// Numbas version: finer_feedback_settings
{"name": "Subtraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Subtraction", "tags": [], "metadata": {"description": "<p>Subtraction questions, whole and decimal, to be carried out by hand. Full worked solutions given in feedback.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Subtraction.</p>\n<p>Work out the following without using a calculator.</p>", "advice": "<p></p>\n<h1>Subtraction by hand</h1>\n<p>To carry out&nbsp;subtraction by hand the easiest way is to line the numbers up one above the other. When lining them up make sure that each place value is lined up correctly. Remember that the first number in the subtraction is placed above the second. The order of subtraction is important and it must be remembered throughout the calculation that it is the top number minus the bottom number.</p>\n<h2>Part a)</h2>\n<h4>i)</h4>\n<p>Lining these up gives</p>\n<p>\\[\\begin{align}\\var{a}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{b}}&amp;\\\\\\end{align}\\]</p>\n<p>Carrying out subtraction down each column in turn starting with the furthest right and placing the answer of these two digits underneath each column gives</p>\n<p>$\\var{z}-\\var{v}=\\var{z-v}$,</p>\n<p>\\[\\begin{align}\\var{a}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{b}}&amp;\\\\\\var{z-v}&amp;\\\\\\end{align}\\]</p>\n<p>And for the other columns $\\var{y}-\\var{u}=\\var{y-u}$ and $\\var{x}-\\var{t}=\\var{x-t}$</p>\n<p>\\[\\begin{align}\\var{a}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{b}}&amp;\\\\\\var{a-b}&amp;\\\\\\end{align}\\]</p>\n<h4>ii)</h4>\n<p>Lining these up gives</p>\n<p>\\[\\begin{align}\\var{c}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{d}}&amp;\\\\\\end{align}\\]</p>\n<p>Carrying out subtraction down each column in turn starting with the furthest right and placing the answer of these two digits underneath each column gives</p>\n<p>$\\var{y}-\\var{u}=\\var{y-u}$,</p>\n<p>\\[\\begin{align}\\var{c}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{d}}&amp;\\\\\\var{y-u}&amp;\\\\\\end{align}\\]</p>\n<p>And for the other columns $\\var{z}-\\var{v}=\\var{z-v}$ and $\\var{x}-\\var{t}=\\var{x-t}$</p>\n<p>\\[\\begin{align}\\var{c}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{d}}&amp;\\\\\\var{c-d}&amp;\\\\\\end{align}\\]</p>\n<p></p>\n<h2>Part b)</h2>\n<p>In part a for each column the number at the top was equal to or greater than the number below it. In this case the subtraction did not need any rearranging or consideration of surrounding columns. This is not often the case with subtraction and when the top number of a column is less than the number beneath it some rearranging may be required. At this point remembering that the column is part of a bigger calculation enables the digits to be changed so that the top number is bigger in each column. It can be helpful to think of it in terms of money and changing a larger domination coin into smaller ones, 10p = 10 x 1p. If the digit at the top of a column is less than the digit below it then this can be changed by looking to the column to the left. If this is not 0 at the top you can change one from this column into 10 in your column.</p>\n<h3>i)</h3>\n<p>Lining up the numbers in the same way gives</p>\n<p>\\[\\begin{align}\\var{f}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{g}}\\\\\\end{align}\\]</p>\n<p>Looking at the furthest right column the calculation we want to perform is $\\var{t}-\\var{z}$,&nbsp;but the number at the top is smaller than the number at the bottom. We need to change this by reducing the number at the top in the column to the left of it by 1 and adding 10 to the top of this column. So the top row now becomes $\\var{y},\\var{x-1},\\var{t+10}$ and the subtraction is performed on the following numbers.