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Add alternative hints to some exercises for A/B testing

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commit 71f1fb3e6f653f86225af6f208feadade8352df4 1 parent 02650e5
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289 exercises/test/counting_1_alternative.html
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+<!DOCTYPE html>
+<html data-require="math spin graphie word-problems">
+<head>
+ <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
+ <title>Counting 1</title>
+ <script src="../khan-exercise.js"></script>
+</head>
+<body>
+ <div class="exercise">
+ <div class="problems">
+ <div id="page-numbers">
+ <div class="vars">
+ <var id="START">randRange( 20, 50 )</var>
+ <var id="ANSWER">randRange( 20, 40 )</var>
+ <var id="END">START + ANSWER - 1</var>
+ <var id="DO, DID, DONE, PROBLEM">spin( "{do,did,done,problem|read,read,read,{page|paper|article}}" ).split( "," )</var>
+ <var id="PROBLEMS">plural( PROBLEM )</var>
+ </div>
+ <p class="spin">
+ <var>person( 1 )</var> {had to <var>DO</var>|was assigned} <var>PROBLEMS</var> <var>START</var> through
+ <var>END</var> for homework {{last {week|night}}|tonight}.
+ </p>
+ <p class="question spin">
+ If <var>person( 1 )</var> <var>DID</var> all of the <var>PROBLEMS</var> <var>he( 1 )</var> was assigned, how
+ many <var>PROBLEMS</var> did <var>he( 1 )</var> <var>DO</var>?
+ </p>
+
+ <div class="solution" data-type="multiple">
+ <p><span class="sol" data-forms="integer"><var>ANSWER</var></span> <var>PROBLEMS</var></p>
+ </div>
+
+ <div class="hints">
+ <p>
+ Instead of counting <span class="hint_green"><var>PROBLEMS</var> <code><var>START</var></code> through
+ <code><var>END</var></code></span>, we can <span class="hint_blue">subtract
+ <code><var>START - 1</var></code></span> from each number so we can count the
+ <var>PROBLEMS</var> <span class="hint_pink">starting from <code>1</code></span>.
+ </p>
+
+ <div>
+ <p>
+ </p>
+ <p><code>
+ \qquad \array{
+ \hspace{40px} &amp;
+ \hspace{40px} &amp;
+ \hspace{40px} &amp;
+ \hspace{40px} &amp; \\
+
+ \color{<var>GREEN</var>}{<var>START</var>} &amp;
+ \color{<var>GREEN</var>}{<var>START + 1</var>} &amp;
+ \color{<var>GREEN</var>}{<var>START + 2</var>} &amp; &amp;
+ \color{<var>GREEN</var>}{<var>END</var>} \\
+
+ \color{<var>BLUE</var>}{\downarrow} &amp;
+ \color{<var>BLUE</var>}{\downarrow} &amp;
+ \color{<var>BLUE</var>}{\downarrow} &amp; &amp;
+ \color{<var>BLUE</var>}{\downarrow} \\
+
+ \color{<var>BLUE</var>}{\llap{-}<var>START-1</var>} &amp;
+ \color{<var>BLUE</var>}{\llap{-}<var>START-1</var>} &amp;
+ \color{<var>BLUE</var>}{\llap{-}<var>START-1</var>} &amp; \cdots\hphantom{-} &amp;
+ \color{<var>BLUE</var>}{\llap{-}<var>START-1</var>} \\
+
+ \color{<var>BLUE</var>}{\downarrow} &amp;
+ \color{<var>BLUE</var>}{\downarrow} &amp;
+ \color{<var>BLUE</var>}{\downarrow} &amp; &amp;
+ \color{<var>BLUE</var>}{\downarrow} \\
+
+ \color{<var>PINK</var>}{1} &amp;
+ \color{<var>PINK</var>}{2} &amp;
+ \color{<var>PINK</var>}{3} &amp; &amp;
+ \color{<var>PINK</var>}{<var>ANSWER</var>} \\
+ }
+ </p></code>
+ </div>
+
+ <p>
+ Now instead of thinking about <span class="hint_green"><var>PROBLEMS</var> <code><var>START</var></code>
+ through <code><var>END</var></code></span>, we can think about how many <var>PROBLEMS</var>
+ <var>person( 1 )</var> would have <var>DONE</var> if <var>he( 1 )</var> were assigned
+ <span class="hint_pink"><var>PROBLEMS</var> <code>1</code> though <code><var>ANSWER</var></code></span>.
+ </p>
+
+ <p>
+ When we think about it this way, we can see that <var>person( 1 )</var> would have <var>DONE</var>
+ <var>ANSWER</var> <var>PROBLEMS</var>. Hopefully it also makes sense why we can't just subtract
+ the first problem number from the last problem number, since if <var>person( 1 )</var> were
+ assigned <var>PROBLEMS</var> 1 through <var>ANSWER</var>, <var>he( 1 )</var> was assigned <var>ANSWER</var>
+ <var>PROBLEMS</var>, not <var>ANSWER-1</var>.
