Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with HTTPS or Subversion.

Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

147 lines (143 sloc) 8.723 kb
<!DOCTYPE html>
<html data-require="math math-format word-problems">
<head>
<meta charset="UTF-8" />
<title>Rate problems 2</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="average-speed">
<div class="vars">
<var id="DISTANCE">randRange( 2, 6 )</var>
<var id="FACTORS">shuffle( getPrimeFactorization( 60 ).concat( getPrimeFactorization( DISTANCE ) ) )</var>
<var id="MULTIPLY">
(function() {
return function( factors ) {
var product = 1;
for ( var i = 0; i &lt; factors.length; i++ ) {
product *= factors[ i ];
}
return product;
}
})()
</var>
<var id="SPLIT">randRange( 3, FACTORS.length - 2 )</var>
<var id="TIME_UP">MULTIPLY( FACTORS.slice( 0, SPLIT ) )</var>
<var id="RATE_UP">MULTIPLY( FACTORS.slice( SPLIT ) )</var>
<var id="K">
(function() {
if ( RATE_UP % 3 === 0 ) {
return 2;
} else {
return 3;
}
})()
</var>
<var id="RATE_DOWN">K * RATE_UP</var>
<var id="TIME_DOWN">60 * DISTANCE / RATE_DOWN</var>
<var id="RATE_AVG">60 * 2 * DISTANCE / ( TIME_UP + TIME_DOWN )</var>
</div>
<p class="problem">Starting at home, <var>person( 1 )</var> traveled uphill to the <var>store( 1 )</var> store for <var>TIME_UP</var> minutes at just <var>RATE_UP</var> mph. <var>He( 1 )</var> then traveled back home along the same path downhill at a speed of <var>K * RATE_UP</var> mph.</p>
<p class="question">What is <var>his( 1 )</var> average speed for the entire trip from home to the <var>store( 1 )</var> store and back?</p>
<div class="solution" data-type="multiple">
<span class="sol"><var>RATE_AVG</var></span> mph
</div>
<div class="hints">
<p>The average speed is not just the average of <var>RATE_UP</var> mph and <var>RATE_DOWN</var> mph.</p>
<p><var>He( 1 )</var> traveled for a longer time uphill (since <var>he( 1 )</var> was going slower), so we can estimate that the average speed is closer to <var>RATE_UP</var> mph than <var>RATE_DOWN</var> mph.</p>
<div>
<p>To calculate the average speed, we will make use of the following:</p>
<p><code>\text{average speed} = \dfrac{\color{<var>KhanUtil.BLUE</var>}{\text{total distance}}}{\color{<var>KhanUtil.ORANGE</var>}{\text{total time}}}</code></p>
<p><code>\text{distance uphill} = \text{distance downhill}</code></p>
</div>
<p>What was the total distance traveled?</p>
<div>
<p><code>\color{<var>KhanUtil.BLUE</var>}{\begin{align*}\text{total distance} &= \text{distance uphill} + \text{distance downhill}\\
&= 2 \times \text{distance uphill}\end{align*}}</code></p>
<p><code>\begin{align*}\text{distance uphill} &= \text{speed uphill} \times \text{time uphill}\\
&= <var>RATE_UP</var>\text{ mph} \times <var>TIME_UP</var>\text{ minutes}\\
&= <var>DISTANCE</var>\text{ miles}\end{align*}</code></p>
</div>
<div>
<p>Substituting to find the total distance:</p>
<p><code>\color{<var>KhanUtil.BLUE</var>}{\text{total distance} = <var>2 * DISTANCE</var>\text{ miles}}</code></p>
</div>
<p>What was the total time spent traveling?</p>
<div>
<p><code>\color{<var>KhanUtil.ORANGE</var>}{\text{total time} = \text{time uphill} + \text{time downhill}}</code></p>
<p><code>\begin{align*}\text{time downhill} &= \dfrac{\text{distance downhill}}{\text{speed downhill}}\\
&= \dfrac{<var>DISTANCE</var>\text{ miles}}{<var>RATE_DOWN</var>\text{ mph}}\\
&= <var>TIME_DOWN</var>\text{ minutes}\end{align*}</code></p>
</div>
<p><code>\color{<var>KhanUtil.ORANGE</var>}{\begin{align*}\text{total time} &= <var>TIME_UP</var>\text{ minutes} + <var>TIME_DOWN</var>\text{ minutes}\\
&= <var>TIME_UP + TIME_DOWN</var>\text{ minutes}\end{align*}}</code></p>
<p>Now that we know both the total distance and total time, we can find the average speed.</p>
<p><code>\begin{align*}\text{average speed} &= \dfrac{\color{<var>KhanUtil.BLUE</var>}{\text{total distance}}}{\color{<var>KhanUtil.ORANGE</var>}{\text{total time}}}\\
&= \dfrac{\color{<var>KhanUtil.BLUE</var>}{<var>2 * DISTANCE</var>\text{ miles}}}{\color{<var>KhanUtil.ORANGE</var>}{<var>TIME_UP + TIME_DOWN</var>\text{ minutes}}}\\
&= <var>RATE_AVG</var>\text{ mph}\end{align*}</code></p>