</p>\n<p>\\[\\begin{align}\\var{y}^{\\;\\var{x-1}}\\rlap{/}\\var{x}^{\\;1}\\var{t}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{t}\\&nbsp; \\;\\;\\var{u}\\ \\;\\;&nbsp;\\var{z}}&amp;\\\\\\end{align}\\]</p>\n<p>So now carrying out the subtraction down each column gives</p>\n<p>$\\var{t+10}-\\var{z}=\\var{t+10-z}$, $\\var{x-1}-\\var{u}=\\var{x-1-u}$, $\\var{y}-\\var{t}=\\var{y-t}$</p>\n<p>\\[\\begin{align}\\var{y}^{\\;\\var{x-1}}\\rlap{/}\\var{x}^{\\;1}\\var{t}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{t}\\ \\;\\;\\var{u}\\ \\;\\;\\var{z}}&amp;\\\\\\var{y-t}\\ \\;\\;\\var{x-1-u}\\ \\;\\;\\var{t+10-z}&amp;\\\\\\end{align}\\]</p>\n<p></p>\n<h3>ii)</h3>\n<p>Again lining up the numbers gives</p>\n<p>\\[\\begin{align}\\var{h}&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{l}}\\\\\\end{align}\\]</p>\n<p>Looking at each column and exchanging along the top&nbsp;where the bottom number is larger the subtraction becomes</p>\n<p>\\[\\begin{align}\\var{x}^{\\;\\var{u-1}}\\rlap{/}\\var{u}^{\\;1}\\var{v}&amp;\\\\\\underline{-{\\hspace 1cm}\\var{y}\\ \\;\\;&nbsp;\\var{z}}&amp;\\\\\\end{align}\\]</p>\n<p>There is still a column with the top number smaller than the bottom, so the process is repeated, carried out in all positions from right to left where the top number is smaller than the bottom.</p>\n<p>\\[\\begin{align}^{\\;\\var{x-1}}\\rlap{/}\\var{x}^{\\;\\var{u-1+10}}\\rlap{/}\\var{u}^{\\;1}\\var{v}&amp;\\\\\\underline{-{\\hspace 1cm}\\var{y}\\ \\;\\;&nbsp;\\var{z}}&amp;\\\\\\end{align}\\]</p>\n<p></p>\n<p>Carrying out subtraction down each column in turn starting with the furthest right and placing the answer of these two digits underneath each column gives</p>\n<p>\\[\\begin{align}^{\\;\\var{x-1}}\\rlap{/}\\var{x}^{\\;\\var{u-1+10}}\\rlap{/}\\var{u}^{\\;1}\\var{v}&amp;\\\\\\underline{-{\\hspace 1cm}\\var{y}\\ \\; \\var{z}}&amp;\\\\\\var{x-1}\\ \\;\\;\\;&nbsp;\\var{u-1+10-y}\\ \\; \\var{v+10-z}&amp;\\\\\\end{align}\\]</p>\n<p></p>\n<h2>Part c)</h2>\n<p>With decimals the key point is to line up the numbers by the decimal point and it can be made easier by adding 0's in to any gaps.</p>\n<h3>i)</h3>\n<p>So lining up the numbers in this case gives</p>\n<p>\\[\\begin{align}\\var{x}\\var{y}.\\var{z}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{t}.\\var{u}\\var{v}}&amp;\\\\\\end{align}\\]</p>\n<p>The same method is used as before, looking at each column in turn and if the top number is smaller than the bottom then change the top numbers by taking one from the column to the left and adding 10 to the current column so the top number becomes more than the bottom in that column.</p>\n<p>\\[\\begin{align}\\var{x}\\ \\;\\;&nbsp;\\var{y}.^{\\var{z-1}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{t}.\\ \\; \\var{u}\\ \\;\\; \\var{v}}&amp;\\\\\\end{align}\\]</p>\n<p>Carrying out the calculation the answer is found. Make sure that the decimal point is placed in the correct place in the answer</p>\n<p>\\[\\begin{align}\\var{x}\\ \\;\\;&nbsp;\\var{y}.^{\\var{z-1}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{t}.\\ \\; \\var{u}\\ \\;\\; \\var{v}}&amp;\\\\\\var{x}\\ \\;\\;&nbsp;\\var{y-t}.\\ \\; \\var{z-1-u}\\ \\;\\; \\var{10-v}&amp;\\\\\\end{align}\\]</p>\n<h3>ii)</h3>\n<p>\\[\\begin{align}\\var{t}\\var{u}.\\var{z}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\var{x}\\var{t}}&amp;\\\\\\end{align}\\]</p>\n<p>The same method is used as before, looking at each column in turn and if the top number is smaller than the bottom then change the top numbers by taking one from the column to the left and adding 10 to the current column so the top number becomes more than the bottom in that column.</p>\n<p>\\[\\begin{align}\\var{t}\\ \\;\\;&nbsp;\\var{u}.