+ </p>
+
+ <p class="final_answer"><var>person( 1 )</var> <var>DID</var> <var>ANSWER</var> <var>PROBLEMS</var>.</p>
+ </div>
+ </div>
+
+ <div id="slices">
+ <div class="vars">
+ <var id="NUM">randRange( 5, 15 )</var>
+ </div>
+ <p class="spin">A baker has a whole {baguette|loaf of bread}.</p>
+ <p class="question spin">How many cuts must {he|she} make to have exactly <var>NUM</var> even slices?</p>
+ <div class="solution" data-type="multiple">
+ <p><span class="sol" data-forms="integer"><var>NUM -1</var></span> cuts</p>
+ </div>
+ <div class="hints">
+ <div>
+ <div class="graphie" id="loaf">
+ init({
+ range: [ [ -1.5, 18.5 ], [ -4, -0.5 ] ],
+ scale: [ 30, 50 ]
+ });
+
+ graph.loaf = raphael.set();
+ graph.labels = [];
+
+ graph.drawLoaf = function( slices ) {
+ var width = 10 + ( slices - 1 ) * 0.5;
+
+ graph.loaf.remove();
+ jQuery.each( graph.labels, function() {
+ this.remove();
+ });
+
+ style({ stroke: "black", fill: ORANGE, opacity: 0.3 }, function() {
+ graph.loaf.push( arc([ 0.6, -2 ], 1, 90, 270, false ) );
+ graph.loaf.push( arc([ width - 0.6, -2 ], 1, 270, 90, false ) );
+ for ( var slice = 0; slice &lt; slices; slice++ ) {
+ if ( slice === 0 ) {
+ graph.loaf.push( path([ [ 0.6, -1 ],
+ [ 10 * ( slice + 1 ) / slices + slice * 0.5, -1 ],
+ [ 10 * ( slice + 1 ) / slices + slice * 0.5, -3 ],
+ [ 0.6, -3 ] ]) );
+ } else if ( slice === slices - 1 ) {
+ graph.loaf.push( path([
+ [ 10 * ( slice + 1 ) / slices + slice * 0.5 - 0.6, -3 ],
+ [ 10 * slice / slices + slice * 0.5, -3 ],
+ [ 10 * slice / slices + slice * 0.5, -1 ],
+ [ 10 * ( slice + 1 ) / slices + slice * 0.5 - 0.6, -1 ] ]) );
+ } else {
+ graph.loaf.push( path([ [ 10 * slice / slices + slice * 0.5, -1 ],
+ [ 10 * ( slice + 1 ) / slices + slice * 0.5, -1 ],
+ [ 10 * ( slice + 1 ) / slices + slice * 0.5, -3 ],
+ [ 10 * slice / slices + slice * 0.5, -3 ],
+ [ 10 * slice / slices + slice * 0.5, -1 ] ]) );
+ }
+ graph.labels.push(
+ label( [ 10 * ( slice + 0.5 ) / slices + slice * 0.5, -2 ], slice+1, "", false )
+ .css({ fontWeight: "bold", opacity: 1 })
+ );
+ if ( slice !== 0 ) {
+ graph.labels.push(
+ label( [ 10 * slice / slices + slice * 0.5 - 0.2, -3 ], slice, "below", false )
+ .addClass( "hint_blue" ).css({ fontWeight: "bold", opacity: 1 })
+ );
+ }
+ }
+ });
+
+ return loaf;
+ };
+
+ graph.drawLoaf( 2 );
+
+ </div>
+ <p><span class="hint_blue">One cut</span> will make two slices.</p>
+ </div>
+
+ <div>
+ <div class="graphie" data-update="loaf">
+ graph.drawLoaf( 3 );
+ </div>
+ <p><span class="hint_blue">Two cuts</span> will make three slices, and so on.</p>
+ </div>
+
+ <div>
+ <div class="graphie" data-update="loaf">
+ graph.drawLoaf( NUM );
+ </div>
+ <p class="final_answer">
+ Therefore, we need <span class="hint_blue"><var>NUM - 1</var> cuts</span> to make <var>NUM</var> slices.
+ </p>
+ </div>
+ </div>
+ </div>
+
+ <div id="fences">
+ <div class="vars">
+ <var id="LENGTH">randRange( 10, 20 )</var>
+ </div>
+ <p><var>person( 1 )</var> is building a straight fence, with posts one meter apart.</p>
+ <p class="question">
+ If the fence is <var>LENGTH</var> meters long, how many fence posts does <var>he( 1 )</var> need?
+ </p>
+ <div class="solution" data-type="multiple">
+ <p><span class="sol" data-forms="integer"><var>LENGTH + 1</var></span> fence posts</p>
+ </div>
+ <div class="hints">
+ <div>
+ <div class="graphie" id="fence">
+ init({
+ range: [ [ -1, LENGTH + 1 ], [ -2, 3.5 ] ],
+ scale: [ 600 / ( LENGTH + 2 ), 20 ]
+ });
+
+ style({ stroke: null, fill: BLUE, opacity: 0.3 }, function() {
+ path([ [ 0, 0 ], [ 0, 1.8 ], [ 1, 1.8 ], [ 1, 0 ], [ 0, 0 ] ]);
+ });
+
+ style({ stroke: GREEN, strokeWidth: 3 }, function() {
+ line( [ 0, 0 ], [ 0, 2 ] );
+ line( [ 1, 0 ], [ 1, 2 ] );
+ });
+ label( [ 0, 1.8 ], "1", "above", false ).css({ color: GREEN });
+ label( [ 1, 1.8 ], "2", "above", false ).css({ color: GREEN });
+
+ style({ stroke: BLUE, strokeWidth: 2 }, function() {
+ graph.brace = path([ [ 0, -0.2 ], [ 0, -0.4 ], [ 0.5, -0.4 ], [ 0.5, -0.6 ], [ 0.5, -0.4 ],
+ [ 1, -0.4 ], [ 1, -0.2 ] ]);
+ });
+ graph.length = label( [ 0.5, -0.4 ], "1 meter", "below", false ).css({ color: BLUE });
+
+ </div>
+ <p>
+ If the fence is <span class="hint_blue">one meter long</span>, <var>he( 1 )</var> needs
+ <span class="hint_green">two posts</span> (one for each end).
+ </p>
+ </div>
+
+ <div>
+ <div class="graphie" data-update="fence">
+ style({ stroke: null, fill: BLUE, opacity: 0.3 }, function() {
+ path([ [ 1, 0 ], [ 1, 1.8 ], [ 2, 1.8 ], [ 2, 0 ], [ 1, 0 ] ]);
+ });
+
+ style({ stroke: GREEN, strokeWidth: 3 }, function() {
+ line( [ 1, 0 ], [ 1, 2 ] );
+ line( [ 2, 0 ], [ 2, 2 ] );
+ });
+ label( [ 2, 1.8 ], "3", "above", false ).css({ color: GREEN });
+
+ graph.brace.remove();
+ graph.length.remove();
+ style({ stroke: BLUE, strokeWidth: 2 }, function() {
+ graph.brace = path([ [ 0, -0.2 ], [ 0, -0.4 ], [ 1, -0.4 ], [ 1, -0.6 ], [ 1, -0.4 ],
+ [ 2, -0.4 ], [ 2, -0.2 ] ]);
+ });
+ graph.length = label( [ 1, -0.4 ], "2 meters", "below", false ).css({ color: BLUE });
+ </div>
+ <p>
+ If the fence is <span class="hint_blue">two meters long</span>, then <var>he( 1 )</var> needs
+ <span class="hint_green">three posts</span>, and so on.
