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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 src="../khan-exercise.js"></script>
+</head>
+<body>
+ <div class="exercise">
+ <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>
+
+ <div class="question">
+ <p class="spin">
+ {<span class = "first"><var>person(1)</var> is <var>A</var> years older than <var>person(2)</var></span>|<span class="first"><var>person(2)</var> is <var>A</var> years younger than <var>person(1)</var></span>}.
+ {For the last {four|3|two} years, <var>person(1)</var> and <var>person(2)</var> have been going to the same school.|<var>person(1)</var> and <var>person(2)</var> first met 3 years ago.|}
+ <span class = "second"><var>Cardinal(B)</var> years ago, <var>person(1)</var> was <var>C</var> times {as old as|older than} <var>person(2)</var>.</span></p>
+
+ <p>How old is <var>person(1)</var> now?</p>
+ </div>
+ <div class="solution"><var>(C * (B + A) - 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>person(1)</var> and <var>person(2)</var>.</p>
+ <p>Let <var>person(1)</var>'s current age be <code><var>personVar(1)</var></code> and <var>person(2)</var>'s current age be <code><var>personVar(2)</var></code>.</p>
+ <div>
+ <p>The information in the first sentence can be expressed in the following equation:</p>
+ <div>
+ <p><code class="hint_blue"><var>personVar(1)</var> = <var>personVar(2)</var> + <var>A</var></code></p>
+ </div>
+ <div class="graphie">
+ $(".first").addClass("hint_blue");
+ </div>
+ </div>
+ <p><var>Cardinal(B)</var> years ago, <var>person(1)</var> was <code><var>personVar(1)</var> - <var>B</var></code> years old, and <var>person(2)</var> was <code><var>personVar(2)</var> - <var>B</var></code> years old.</p>
+ <div>
+ <p>The information in the second sentence can be expressed in the following equation:</p>
+ <div>
+ <p><code class="hint_red"><var>personVar(1)</var> - <var>B</var> = <var>C</var>(<var>personVar(2)</var> - <var>B</var>)</code></p>
+ </div>
+ <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>personVar(1)</var></code>, it might be easiest to solve our first equation for <code><var>personVar(2)</var></code> and substitute it into our second equation.</p>
+ <div>
+ <p>Solving our first equation for <code><var>personVar(2)</var></code>, we get: <code class="hint_blue"><var>personVar(2)</var> = <var>personVar(1)</var> - <var>A</var></code>. Substituting this into our second equation, we get the equation:</p>
+ <div>
+ <p><code class="hint_red"><var>personVar(1)</var> - <var>B</var> = <var>C</var>(</code><code class="hint_blue">(<var>personVar(1)</var> - <var>A</var>)</code></span> <code class="hint_red"> -</code> <code class="hint_red"> <var>B</var>)</code></p>
+ </div>
+ <p>which combines the information about <code><var>personVar(1)</var></code> from both of our original equations.</p>
+ </div>
+ <p>Simplifying the right side of this equation, we get: <code><var>personVar(1)</var> - <var>B</var> = <var>C</var><var>personVar(1)</var> - <var>C * (A + B)</var></code>.</p>
+ <p>Solving for <code><var>personVar(1)</var></code>, we get: <code><var>C - 1</var> <var>personVar(1)</var> = <var>C * (A + B) - B</var></code>.</p>
+ <p><code><var>personVar(1)</var> = <var>(C * (B + A) - B) / (C - 1)</var></code>.</p>
+ </div>
+ </div>
+
+ <div id="solve-younger-1" data-type="solve-older-1">
+ <div class="question">
+ <p><span class="first"><var>person(1)</var> is <var>A</var> years older than
+ <var>person(2)</var>.</span> <span class="second"><var>Cardinal(B)</var> years ago, <var>person(1)</var>
+ was <var>C</var> times as old as <var>person(2)</var>.<span></p>
+
+ <p>How old is <var>person(2)</var> now?</p>
+ </div>
