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

398 lines (366 sloc) 23.006 kb
<!DOCTYPE html>
<html data-require="math math-format word-problems spin graphie">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Age word problems</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars" data-ensure="V1 !== 'o' && V2 !== 'o'">
<var id="I1">randRange(1, 10)</var>
<var id="I2">randRangeExclude(1, 10, [I1])</var>
<var id="P1">person(I1)</var>
<var id="P2">person(I2)</var>
<var id="V1">personVar(I1)</var>
<var id="V2">personVar(I2)</var>
</div>
<div class="problems">
<div id="solve-older-1">
<div class="vars">
<var id="C">randRange(3, 5)</var>
<var id="B">randRange(2, 20)</var>
<var id="A">randRange(1, 10) * (C - 1)</var>
</div>
<p class="problem spin">
{<span class="first"><var>P1</var> is <code><var>A</var></code> years older than <var>P2</var></span>|
<span class="first"><var>P2</var> is <code><var>A</var></code> years younger than <var>P1</var></span>}.
{For the last {four|<code>3</code>|two} years, <var>P1</var> and <var>P2</var> have been friends.|
<var>P1</var> and <var>P2</var> first met {four|<code>3</code>|two} years ago.|}
<span class="second">
<var>CardinalThrough20(B)</var> years ago, <var>P1</var> was <code><var>C</var></code> times as old as <var>P2</var>.
</span>
</p>
<p class="question">How old is <var>P1</var> now?</p>
<div class="solution"><var>(C * (B + A) - B) / (C - 1)</var></div>
<div class="hints" data-apply="appendContents">
<div>
<p>The information in the first sentence can be expressed in the following equation:</p>
<p><code>\blue{<var>V1</var> = <var>V2</var> + <var>A</var>}</code></p>
<div class="graphie">
$(".first").addClass("hint_blue");
</div>
</div>
<p>
<var>CardinalThrough20(B)</var> years ago, <var>P1</var> was <code><var>V1</var> - <var>B</var></code> years old,
and <var>P2</var> was <code><var>V2</var> - <var>B</var></code> years old.
</p>
<div>
<p>The information in the second sentence can be expressed in the following equation:</p>
<p><code>\red{<var>V1</var> - <var>B</var> = <var>C</var>(<var>V2</var> - <var>B</var>)}</code></p>
<div class="graphie">
$(".second").addClass("hint_red");
</div>
</div>
<p>Now we have two independent equations, and we can solve for our two unknowns.</p>
<p>
Because we are looking for <code><var>V1</var></code>,
it might be easiest to solve our first equation for <code><var>V2</var></code>
and substitute it into our second equation.
</p>
<div>
<p>
Solving our first equation for <code><var>V2</var></code>, we get:
<code>\blue{<var>V2</var> = <var>V1</var> - <var>A</var>}</code>.
Substituting this into our second equation, we get the equation:
</p>
<p>
<code>\red{<var>V1</var> - <var>B</var> = <var>C</var>(}
\blue{(<var>V1</var> - <var>A</var>)}\red{ - <var>B</var>)}</code>
</p>
<p>which combines the information about <code><var>V1</var></code> from both of our original equations.</p>
</div>
<p>
Simplifying the right side of this equation, we get:
<code><var>V1</var> - <var>B</var> = <var>C</var><var>V1</var> - <var>C * (A + B)</var></code>.
</p>
<p>
Solving for <code><var>V1</var></code>, we get:
<code><var>C - 1</var> <var>V1</var> = <var>C * (A + B) - B</var></code>.
</p>
<p><code><var>V1</var> = <var>(C * (B + A) - B) / (C - 1)</var></code>.</p>
</div>
</div>
<div id="solve-younger-1" data-type="solve-older-1">
<p class="question">How old is <var>P2</var> now?</p>
<div class="solution"><var>(A - B + C * B) / (C - 1)</var></div>
<div class="hints">
<p>
We can use the given information to write down two equations that describe the ages of
<var>P1</var> and <var>P2</var>.
