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. \(\mathrm{YXY}-253\)