$$ from $$7$$ and find the answer $\begin{array}{cccc} 7 & 9 & 13 &\\ 2 & 5 & \mathrm{Y} & - \\ \hline 5 & 4 & 4 \end{array}$

### Exercises

Perform the following subtractions:
1. $$302 - 239$$
2. $$1029-62\mathrm{X}$$
3. $$2045-10\mathrm{Y}0$$
4. $$409-502$$
5. $$\mathrm{YXY}-253$$