</p>
<p>
Let <var>P1</var>'s current age be <code><var>V1</var></code>
and <var>P2</var>'s current age be <code><var>V2</var></code>.
</p>
<div>
<p>The information in the first sentence can be expressed in the following equation:</p>
<p><code>\blue{<var>V1</var> = <var>V2</var> + <var>A</var>}</code></p>
<div class="graphie">
$(".first").addClass("hint_blue");
</div>
</div>
<p>
<var>CardinalThrough20(B)</var> years ago, <var>P1</var> was <code><var>V1</var> - <var>B</var></code> years old,
and <var>P2</var> was <code><var>V2</var> - <var>B</var></code> years old.
</p>
<div>
<p>The information in the second sentence can be expressed in the following equation:</p>
<p><code>\red{<var>V1</var> - <var>B</var> = <var>C</var>(<var>V2</var> - <var>B</var>)}</code></p>
<div class="graphie">
$(".second").addClass("hint_red");
</div>
</div>
<p>Now we have two independent equations, and we can solve for our two unknowns.</p>
<p>
Because we are looking for <code><var>V2</var></code>,
it might be easiest to use our first equation for <code><var>V1</var></code>
and substitute it into our second equation.
</p>
<div>
<p>
Our first equation is: <code>\blue{<var>V1</var> = <var>V2</var> + <var>A</var>}</code>.
Substituting this into our second equation, we get the equation:
</p>
<p><code>
\blue{(<var>V2</var> + <var>A</var>)}\red{-<var>B</var> = <var>C</var>(<var>V2</var> - <var>B</var>)}
</code></p>
<p>which combines the information about <code><var>V2</var></code> from both of our original equations.</p>
</div>
<p>Simplifying both sides of this equation, we get: <code><var>V2</var> + <var>A - B</var> = <var>C</var> <var>V2</var> - <var>C * B</var></code>.</p>
<p>Solving for <code><var>V2</var></code>, we get: <code><var>C - 1</var> <var>V2</var> = <var>A - B + C * B</var></code>.</p>
<p><code><var>V2</var> = <var>(A - B + C * B) / (C - 1)</var></code>.</p>
</div>
</div>
<div id="solve-older-2">
<div class="vars">
<var id="C">randRange(3, 5)</var>
<var id="A">randRange(2, 10) * (C - 1)</var>
</div>
<p class="problem">
<span class="first"><var>P1</var> is <code><var>C</var></code> times as old as <var>P2</var></span> and
<span class="second">is also <code><var>A</var></code> years older than <var>P2</var></span>.
</p>
<p class="question">How old is <var>P1</var>?</p>
<div class="solution"><var>A * C / (C - 1)</var></div>
<div class="hints" data-apply="appendContents">
<div>
<p><code>\blue{<var>V1</var> = <var>C</var><var>V2</var>}</code></p>
<p><code>\red{<var>V1</var> = <var>V2</var> + <var>A</var>}</code></p>
<div class="graphie">
$(".first").addClass("hint_blue");
$(".second").addClass("hint_red");
</div>
</div>
<p>Now we have two independent equations, and we can solve for our two unknowns.</p>
<p>One way to solve for <code><var>V1</var></code> is to solve the second equation for <code><var>V2</var></code> and substitute that value into the first equation.</p>
<div>
<p>
Solving our second equation for <code><var>V2</var></code>, we get:
<code>\red{<var>V2</var> = <var>V1</var> - <var>A</var>}</code>.
