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<!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>
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