+ </p>
+ </div>
+
+ <div>
+ <div class="graphie" data-update="fence">
+ for ( var x = 2; x &lt; LENGTH; x++ ) {
+ style({ stroke: null, fill: BLUE, opacity: 0.3 }, function() {
+ path([ [ x, 0 ], [ x, 1.8 ], [ x + 1, 1.8 ], [ x + 1, 0 ], [ x, 0 ] ]);
+ });
+
+ style({ stroke: GREEN, strokeWidth: 3 }, function() {
+ line( [ x, 0 ], [ x, 2 ] );
+ line( [ x + 1, 0 ], [ x + 1, 2 ] );
+ });
+ label( [ x + 1, 1.8 ], x + 2, "above", false ).css({ color: GREEN });
+ }
+
+ graph.brace.remove();
+ graph.length.remove();
+ style({ stroke: BLUE, strokeWidth: 2 }, function() {
+ graph.brace = path([ [ 0, -0.2 ], [ 0, -0.4 ], [ LENGTH/2, -0.4 ], [ LENGTH/2, -0.6 ],
+ [ LENGTH/2, -0.4 ], [ LENGTH, -0.4 ], [ LENGTH, -0.2 ] ]);
+ });
+ graph.length = label( [ LENGTH / 2, -0.4 ], LENGTH + " meters", "below", false ).css({ color: BLUE });
+ </div>
+ <p class="final_answer">
+ Therefore, for <var>his( 1 )</var> <span class="hint_blue"><var>LENGTH</var> meter</span> fence,
+ <var>person( 1 )</var> needs <span class="hint_green"><var>LENGTH + 1</var> posts</span>.
+ </p>
+ </div>
+ </div>
+ </div>
+
+ </div>
+ </div>
+</body>
+</html>
View
1,026 exercises/test/dividing_fractions_alternative.html
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+
+
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+<span id="L117" rel="#L117">117</span>
+</pre>
+ </td>
+ <td width="100%">
+ <div class="highlight"><pre><div class='line' id='LC1'><span class="cp">&lt;!DOCTYPE html&gt;</span></div><div class='line' id='LC2'><span class="nt">&lt;html</span> <span class="na">data-require=</span><span class="s">&quot;math math-format&quot;</span><span class="nt">&gt;</span></div><div class='line' id='LC3'><span class="nt">&lt;head&gt;</span></div><div class='line' id='LC4'> <span class="nt">&lt;meta</span> <span class="na">http-equiv=</span><span class="s">&quot;Content-Type&quot;</span> <span class="na">content=</span><span class="s">&quot;text/html; charset=UTF-8&quot;</span><span class="nt">&gt;</span></div><div class='line' id='LC5'> <span class="nt">&lt;title&gt;</span>Dividing fractions<span class="nt">&lt;/title&gt;</span></div><div class='line' id='LC6'> <span class="nt">&lt;script </span><span class="na">src=</span><span class="s">&quot;../khan-exercise.js&quot;</span><span class="nt">&gt;&lt;/script&gt;</span></div><div class='line' id='LC7'><span class="nt">&lt;/head&gt;</span></div><div class='line' id='LC8'><span class="nt">&lt;body&gt;</span></div><div class='line' id='LC9'> <span class="nt">&lt;div</span> <span class="na">class=</span><span class="s">&quot;exercise&quot;</span><span class="nt">&gt;</span></div><div class='line' id='LC10'> <span class="nt">&lt;div</span> <span class="na">class=</span><span class="s">&quot;vars&quot;</span> <span class="na">data-ensure=</span><span class="s">&quot;getGCD(N1, D1) === 1 &amp;&amp; getGCD(N2, D2) === 1&quot;</span><span class="nt">&gt;</span></div><div class='line' id='LC11'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;NEG1&quot;</span><span class="nt">&gt;</span>randFromArray([1, -1])<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC12'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;NEG1S&quot;</span><span class="nt">&gt;</span>NEG1 === -1 ? &quot;-&quot; : &quot;&quot;<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC13'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;N1&quot;</span><span class="nt">&gt;</span>randRange(1, 9)<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC14'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;D1&quot;</span><span class="nt">&gt;</span>randRange(2, 9)<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC15'><br/></div><div class='line' id='LC16'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;NEG2&quot;</span><span class="nt">&gt;</span>randFromArray([1, -1])<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC17'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;NEG2S&quot;</span><span class="nt">&gt;</span>NEG2 === -1 ? &quot;-&quot; : &quot;&quot;<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC18'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;N2&quot;</span><span class="nt">&gt;</span>randRange(1, 9)<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC19'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;D2&quot;</span><span class="nt">&gt;</span>randRange(2, 9)<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC20'><br/></div><div class='line' id='LC21'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;GCD1&quot;</span><span class="nt">&gt;</span>getGCD( N1, N2 )<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC22'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;SIMP_N1&quot;</span><span class="nt">&gt;</span>N1 / GCD1<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC23'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;SIMP_N2&quot;</span><span class="nt">&gt;</span>N2 / GCD1<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC24'><br/></div><div class='line' id='LC25'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;GCD2&quot;</span><span class="nt">&gt;</span>getGCD( D1, D2 )<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC26'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;SIMP_D1&quot;</span><span class="nt">&gt;</span>D1 / GCD2<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC27'> <span class="nt">&lt;var</span> <span class="na">id=</span><span class="s">&quot;SIMP_D2&quot;</span><span class="nt">&gt;</span>D2 / GCD2<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC28'><br/></div><div class='line' id='LC29'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC30'><br/></div><div class='line' id='LC31'> <span class="nt">&lt;div</span> <span class="na">class=</span><span class="s">&quot;problems&quot;</span><span class="nt">&gt;</span></div><div class='line' id='LC32'> <span class="nt">&lt;div&gt;</span></div><div class='line' id='LC33'> <span class="nt">&lt;div</span> <span class="na">class=</span><span class="s">&quot;problem&quot;</span><span class="nt">&gt;</span>Reduce to lowest terms:<span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC34'> <span class="nt">&lt;div</span> <span class="na">class=</span><span class="s">&quot;question&quot;</span><span class="nt">&gt;</span></div><div class='line' id='LC35'> <span class="nt">&lt;p&gt;&lt;code&gt;&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> \dfrac{<span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span>} \div <span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \dfrac{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>} = {?