^{\\var{z-1}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\ \\; \\var{x}\\ \\;\\; \\var{t}}&amp;\\\\\\end{align}\\]</p>\n<p>Carying out the calculation the answer is found . Make sure that the decimal point is placed in the correct place in the answer.</p>\n<div data-jme-visible=\"z-1&lt;x\">\n<p>\\[\\begin{align}\\var{t}\\ \\;\\;^{\\var{u-1}}&nbsp;\\rlap{/}\\var{u}.^{\\var{z-1+10}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\;\\ \\;&nbsp;\\var{x}\\ \\;\\; \\var{t}}&amp;\\\\\\end{align}\\]</p>\n<p></p>\n<p>\\[\\begin{align}^{\\var{t-1}}\\rlap{/}\\var{t}\\ \\;\\;^{\\var{u-1+10}}&nbsp;\\rlap{/}\\var{u}.^{\\var{z-1+10}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\;\\ \\; \\var{x}\\ \\;\\; \\var{t}}&amp;\\\\\\end{align}\\]</p>\n<p>Now carrying out the calculations in each column gives</p>\n<p>\\[\\begin{align}^{\\var{t-1}}\\rlap{/}\\var{t}\\ \\;\\;^{\\var{u-1+10}}&nbsp;\\rlap{/}\\var{u}.^{\\var{z-1+10}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\;\\ \\; \\var{x}\\ \\;\\; \\var{t}}&amp;\\\\\\var{t-1}\\ \\;\\;&nbsp;\\var{u+9-y}.\\;\\ \\; \\var{z+9-x}\\ \\;\\; \\var{10-t}&amp;\\\\\\end{align}\\]</p>\n</div>\n<p></p>\n<div data-jme-visible=\"z-1&gt;x\">\n<p>\\[\\begin{align}^{\\var{t-1}}\\rlap{/}\\var{t}\\ \\;\\;^{\\var{1}} \\var{u}.^{\\var{z-1}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\ \\; \\var{x}\\ \\;\\; \\var{t}}&amp;\\\\\\end{align}\\]<b></b><i></i><u></u><sub></sub><sup></sup><span style=\"text-decoration: line-through;\"></span></p>\n<p>Now carrying out the calculations in each column gives</p>\n<p>\\[\\begin{align}^{\\var{t-1}}\\rlap{/}\\var{t}\\ \\;\\;^{\\var{1}} \\var{u}.^{\\var{z-1}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\ \\; \\var{x}\\ \\;\\; \\var{t}}&amp;\\\\\\var{t-1}\\ \\;\\; \\var{u+10-y}.\\ \\; \\var{z-1-x}\\ \\;\\; \\var{10-t}&amp;\\\\\\end{align}\\]</p>\n</div>\n<p></p>\n<div data-jme-visible=\"z-1=x\">\n<p>\\[\\begin{align}^{\\var{t-1}}\\rlap{/}\\var{t}\\ \\;\\;^{\\var{1}} \\var{u}.^{\\var{z-1}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\ \\; \\var{x}\\ \\;\\; \\var{t}}&amp;\\\\\\end{align}\\]<b></b><i></i><u></u><sub></sub><sup></sup><span style=\"text-decoration: line-through;\"></span></p>\n<p>Now carrying out the calculations in each column gives</p>\n<p>\\[\\begin{align}^{\\var{t-1}}\\rlap{/}\\var{t}\\ \\;\\;^{\\var{1}} \\var{u}.^{\\var{z-1}}\\rlap{/}\\var{z}^{\\ \\;1}0&amp;\\\\\\underline{-{\\hspace 0.5cm}\\var{y}.\\ \\; \\var{x}\\ \\;\\; \\var{t}}&amp;\\\\\\var{t-1}\\ \\;\\; \\var{u+10-y}.\\ \\; \\var{z-1-x}\\ \\;\\; \\var{10-t}&amp;\\\\\\end{align}\\]</p>\n</div>\n<p></p>\n<h2>Additional resources</h2>\n<p></p>\n<p>For further examples and questions</p>\n<p><a href=\"https://www.khanacademy.org/math/arithmetic/arith-review-add-subtract/arith-review-regrouping-3-dig/v/basic-regrouping-or-borrowing-when-subtracting-three-digit-numbers\">Multi Digit Subtraction</a>&nbsp;- Khan Academy</p>\n<p><a href=\"https://www.khanacademy.org/math/arithmetic/arith-decimals/arith-review-sub-decimals/v/strategies-for-subtracting-more-complex-decimals-with-tenths\">Subtracting Decimals</a>&nbsp;- Khan Academy</p>", "rulesets": {}, "extensions": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "a/10", "description": "<p>xy.z</p>", "templateType": "anything"}, "f2": {"name": "f2", "group": "Ungrouped variables", "definition": "f/100", "description": "<p>x.yt</p>", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "b/100", "description": "<p>t.uv</p>", "templateType": "anything"}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "100*t+10*u+z", "description": "<p>tuz</p>", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "b/10", "description": "<p>tu.v</p>", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": 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