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<!DOCTYPE html>
<html data-require="math word-problems">
<head>
<meta charset="UTF-8" />
<title>Rate problems 1</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="SPEED1">11 + rand(9)</var>
<var id="SPEED2">41 + rand(9)</var>
<var id="AVG">31 + rand(9)</var>
<var id="TIME">1 + rand(9)</var>
<var id="DIST">AVG * TIME</var>
<var id="DIST2">(SPEED1 * SPEED2 * TIME - DIST * SPEED2) / (SPEED1 - SPEED2)</var>
<var id="DIST1">DIST - DIST2</var>
</div>
<div class="problems">
<div id="original">
<div class="problem">
<p><var>person(1)</var> traveled by <var>vehicle(1)</var> at an average speed of <var>SPEED1</var> miles per hour.</p>
<p>Then, <var>he(1)</var> traveled by <var>vehicle(2)</var> at an average speed of <var>SPEED2</var> miles per hour.</p>
<p>In total, <var>he(1)</var> traveled <var>DIST</var> miles for <var>plural(TIME, "hour")</var>.</p>
</div>
<p class="question">How many miles did <var>person(1)</var> travel by <var>vehicle(1)</var>? (Round to the nearest mile.)</p>
<p class="solution"><var>round(DIST1)</var></p>
<div class="hints" data-apply="appendContents">
<p id="final-hint"><code>d_<var>vehicleVar(1)</var> = <var>round(DIST1)</var></code> miles</p>
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</div>
<div data-type="original">
<p class="question">How many miles did <var>person(1)</var> travel by <var>vehicle(2)</var>? (Round to the nearest mile.)</p>
<p class="solution"><var>round(DIST2)</var></p>
<div class="hints" data-apply="appendContents">
<p id="final-hint"><code>d_<var>vehicleVar(2)</var> = <var>round(DIST2)</var></code> miles</p>
</div>
</div>
<div data-type="original">
<p class="question">How many minutes did <var>person(1)</var> travel by <var>vehicle(1)</var>? (Round to the nearest minute.)</p>
<p class="solution"><var>round(DIST1 / SPEED1 * 60)</var></p>
<div class="hints" data-apply="appendContents">
<p id="final-hint"><code>t_<var>vehicleVar(1)</var> = <var>round(DIST1 / SPEED1 * 60)</var></code> minutes</p>
</div>
</div>
<div data-type="original">
<p class="question">How many minutes did <var>person(1)</var> travel by <var>vehicle(2)</var>? (Round to the nearest minute.)</p>
<p class="solution"><var>round(DIST2 / SPEED2 * 60)</var></p>
<div class="hints" data-apply="appendContents">
<p id="final-hint"><code>t_<var>vehicleVar(2)</var> = <var>round(DIST2 / SPEED2 * 60)</var></code> minutes</p>
</div>
</div>
</div>
<div class="hints">
<p>Remember that <code>d = r * t</code>, or written another way, <code>t = d / r</code>.</p>
<div>
<p><code>d_<var>vehicleVar(1)</var> =</code> distance that <var>person(1)</var> traveled by <var>vehicle(1)</var></p>
<p><code>d_<var>vehicleVar(2)</var> =</code> distance that <var>person(1)</var> traveled by <var>vehicle(2)</var></p>
<p>Total distance: <code class="hint_orange">d_<var>vehicleVar(1)</var> + d_<var>vehicleVar(2)</var> = <var>DIST</var></code></p>
</div>
<p>Total time: <code class="hint_blue">t_<var>vehicleVar(1)</var> + t_<var>vehicleVar(2)</var> = <var>TIME</var></code></p>
<p><code class="hint_blue">t_<var>vehicleVar(1)</var> = (d_<var>vehicleVar(1)</var> / <var>SPEED1</var>)</code> and
<code class="hint_blue">t_<var>vehicleVar(2)</var> = (d_<var>vehicleVar(2)</var> / <var>SPEED2</var>)</code></p>
<p>Substitute the blue equations for:
<code>(d_<var>vehicleVar(1)</var> / <var>SPEED1</var>) + (d_<var>vehicleVar(2)</var> / <var>SPEED2</var>) = <var>TIME</var></code></p>
<p>Multiply the above equation by <var>-1 * SPEED1</var>:
<code class="hint_orange">-d_<var>vehicleVar(1)</var> - (<var>SPEED1</var> / <var>SPEED2</var>) * d_<var>vehicleVar(2)</var> = <var>-SPEED1 * TIME</var></code></p>
<p>Add the two orange equations for:
<code>(<var>SPEED2 - SPEED1</var> / <var>SPEED2</var>) * d_<var>vehicleVar(2)</var> = <var>DIST - (SPEED1 * TIME)</var></code></p>
<p>Simplify and round to the nearest integer:<br>
<code>d_<var>vehicleVar(2)</var> = <var>round(DIST2)</var></code> miles<br>
<code>d_<var>vehicleVar(1)</var> = <var>round(DIST1)</var></code> miles<br>
<code>t_<var>vehicleVar(1)</var> = <var>round(DIST1 / SPEED1 * 60)</var></code> minutes<br>
<code>t_<var>vehicleVar(2)</var> = <var>round(DIST2 / SPEED2 * 60)</var></code> minutes</p>
</div>
</div>
</body>
</html>
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