}<span class="nt">&lt;/code&gt;&lt;/p&gt;</span></div><div class='line' id='LC36'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC37'> <span class="nt">&lt;div</span> <span class="na">class=</span><span class="s">&quot;solution&quot;</span> <span class="na">data-type=</span><span class="s">&quot;rational&quot;</span><span class="nt">&gt;&lt;var&gt;</span>(NEG1 * N1 * NEG2 * D2) / (D1 * N2)<span class="nt">&lt;/var&gt;&lt;/div&gt;</span></div><div class='line' id='LC38'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC39'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC40'><br/></div><div class='line' id='LC41'> <span class="nt">&lt;div</span> <span class="na">class=</span><span class="s">&quot;hints&quot;</span><span class="nt">&gt;</span></div><div class='line' id='LC42'> <span class="nt">&lt;p&gt;</span>Dividing by a fraction is the same as multiplying by the reciprocal.<span class="nt">&lt;/p&gt;</span></div><div class='line' id='LC43'> <span class="nt">&lt;p&gt;</span>The reciprocal of <span class="nt">&lt;code</span> <span class="na">class=</span><span class="s">&quot;hint_green&quot;</span><span class="nt">&gt;&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \dfrac{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}<span class="nt">&lt;/code&gt;</span> is <span class="nt">&lt;code</span> <span class="na">class=</span><span class="s">&quot;hint_blue&quot;</span><span class="nt">&gt;&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \dfrac{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}<span class="nt">&lt;/code&gt;</span>. We just flipped the numerator and denominator.</div><div class='line' id='LC44'><br/></div><div class='line' id='LC45'> <span class="nt">&lt;div&gt;</span></div><div class='line' id='LC46'> <span class="nt">&lt;p&gt;</span></div><div class='line' id='LC47'> Since multiplying by the reciprocal is the same as dividing, lets use the reciprocal to change the problem into a multiplication problem:</div><div class='line' id='LC48'> <span class="nt">&lt;/p&gt;</span></div><div class='line' id='LC49'> <span class="nt">&lt;p&gt;&lt;code&gt;</span></div><div class='line' id='LC50'> \begin{eqnarray}</div><div class='line' id='LC51'> <span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span>} \color{<span class="nt">&lt;var&gt;</span>GREEN<span class="nt">&lt;/var&gt;</span>}{\div <span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}}</div><div class='line' id='LC52'> <span class="ni">&amp;amp;</span> \qquad = \qquad <span class="ni">&amp;amp;</span></div><div class='line' id='LC53'> <span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span>} \color{<span class="nt">&lt;var&gt;</span>BLUE<span class="nt">&lt;/var&gt;</span>}{\times <span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}}</div><div class='line' id='LC54'> \end{eqnarray}</div><div class='line' id='LC55'> <span class="nt">&lt;/code&gt;&lt;/p&gt;</span></div><div class='line' id='LC56'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC57'><br/></div><div class='line' id='LC58'> <span class="nt">&lt;div&gt;</span></div><div class='line' id='LC59'> <span class="nt">&lt;p&gt;</span></div><div class='line' id='LC60'> Because we&#39;re now multiplying fractions instead of dividing them, all we have to do is multiply the numerators and the denominators:</div><div class='line' id='LC61'> <span class="nt">&lt;/p&gt;</span></div><div class='line' id='LC62'> <span class="nt">&lt;p&gt;&lt;code&gt;</span></div><div class='line' id='LC63'> \begin{eqnarray}</div><div class='line' id='LC64'> \hphantom{<span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span>} \div \color{<span class="nt">&lt;var&gt;</span>GREEN<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}}}</div><div class='line' id='LC65'> <span class="ni">&amp;amp;</span> \qquad = \qquad <span class="ni">&amp;amp;</span></div><div class='line' id='LC66'> \dfrac{<span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> <span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span> \times \color{<span class="nt">&lt;var&gt;</span>BLUE<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> <span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span> \times \color{<span class="nt">&lt;var&gt;</span>BLUE<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}}</div><div class='line' id='LC67'> \end{eqnarray}</div><div class='line' id='LC68'> <span class="nt">&lt;/code&gt;&lt;/p&gt;</span></div><div class='line' id='LC69'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC70'><br/></div><div class='line' id='LC71'> <span class="nt">&lt;div</span> <span class="na">data-if=</span><span class="s">&quot;GCD1 !== 1 || GCD2 !== 1&quot;</span> <span class="na">data-unwrap</span><span class="nt">&gt;</span></div><div class='line' id='LC72'> <span class="nt">&lt;p&gt;</span></div><div class='line' id='LC73'> We could just multiply everything to get <span class="nt">&lt;code&gt;</span>\frac{<span class="nt">&lt;var&gt;</span>NEG1 * NEG2 * N1 * D2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>N2 * D1<span class="nt">&lt;/var&gt;</span>}<span class="nt">&lt;/code&gt;</span></div><div class='line' id='LC74'> and then try to reduce that to get the final answer, but it&#39;s easier if we can find and reduce some common factors before we multiply.