Substituting this into our first equation, we get the equation:
</p>
<p><code>\blue{<var>V1</var> = <var>C</var>}\red{(<var>V1</var> - <var>A</var>)}</code></p>
<p>which combines the information about <code><var>V1</var></code> from both of our original equations.</p>
</div>
<p>Simplifying the right side of this equation, we get: <code><var>V1</var> = <var>C</var><var>V1</var> - <var>C * A</var></code>.</p>
<p>Solving for <code><var>V1</var></code>, we get: <code><var>C - 1</var> <var>V1</var> = <var>A * C</var></code>.</p>
<p><code><var>V1</var> = <var>A * C / (C - 1)</var></code>.</p>
</div>
</div>
<div id="solve-younger-2" data-type="solve-older-2">
<p class="question">How old is <var>P2</var>?</p>
<div class="solution"><var>A / (C - 1)</var></div>
<div class="hints">
<p>
We can use the given information to write down two equations that describe the ages of
<var>P1</var> and <var>P2</var>.
</p>
<p>
Let <var>P1</var>'s current age be <code><var>V1</var></code>
and <var>P2</var>'s current age be <code><var>V2</var></code>.
</p>
<p>We can use the given information to write down two equations that describe the ages of <var>P1</var> and <var>P2</var>.</p>
<p>Let <var>P1</var>'s current age be <code><var>V1</var></code> and <var>P2</var>'s current age be <code><var>V2</var></code>.</p>
<div>
<p><code>\blue{<var>V1</var> = <var>C</var><var>V2</var>}</code></p>
<p><code>\red{<var>V1</var> = <var>V2</var> + <var>A</var>}</code></p>
<div class="graphie">
$(".first").addClass("hint_blue");
$(".second").addClass("hint_red");
</div>
</div>
<p>Now we have two independent equations, and we can solve for our two unknowns.</p>
<p>Since we are looking for <code><var>V2</var></code>, and both of our equations have <code><var>V1</var></code> alone on one side, this is a convenient time to use elimination.</p>
<div>
<p>Subtracting the second equation from the first equation, we get:</p>
<p><code>0 = \blue{<var>C</var><var>V2</var>} -\red{(<var>V2</var> + <var>A</var>)}</code></p>
<p>which combines the information about <code><var>V2</var></code> from both of our original equations.</p>
</div>
<p>Solving for <code><var>V2</var></code>, we get: <code><var>C - 1</var> <var>V2</var> = <var>A</var></code>.</p>
<p><code><var>V2</var> = <var>A / (C - 1)</var></code>.</p>
</div>
</div>
<div id="solve-older-3">
<div class="vars" data-ensure="C - A !== A &amp;&amp; A * B * (C - 1) &lt; 100 * (C - A)">
<var id="A">randRange(2, 5)</var>
<var id="C">randRange(A + 2, 9)</var>
<var id="B">randRange(2, 7) * (C - A)</var>
</div>
<p class="problem">
<span class="first"><var>P1</var> is <code><var>A</var></code> times as old as <var>P2</var>.</span>
<span class="second">
<code><var>B</var></code> years ago, <var>P1</var> was <code><var>C</var></code> times as old as <var>P2</var>.
</span>
</p>
<p class="question">How old is <var>P1</var> now?</p>
<div class="solution"><var>A * B * (C - 1) / (C - A)</var></div>
<div class="hints" data-apply="appendContents">
<div>
<p>The information in the first sentence can be expressed in the following equation:</p>
<p><code>\blue{<var>V1</var> = <var>A</var><var>V2</var>}</code></p>
<div class="graphie">
$(".first").addClass("hint_blue");
</div>
</div>
<p><var>CardinalThrough20(B)</var> years ago, <var>P1</var> was <code><var>V1</var> - <var>B</var></code> years old, and <var>P2</var> was <code><var>V2</var> - <var>B</var></code> years old.</p>
<div>
<p>The information in the second sentence can be expressed in the following equation:</p>
<p><code>\red{<var>V1</var> - <var>B</var> = <var>C</var>(<var>V2</var> - <var>B</var>)}</code></p>
<div class="graphie">
$(".second").addClass("hint_red");
</div>
</div>
<p>Now we have two independent equations, and we can solve for our two unknowns.</p>
<p>Because we are looking for <code><var>V1</var></code>, it might be easiest to solve our first equation for <code><var>V2</var></code> and substitute it into our second equation.</p>
<div>
<p>
Solving our first equation for <code><var>V2</var></code>, we get:
<code>\blue{<var>V2</var> = \dfrac{<var>V1</var>}{<var>A</var>}}</code>.