<p>The average speed is <var>RATE_AVG</var> mph, and which is closer to <var>RATE_UP</var> mph than <var>RATE_DOWN</var> mph as we expected.</p>
</div>
</div>
<div id="collective-action-same">
<div class="vars">
<var id="PEOPLE_INIT">randRange( 5, 10 )</var>
<var id="WALL_INIT">PEOPLE_INIT</var>
<var id="TIME_INIT">randRange( 20, 59 )</var>
<var id="PEOPLE_FINAL">randRange( PEOPLE_INIT + 3, 20 )</var>
<var id="WALL_FINAL">PEOPLE_FINAL</var>
<var id="TIME_FINAL">TIME_INIT</var>
</div>
<p class="problem">It takes <var>TIME_INIT</var> minutes for <var>PEOPLE_INIT</var> people to paint <var>WALL_INIT</var> walls.</p>
<p class="question">How many minutes does it take <var>PEOPLE_FINAL</var> people to paint <var>WALL_FINAL</var> walls?</p>
<div class="solution" data-type="multiple">
<span class="sol"><var>TIME_FINAL</var></span> minutes
</div>
<div class="hints">
<p>Imagine that each person is assigned one wall, and all <var>PEOPLE_INIT</var> people begin painting at the same time.</p>
<p>Since everyone will finish painting their assigned wall after <var>TIME_INIT</var> minutes, it takes one person <var>TIME_INIT</var> minutes to paint one wall.</p>
<p>If we have <var>PEOPLE_FINAL</var> people and <var>WALL_FINAL</var> walls, we can again assign one wall to each person.</p>
<p>Everyone will take <var>TIME_FINAL</var> minutes to paint their assigned wall.</p>
<p>In other words, it takes <var>TIME_FINAL</var> minutes for <var>PEOPLE_FINAL</var> people to paint <var>WALL_FINAL</var> walls.</p>
</div>
</div>
<div id="collective-action-calculate">
<div class="vars">
<var id="PEOPLE_INIT, WALL_INIT">randRangeUnique( 3, 7, 2 )</var>
<var id="TIME_INIT">randRange( 30, 50 )</var>
<var id="PEOPLE_FINAL">randRange( PEOPLE_INIT + 1, 10 )</var>
<var id="WALL_FINAL">randRange( WALL_INIT + 1, 10 )</var>
<var id="TIME_FINAL">round( WALL_FINAL * TIME_INIT * PEOPLE_INIT / ( WALL_INIT * PEOPLE_FINAL ) )</var>
<var id="NEED_TO_ROUND">getGCD( WALL_FINAL * TIME_INIT * PEOPLE_INIT, WALL_INIT * PEOPLE_FINAL ) !== (WALL_INIT * PEOPLE_FINAL)</var>
</div>
<p class="problem"><var>PEOPLE_INIT</var> people can paint <var>WALL_INIT</var> walls in <var>TIME_INIT</var> minutes.</p>
<p class="question">How many minutes will it take for <var>PEOPLE_FINAL</var> people to paint <var>WALL_FINAL</var> walls? Round to the nearest minute.</p>
<div class="solution" data-type="multiple">
<span class="sol"><var>TIME_FINAL</var></span> minutes
<p class="example">the number of minutes, rounded to the nearest minute</p>
</div>
<div class="hints">
<div>
<p>We know the following about the number of walls <code>w</code> painted by <code>p</code> people in <code>t</code> minutes at a constant rate <code>r</code>.</p>
<p><code>w = r \cdot t \cdot p</code></p>
<p><code>\begin{align*}w &= <var>WALL_INIT</var>\text{ walls}\\
p &= <var>PEOPLE_INIT</var>\text{ people}\\
t &= <var>TIME_INIT</var>\text{ minutes}\end{align*}</code></p>
</div>
<div>
<p>Substituting known values and solving for <code>r</code>:</p>
<p><code>r = \dfrac{w}{t \cdot p}= \dfrac{<var>WALL_INIT</var>}{<var>TIME_INIT</var> \cdot <var>PEOPLE_INIT</var>} = <var>fractionReduce( WALL_INIT, TIME_INIT * PEOPLE_INIT )</var>\text{ walls painted per minute per person}</code></p>
</div>
<p>We can now calculate the amount of time to paint <var>WALL_FINAL</var> walls with <var>PEOPLE_FINAL</var> people.</p>
<p><code>t = \dfrac{w}{r \cdot p} = \dfrac{<var>WALL_FINAL</var>}{<var>fractionReduce( WALL_INIT, TIME_INIT * PEOPLE_INIT )</var> \cdot <var>PEOPLE_FINAL</var>} = \dfrac{<var>WALL_FINAL</var>}{<var>fractionReduce( WALL_INIT * PEOPLE_FINAL, TIME_INIT * PEOPLE_INIT )</var>} = <var>fractionReduce( WALL_FINAL * TIME_INIT * PEOPLE_INIT, WALL_INIT * PEOPLE_FINAL )</var>\text{ minutes}</code><span data-if="NEED_TO_ROUND"><code>= <var>mixedFractionFromImproper( WALL_FINAL * TIME_INIT * PEOPLE_INIT, WALL_INIT * PEOPLE_FINAL, true, true )</var>\text{ minutes}</code></span></p>
<div data-if="NEED_TO_ROUND">
<p>Round to the nearest minute:</p>
<p><code>t = <var>TIME_FINAL</var>\text{ minutes}</code></p>
</div>
</div>
</div>
</div>
</div>
</body>
</html>
Jump to Line
Something went wrong with that request. Please try again.