</div><div class='line' id='LC75'> <span class="nt">&lt;/p&gt;</span></div><div class='line' id='LC76'> <span class="nt">&lt;div&gt;</span></div><div class='line' id='LC77'> <span class="nt">&lt;p&gt;</span>In this case, we can <span class="nt">&lt;span</span> <span class="na">class=</span><span class="s">&quot;hint_pink&quot;</span><span class="nt">&gt;</span>divide<span class="nt">&lt;/span&gt;</span> the <span class="nt">&lt;code</span> <span class="na">class=</span><span class="s">&quot;hint_blue&quot;</span><span class="nt">&gt;&lt;var&gt;</span>NEG2 * D2<span class="nt">&lt;/var&gt;&lt;/code&gt;</span> in the numerator and the <span class="nt">&lt;code&gt;&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;&lt;/code&gt;</span></div><div class='line' id='LC78'> in the denominator <span class="nt">&lt;span</span> <span class="na">class=</span><span class="s">&quot;hint_pink&quot;</span><span class="nt">&gt;</span>by <span class="nt">&lt;code&gt;&lt;var&gt;</span>GCD2<span class="nt">&lt;/var&gt;&lt;/code&gt;&lt;/span&gt;</span>:<span class="nt">&lt;/p&gt;</span></div><div class='line' id='LC79'> <span class="nt">&lt;p&gt;&lt;code&gt;</span></div><div class='line' id='LC80'> \begin{eqnarray}</div><div class='line' id='LC81'> \hphantom{<span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span>} \div \color{<span class="nt">&lt;var&gt;</span>GREEN<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}}}</div><div class='line' id='LC82'> <span class="ni">&amp;amp;</span> \qquad = \qquad <span class="ni">&amp;amp;</span></div><div class='line' id='LC83'> \dfrac{<span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> <span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span> \times \color{<span class="nt">&lt;var&gt;</span>PINK<span class="nt">&lt;/var&gt;</span>}{\cancel{\color{<span class="nt">&lt;var&gt;</span>BLUE<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> <span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}}^{<span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;&lt;var&gt;</span>SIMP_D2<span class="nt">&lt;/var&gt;</span>}}}{\color{<span class="nt">&lt;var&gt;</span>PINK<span class="nt">&lt;/var&gt;</span>}{\cancel{\color{black}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span>}}^{<span class="nt">&lt;var&gt;</span>SIMP_D1<span class="nt">&lt;/var&gt;</span>}} \times \color{<span class="nt">&lt;var&gt;</span>BLUE<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}} \\ \\</div><div class='line' id='LC84'> <span class="ni">&amp;amp;</span> \qquad = \qquad <span class="ni">&amp;amp;</span></div><div class='line' id='LC85'> \dfrac{<span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> <span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span> \times \color{<span class="nt">&lt;var&gt;</span>PINK<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;&lt;var&gt;</span>SIMP_D2<span class="nt">&lt;/var&gt;</span>}}{\color{<span class="nt">&lt;var&gt;</span>PINK<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>SIMP_D1<span class="nt">&lt;/var&gt;</span>} \times \color{<span class="nt">&lt;var&gt;</span>BLUE<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}}</div><div class='line' id='LC86'> \end{eqnarray}</div><div class='line' id='LC87'> <span class="nt">&lt;/code&gt;&lt;/p&gt;</span></div><div class='line' id='LC88'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC89'><br/></div><div class='line' id='LC90'> <span class="nt">&lt;div&gt;</span></div><div class='line' id='LC91'> <span class="nt">&lt;p</span> <span class="na">class=</span><span class="s">&quot;final_answer&quot;</span><span class="nt">&gt;</span>After reducing the common factors, it&#39;s easier to multiply and get the simplified answer:<span class="nt">&lt;/p&gt;</span></div><div class='line' id='LC92'> <span class="nt">&lt;p&gt;&lt;code&gt;</span></div><div class='line' id='LC93'> \begin{eqnarray}</div><div class='line' id='LC94'> \hphantom{\color{gray}{<span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span>} \div <span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}}}</div><div class='line' id='LC95'> <span class="ni">&amp;amp;</span> \qquad = \qquad <span class="ni">&amp;amp;</span></div><div class='line' id='LC96'> <span class="nt">&lt;var&gt;</span>fractionReduce(NEG1 * N1 * NEG2 * D2, D1 * N2)<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC97'> \end{eqnarray}</div><div class='line' id='LC98'> <span class="nt">&lt;/code&gt;&lt;/p&gt;</span></div><div class='line' id='LC99'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC100'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC101'><br/></div><div class='line' id='LC102'> <span class="nt">&lt;div</span> <span class="na">data-else</span><span class="nt">&gt;</span></div><div class='line' id='LC103'> <span class="nt">&lt;p</span> <span class="na">class=</span><span class="s">&quot;final_answer&quot;</span><span class="nt">&gt;</span>Just multiply to get the final answer. Double-check that it&#39;s simplified:<span class="nt">&lt;/p&gt;</span></div><div class='line' id='LC104'> <span class="nt">&lt;p&gt;&lt;code&gt;</span></div><div class='line' id='LC105'> \begin{eqnarray}</div><div class='line' id='LC106'> \hphantom{\color{gray}{<span class="nt">&lt;var&gt;</span>NEG1S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N1<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D1<span class="nt">&lt;/var&gt;</span>} \div <span class="nt">&lt;var&gt;</span>NEG2S<span class="nt">&lt;/var&gt;</span> \frac{<span class="nt">&lt;var&gt;</span>N2<span