Substituting this into our second equation, we get:
</p>
<p><code>
\red{<var>V1</var> - <var>B</var> = <var>C</var>
(}\blue{\frac{<var>V1</var>}{<var>A</var>}} \red{- <var>B</var>)}
</code></p>
<p>which combines the information about <code><var>V1</var></code> from both of our original equations.</p>
</div>
<p>Simplifying the right side of this equation, we get: <code><var>V1</var> - <var>B</var> = <var>fractionReduce(C, A)</var> <var>V1</var> - <var>C * B</var></code>.</p>
<p>Solving for <code><var>V1</var></code>, we get: <code><var>fractionReduce(C - A, A)</var> <var>V1</var> = <var>B * (C - 1)</var></code>.</p>
<p><code><var>V1</var> = <var>fractionReduce(A, C - A)</var> \cdot <var>B * (C - 1)</var> = <var>A * B * (C - 1) / (C - A)</var></code>.</p>
</div>
</div>
<div id="solve-younger-3" data-type="solve-older-3">
<p class="question">How old is <var>P2</var> now?</p>
<div class="solution"><var>B * (C - 1) / (C - A)</var></div>
<div class="hints">
<p>
We can use the given information to write down two equations that describe the ages of
<var>P1</var> and <var>P2</var>.
</p>
<p>
Let <var>P1</var>'s current age be <code><var>V1</var></code>
and <var>P2</var>'s current age be <code><var>V2</var></code>.
</p>
<div>
<p>The information in the first sentence can be expressed in the following equation:</p>
<p><code>\blue{<var>V1</var> = <var>A</var><var>V2</var>}</code></p>
<div class="graphie">
$(".first").addClass("hint_blue");
</div>
</div>
<p><var>CardinalThrough20(B)</var> years ago, <var>P1</var> was <code><var>V1</var> - <var>B</var></code> years old, and <var>P2</var> was <code><var>V2</var> - <var>B</var></code> years old.</p>
<div>
<p>The information in the second sentence can be expressed in the following equation:</p>
<p><code>\red{<var>V1</var> - <var>B</var> = <var>C</var>(<var>V2</var> - <var>B</var>)}</code></p>
<div class="graphie">
$(".second").addClass("hint_red");
</div>
</div>
<p>Now we have two independent equations, and we can solve for our two unknowns.</p>
<p>Because we are looking for <code><var>V2</var></code>, it might be easiest to use our first equation for <code><var>V1</var></code> and substitute it into our second equation.</p>
<div>
<p>
Our first equation is: <code>\blue{<var>V1</var> = <var>A</var><var>V2</var>}</code>.
Substituting this into our second equation, we get:
</p>
<p>
<code>\blue{<var>A</var><var>V2</var>} \red {-<var>B</var> =
<var>C</var>(<var>V2</var> - <var>B</var>)}</code>
</p>
<p>which combines the information about <code><var>V2</var></code> from both of our original equations.</p>
</div>
<p>Simplifying the right side of this equation, we get: <code><var>A</var> <var>V2</var> - <var>B</var> = <var>C</var> <var>V2</var> - <var>B * C</var></code>.</p>
<p>Solving for <code><var>V2</var></code>, we get: <code><var>C - A</var> <var>V2</var> = <var>B * (C - 1)</var>.</code>
</p><p><code><var>V2</var> = <var>B * (C - 1) / (C - A)</var></code>.</p>
</div>
</div>
<div id="solve-single-4" data-weight="2">
<div class="vars" data-ensure="B &lt;= 60">
<var id="A">randRange(3, 20)</var>
<var id="B">randRange(7, 24) * (A - 1)</var>
</div>
<p class="problem" data-if="isMale(I1)">
In <code><var>B</var></code> years, <var>P1</var> will be <code><var>A</var></code> times as old as he is right now.