class="nt">&lt;/var&gt;</span>}{<span class="nt">&lt;var&gt;</span>D2<span class="nt">&lt;/var&gt;</span>}}}</div><div class='line' id='LC107'> <span class="ni">&amp;amp;</span> \qquad = \qquad <span class="ni">&amp;amp;</span></div><div class='line' id='LC108'> <span class="nt">&lt;var&gt;</span>fractionReduce(NEG1 * N1 * NEG2 * D2, D1 * N2)<span class="nt">&lt;/var&gt;</span></div><div class='line' id='LC109'> \end{eqnarray}</div><div class='line' id='LC110'> <span class="nt">&lt;/code&gt;&lt;/p&gt;</span></div><div class='line' id='LC111'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC112'><br/></div><div class='line' id='LC113'><br/></div><div class='line' id='LC114'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC115'> <span class="nt">&lt;/div&gt;</span></div><div class='line' id='LC116'><span class="nt">&lt;/body&gt;</span></div><div class='line' id='LC117'><span class="nt">&lt;/html&gt;</span></div></pre></div>
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+ </div>
+
+ <div id="footer-push"></div><!-- hack for sticky footer -->
+ </div><!-- end of wrapper - hack for sticky footer -->
+
+ <!-- footer -->
+ <div id="footer" >
+
+ <div class="upper_footer">
+ <div class="container clearfix">
+
+ <!--[if IE]><h4 id="blacktocat_ie">GitHub Links</h4><![endif]-->
+ <![if !IE]><h4 id="blacktocat">GitHub Links</h4><![endif]>
+
+ <ul class="footer_nav">
+ <h4>GitHub</h4>
+ <li><a href="https://github.com/about">About</a></li>
+ <li><a href="https://github.com/blog">Blog</a></li>
+ <li><a href="https://github.com/features">Features</a></li>
+ <li><a href="https://github.com/contact">Contact &amp; Support</a></li>
+ <li><a href="https://github.com/training">Training</a></li>
+ <li><a href="http://enterprise.github.com/">GitHub Enterprise</a></li>
+ <li><a href="http://status.github.com/">Site Status</a></li>
+ </ul>
+
+ <ul class="footer_nav">
+ <h4>Clients</h4>
+ <li><a href="http://mac.github.com/">GitHub for Mac</a></li>
+ <li><a href="http://windows.github.com/">GitHub for Windows</a></li>
+ <li><a href="http://eclipse.github.com/">GitHub for Eclipse</a></li>
+ <li><a href="http://mobile.github.com/">GitHub Mobile Apps</a></li>
+ </ul>
+
+ <ul class="footer_nav">
+ <h4>Tools</h4>
+ <li><a href="http://get.gaug.es/">Gauges: Web analytics</a></li>
+ <li><a href="http://speakerdeck.com">Speaker Deck: Presentations</a></li>
+ <li><a href="https://gist.github.com">Gist: Code snippets</a></li>
+
+ <h4 class="second">Extras</h4>
+ <li><a href="http://jobs.github.com/">Job Board</a></li>
+ <li><a href="http://shop.github.com/">GitHub Shop</a></li>
+ <li><a href="http://octodex.github.com/">The Octodex</a></li>
+ </ul>
+
+ <ul class="footer_nav">
+ <h4>Documentation</h4>
+ <li><a href="http://help.github.com/">GitHub Help</a></li>
+ <li><a href="http://developer.github.com/">Developer API</a></li>
+ <li><a href="http://github.github.com/github-flavored-markdown/">GitHub Flavored Markdown</a></li>
+ <li><a href="http://pages.github.com/">GitHub Pages</a></li>
+ </ul>
+
+ </div><!-- /.site -->
+ </div><!-- /.upper_footer -->
+
+<div class="lower_footer">
+ <div class="container clearfix">
+ <!--[if IE]><div id="legal_ie"><![endif]-->
+ <![if !IE]><div id="legal"><![endif]>
+ <ul>
+ <li><a href="https://github.com/site/terms">Terms of Service</a></li>
+ <li><a href="https://github.com/site/privacy">Privacy</a></li>
+ <li><a href="https://github.com/security">Security</a></li>
+ </ul>
+
+ <p>&copy; 2012 <span title="0.13688s from fe19.rs.github.com">GitHub</span> Inc. All rights reserved.</p>
+ </div><!-- /#legal or /#legal_ie-->
+
+ </div><!-- /.site -->
+</div><!-- /.lower_footer -->
+
+ </div><!-- /#footer -->
+
+
+
+<div id="keyboard_shortcuts_pane" class="instapaper_ignore readability-extra" style="display:none">
+ <h2>Keyboard Shortcuts <small><a href="#" class="js-see-all-keyboard-shortcuts">(see all)</a></small></h2>
+
+ <div class="columns threecols">
+ <div class="column first">
+ <h3>Site wide shortcuts</h3>
+ <dl class="keyboard-mappings">
+ <dt>s</dt>
+ <dd>Focus site search</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>?</dt>
+ <dd>Bring up this help dialog</dd>
+ </dl>
+ </div><!-- /.column.first -->
+
+ <div class="column middle" style='display:none'>
+ <h3>Commit list</h3>
+ <dl class="keyboard-mappings">
+ <dt>j</dt>
+ <dd>Move selection down</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>k</dt>
+ <dd>Move selection up</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>c <em>or</em> o <em>or</em> enter</dt>
+ <dd>Open commit</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>y</dt>
+ <dd>Expand URL to its canonical form</dd>
+ </dl>
+ </div><!-- /.column.first -->
+
+ <div class="column last js-hidden-pane" style='display:none'>
+ <h3>Pull request list</h3>
+ <dl class="keyboard-mappings">
+ <dt>j</dt>
+ <dd>Move selection down</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>k</dt>
+ <dd>Move selection up</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>o <em>or</em> enter</dt>
+ <dd>Open issue</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt><span class="platform-mac">⌘</span><span class="platform-other">ctrl</span> <em>+</em> enter</dt>
+ <dd>Submit comment</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt><span class="platform-mac">⌘</span><span class="platform-other">ctrl</span> <em>+</em> shift p</dt>
+ <dd>Preview comment</dd>
+ </dl>
+ </div><!-- /.columns.last -->
+
+ </div><!-- /.columns.equacols -->
+
+ <div class="js-hidden-pane" style='display:none'>
+ <div class="rule"></div>
+
+ <h3>Issues</h3>
+
+ <div class="columns threecols">
+ <div class="column first">
+ <dl class="keyboard-mappings">
+ <dt>j</dt>
+ <dd>Move selection down</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>k</dt>
+ <dd>Move selection up</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>x</dt>
+ <dd>Toggle selection</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>o <em>or</em> enter</dt>
+ <dd>Open issue</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt><span class="platform-mac">⌘</span><span class="platform-other">ctrl</span> <em>+</em> enter</dt>
+ <dd>Submit comment</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt><span class="platform-mac">⌘</span><span class="platform-other">ctrl</span> <em>+</em> shift p</dt>
+ <dd>Preview comment</dd>
+ </dl>
+ </div><!-- /.column.first -->
+ <div class="column last">
+ <dl class="keyboard-mappings">
+ <dt>c</dt>
+ <dd>Create issue</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>l</dt>
+ <dd>Create label</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>i</dt>
+ <dd>Back to inbox</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>u</dt>
+ <dd>Back to issues</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>/</dt>
+ <dd>Focus issues search</dd>
+ </dl>
+ </div>
+ </div>
+ </div>
+
+ <div class="js-hidden-pane" style='display:none'>
+ <div class="rule"></div>
+
+ <h3>Issues Dashboard</h3>
+
+ <div class="columns threecols">
+ <div class="column first">
+ <dl class="keyboard-mappings">
+ <dt>j</dt>
+ <dd>Move selection down</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>k</dt>
+ <dd>Move selection up</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>o <em>or</em> enter</dt>
+ <dd>Open issue</dd>
+ </dl>
+ </div><!-- /.column.first -->
+ </div>
+ </div>
+
+ <div class="js-hidden-pane" style='display:none'>
+ <div class="rule"></div>
+
+ <h3>Network Graph</h3>
+ <div class="columns equacols">
+ <div class="column first">
+ <dl class="keyboard-mappings">
+ <dt><span class="badmono">←</span> <em>or</em> h</dt>
+ <dd>Scroll left</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt><span class="badmono">→</span> <em>or</em> l</dt>
+ <dd>Scroll right</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt><span class="badmono">↑</span> <em>or</em> k</dt>
+ <dd>Scroll up</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt><span class="badmono">↓</span> <em>or</em> j</dt>
+ <dd>Scroll down</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>t</dt>
+ <dd>Toggle visibility of head labels</dd>
+ </dl>
+ </div><!-- /.column.first -->
+ <div class="column last">
+ <dl class="keyboard-mappings">
+ <dt>shift <span class="badmono">←</span> <em>or</em> shift h</dt>
+ <dd>Scroll all the way left</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>shift <span class="badmono">→</span> <em>or</em> shift l</dt>
+ <dd>Scroll all the way right</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>shift <span class="badmono">↑</span> <em>or</em> shift k</dt>
+ <dd>Scroll all the way up</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>shift <span class="badmono">↓</span> <em>or</em> shift j</dt>
+ <dd>Scroll all the way down</dd>
+ </dl>
+ </div><!-- /.column.last -->
+ </div>
+ </div>
+
+ <div class="js-hidden-pane" >
+ <div class="rule"></div>
+ <div class="columns threecols">
+ <div class="column first js-hidden-pane" >
+ <h3>Source Code Browsing</h3>
+ <dl class="keyboard-mappings">
+ <dt>t</dt>
+ <dd>Activates the file finder</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>l</dt>
+ <dd>Jump to line</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>w</dt>
+ <dd>Switch branch/tag</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>y</dt>
+ <dd>Expand URL to its canonical form</dd>
+ </dl>
+ </div>
+ </div>
+ </div>
+
+ <div class="js-hidden-pane" style='display:none'>
+ <div class="rule"></div>
+ <div class="columns threecols">
+ <div class="column first">
+ <h3>Browsing Commits</h3>
+ <dl class="keyboard-mappings">
+ <dt><span class="platform-mac">⌘</span><span class="platform-other">ctrl</span> <em>+</em> enter</dt>
+ <dd>Submit comment</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>escape</dt>
+ <dd>Close form</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>p</dt>
+ <dd>Parent commit</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>o</dt>
+ <dd>Other parent commit</dd>
+ </dl>
+ </div>
+ </div>
+ </div>
+
+ <div class="js-hidden-pane" style='display:none'>
+ <div class="rule"></div>
+ <h3>Notifications</h3>
+
+ <div class="columns threecols">
+ <div class="column first">
+ <dl class="keyboard-mappings">
+ <dt>j</dt>
+ <dd>Move selection down</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>k</dt>
+ <dd>Move selection up</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>o <em>or</em> enter</dt>
+ <dd>Open notification</dd>
+ </dl>
+ </div><!-- /.column.first -->
+
+ <div class="column second">
+ <dl class="keyboard-mappings">
+ <dt>e <em>or</em> shift i <em>or</em> y</dt>
+ <dd>Mark as read</dd>
+ </dl>
+ <dl class="keyboard-mappings">
+ <dt>shift m</dt>
+ <dd>Mute thread</dd>
+ </dl>
+ </div><!-- /.column.first -->
+ </div>
+ </div>
+
+</div>
+
+ <div id="markdown-help" class="instapaper_ignore readability-extra">
+ <h2>Markdown Cheat Sheet</h2>
+
+ <div class="cheatsheet-content">
+
+ <div class="mod">
+ <div class="col">
+ <h3>Format Text</h3>
+ <p>Headers</p>
+ <pre>
+# This is an &lt;h1&gt; tag
+## This is an &lt;h2&gt; tag
+###### This is an &lt;h6&gt; tag</pre>
+ <p>Text styles</p>
+ <pre>
+*This text will be italic*
+_This will also be italic_
+**This text will be bold**
+__This will also be bold__
+
+*You **can** combine them*
+</pre>
+ </div>
+ <div class="col">
+ <h3>Lists</h3>
+ <p>Unordered</p>
+ <pre>
+* Item 1
+* Item 2
+ * Item 2a
+ * Item 2b</pre>
+ <p>Ordered</p>
+ <pre>
+1. Item 1
+2. Item 2
+3. Item 3
+ * Item 3a
+ * Item 3b</pre>
+ </div>
+ <div class="col">
+ <h3>Miscellaneous</h3>
+ <p>Images</p>
+ <pre>
+![GitHub Logo](/images/logo.png)
+Format: ![Alt Text](url)
+</pre>
+ <p>Links</p>
+ <pre>
+http://github.com - automatic!