</p><p data-else="">
In <code><var>B</var></code> years, <var>P1</var> will be <code><var>A</var></code> times as old as she is right now.
</p>
<p class="question" data-if="isMale(I1)">How old is he right now?</p>
<p class="question" data-else="">How old is she right now?</p>
<div class="solution"><var>B / (A - 1)</var></div>
<div class="hints">
<p>We can use the given information to write down an equation about <var>P1</var>'s age.</p>
<p>Let <var>P1</var>'s age be <code><var>V1</var></code>.</p>
<p data-if="isMale(I1)">In <code><var>B</var></code> years, he will be <code><var>V1</var> + <var>B</var></code> years old.</p>
<p data-else="">In <code><var>B</var></code> years, she will be <code><var>V1</var> + <var>B</var></code> years old.</p>
<p data-if="isMale(I1)">At that time, he will also be <code><var>A</var> <var>V1</var></code> years old.</p>
<p data-else="">At that time, she will also be <code><var>A</var> <var>V1</var></code> years old.</p>
<div>
<p>Writing this information as an equation, we get:</p>
<p><code><var>V1</var> + <var>B</var> = <var>A</var> <var>V1</var></code></p>
</div>
<p>Solving for <code><var>V1</var></code>, we get: <code><var>A - 1</var> <var>V1</var> = <var>B</var></code>.</p>
<p><code><var>V1</var> = <var>B / (A - 1)</var></code>.</p>
</div>
</div>
<div id="solve-single-5" data-weight="2">
<div class="vars" data-ensure="A &lt;= 80 &amp;&amp; B &gt;= 2 &amp;&amp; (A - B * C) &gt; (C - 1)">
<var id="C">randRange(3, 5)</var>
<var id="B">randRange(1, 10) * (C - 1)</var>
<var id="A">randRange(C * B + 1, 15) * (C - 1)</var>
</div>
<p class="problem">
<var>P1</var> is <code><var>A</var></code> years old and <var>P2</var> is <code><var>B</var></code> years old.
</p>
<p class="question">
How many years will it take until <var>P1</var> is only <code><var>C</var></code> times as old as <var>P2</var>?
</p>
<div class="solution"><var>(A - B * C) / (C - 1)</var></div>
<div class="hints">
<p>We can use the given information to write down an equation about how many years it will take.</p>
<p>Let <code>y</code> be the number of years that it will take.</p>
<p>In <code>y</code> years, <var>P1</var> will be <code><var>A</var> + y</code> years old and <var>P2</var> will be <code><var>B</var> + y</code> years old.</p>
<p>At that time, <var>P1</var> will be <var>C</var> times as old as <var>P2</var>.</p>
<div>
<p>Writing this information as an equation, we get:</p>
<p><code><var>A</var> + y = <var>C</var> (<var>B</var> + y)</code></p>
</div>
<p>Simplifying the right side of this equation, we get: <code><var>A</var> + y = <var>C * B</var> + <var>C</var> y</code>.</p>
<p>Solving for <code>y</code>, we get: <code><var>C - 1</var> y = <var>A - C * B</var></code>.</p>
<p><code>y = <var>(A - C * B) / (C - 1)</var></code>.</p>
</div>
</div>
</div>
<div class="hints">
<p>
We can use the given information to write down two equations that describe the ages of
<var>P1</var> and <var>P2</var>.
</p>
<p>
Let <var>P1</var>'s current age be <code><var>V1</var></code>
and <var>P2</var>'s current age be <code><var>V2</var></code>.
</p>
</div>
</div>
</body>
</html>
Jump to Line
Something went wrong with that request. Please try again.