+[GitHub](http://github.com)</pre>
+<p>Blockquotes</p>
+ <pre>
+As Kanye West said:
+
+> We're living the future so
+> the present is our past.
+</pre>
+ </div>
+ </div>
+ <div class="rule"></div>
+
+ <h3>Code Examples in Markdown</h3>
+ <div class="col">
+ <p>Syntax highlighting with <a href="http://github.github.com/github-flavored-markdown/" title="GitHub Flavored Markdown" target="_blank">GFM</a></p>
+ <pre>
+```javascript
+function fancyAlert(arg) {
+ if(arg) {
+ $.facebox({div:'#foo'})
+ }
+}
+```</pre>
+ </div>
+ <div class="col">
+ <p>Or, indent your code 4 spaces</p>
+ <pre>
+Here is a Python code example
+without syntax highlighting:
+
+ def foo:
+ if not bar:
+ return true</pre>
+ </div>
+ <div class="col">
+ <p>Inline code for comments</p>
+ <pre>
+I think you should use an
+`&lt;addr&gt;` element here instead.</pre>
+ </div>
+ </div>
+
+ </div>
+</div>
+
+
+ <div id="ajax-error-message" class="flash flash-error">
+ <span class="mini-icon mini-icon-exclamation"></span>
+ Something went wrong with that request. Please try again.
+ <a href="#" class="mini-icon mini-icon-remove-close ajax-error-dismiss"></a>
+ </div>
+
+ <div id="logo-popup">
+ <h2>Looking for the GitHub logo?</h2>
+ <ul>
+ <li>
+ <h4>GitHub Logo</h4>
+ <a href="http://github-media-downloads.s3.amazonaws.com/GitHub_Logos.zip"><img alt="Github_logo" src="https://a248.e.akamai.net/assets.github.com/images/modules/about_page/github_logo.png?1340935010" /></a>
+ <a href="http://github-media-downloads.s3.amazonaws.com/GitHub_Logos.zip" class="minibutton download">Download</a>
+ </li>
+ <li>
+ <h4>The Octocat</h4>
+ <a href="http://github-media-downloads.s3.amazonaws.com/Octocats.zip"><img alt="Octocat" src="https://a248.e.akamai.net/assets.github.com/images/modules/about_page/octocat.png?1340935010" /></a>
+ <a href="http://github-media-downloads.s3.amazonaws.com/Octocats.zip" class="minibutton download">Download</a>
+ </li>
+ </ul>
+ </div>
+
+
+
+ <span id='server_response_time' data-time='0.13920' data-host='fe19'></span>
+
+ </body>
+</html>
+
View
1,107 exercises/test/exponents_2_alternative.html
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+
+
+
+<!DOCTYPE html>
+<html>
+ <head prefix="og: http://ogp.me/ns# fb: http://ogp.me/ns/fb# githubog: http://ogp.me/ns/fb/githubog#">
+ <meta charset='utf-8'>
+ <meta http-equiv="X-UA-Compatible" content="IE=edge">
+ <title>khan-exercises/exercises/exponents_2.html at 85580f5e16aa7a65d392c1cf6d227be952a73441 · Khan/khan-exercises · GitHub</title>
+ <link rel="search" type="application/opensearchdescription+xml" href="/opensearch.xml" title="GitHub" />
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+
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+
+
+
+ <link rel='permalink' href='/Khan/khan-exercises/blob/85580f5e16aa7a65d392c1cf6d227be952a73441/exercises/exponents_2.html'>
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+ <meta property="og:description" content="khan-exercises - A framework for building exercises to work with Khan Academy."/>
+
+ <meta name="description" content="khan-exercises - A framework for building exercises to work with Khan Academy." />
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+
+ </head>
+
+
+ <body class="logged_out page-blob vis-public env-production ">
+ <div id="wrapper">
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+ <a class="site-logo " href="https://github.com/">
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+ <!--
+ make sure to use fully qualified URLs here since this nav
+ is used on error pages on other domains
+ -->
+ <ul class="top-nav logged_out">
+ <li class="pricing"><a href="https://github.com/plans">Signup and Pricing</a></li>
+ <li class="explore"><a href="https://github.com/explore">Explore GitHub</a></li>
+ <li class="features"><a href="https://github.com/features">Features</a></li>
+ <li class="blog"><a href="https://github.com/blog">Blog</a></li>
+ <li class="login"><a href="https://github.com/login?return_to=%2FKhan%2Fkhan-exercises%2Fblob%2F85580f5e16aa7a65d392c1cf6d227be952a73441%2Fexercises%2Fexponents_2.html">Sign in</a></li>
+ </ul>
+
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+
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+ <div class="container hentry">
+
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+ </li>
+ </ul>
+
+ <h1 itemscope itemtype="http://data-vocabulary.org/Breadcrumb" class="entry-title public">
+ <span class="repo-label"><span>public</span></span>
+ <span class="mega-icon mega-icon-public-repo"></span>
+ <span class="author vcard">
+<a href="/Khan" class="url fn" itemprop="url" rel="author"> <span itemprop="title">Khan</span>
+ </a></span> /
+ <strong><a href="/Khan/khan-exercises" class="js-current-repository">khan-exercises</a></strong>
+ </h1>
+ </div>
+
+
+
+ <ul class="tabs">
+ <li><a href="/Khan/khan-exercises/tree/" class="selected" highlight="repo_sourcerepo_downloadsrepo_commitsrepo_tagsrepo_branches">Code</a></li>
+ <li><a href="/Khan/khan-exercises/network" highlight="repo_network">Network</a></li>
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+<span id="L90" rel="#L90">90</span>
+<span id="L91" rel="#L91">91</span>
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