+ <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>person(1)</var> and <var>person(2)</var>.</p>
+ <p>Let <var>person(1)</var>'s current age be <code><var>personVar(1)</var></code> and <var>person(2)</var>'s current age be <code><var>personVar(2)</var></code>.</p>
+ <div>
+ <p>The information in the first sentence can be expressed in the following equation:</p>
+ <div>
+ <p><code class="hint_blue"><var>personVar(1)</var> = <var>personVar(2)</var> + <var>A</var></code></p>
+ </div>
+ <div class="graphie">
+ $(".first").addClass("hint_blue");
+ </div>
+ </div>
+ <p><var>Cardinal(B)</var> years ago, <var>person(1)</var> was <code><var>personVar(1)</var> - <var>B</var></code> years old, and <var>person(2)</var> was <code><var>personVar(2)</var> - <var>B</var></code> years old.</p>
+ <div>
+ <p>The information in the second sentence can be expressed in the following equation:</p>
+ <div>
+ <p><code class="hint_red"><var>personVar(1)</var> - <var>B</var> = <var>C</var>(<var>personVar(2)</var> - <var>B</var>)</code></p>
+ </div>
+ <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>personVar(2)</var></code>, it might be easiest to use our first equation for <code><var>personVar(1)</var></code> and substitute it into our second equation.</p>
+ <div>
+ <p>Our first equation is: <code class="hint_blue"><var>personVar(1)</var> = <var>personVar(2)</var> + <var>A</var></code>. Substituting this into our second equation, we get the equation:</p>
+ <div>
+ <p><code class="hint_blue">(<var>personVar(2)</var> + <var>A</var>)</code> <code class="hint_red">-</code> <code class="hint_red"><var>B</var> = <var>C</var>(<var>personVar(2)</var> - <var>B</var>)</code></p>
+ </div>
+ <p>which combines the information about <code><var>personVar(2)</var></code> from both of our original equations.</p>
+ </div>
+ <p>Simplifying both sides of this equation, we get: <code><var>personVar(2)</var> + <var>A - B</var> = <var>C</var> <var>personVar(2)</var> - <var>C * B</var></code>.</p>
+ <p>Solving for <code><var>personVar(2)</var></code>, we get: <code><var>C - 1</var> <var>personVar(2)</var> = <var>A - B + C * B</var></code>.</p>
+ <p><code><var>personVar(2)</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>
+
+ <div class="question">
+ <p><span class="first"><var>person(1)</var> is <var>C</var> times as old as
+ <var>person(2)</var></span> and <span class="second">is also <var>A</var>
+ years older than <var>person(2)</var></span>.</p>
+
+ <p>How old is <var>person(1)</var>?</p>
+ </div>
+ <div class="solution"><var>A * C / (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>person(1)</var> and <var>person(2)</var>.</p>
+ <p>Let <var>person(1)</var>'s current age be <code><var>personVar(1)</var></code> and <var>person(2)</var>'s current age be <code><var>personVar(2)</var></code>.</p>
+ <div>
+ <div>
+ <p><code class="hint_blue"><var>personVar(1)</var> = <var>C</var><var>personVar(2)</var></code></p>
+ </div>
+ <div>
+ <p><code class="hint_red"><var>personVar(1)</var> = <var>personVar(2)</var> + <var>A</var></code></p>
+ </div>
+ <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>personVar(1)</var></code> is to solve the second equation for <code><var>personVar(2)</var></code> and substitute that value into the first equation.</p>
+ <div>
+ <p>Solving our second equation for <code><var>personVar(2)</var></code>, we get: <code class="hint_red"><var>personVar(2)</var> = <var>personVar(1)</var> - <var>A</var></code>. Substituting this into our first equation, we get the equation: </p>
+ <div>
+ <p><code class="hint_blue"><var>personVar(1)</var> = <var>C</var></code><code class="hint_red">(<var>personVar(1)</var> - <var>A</var>)</code></p>
+ </div>
+ <p>which combines the information about <code><var>personVar(1)</var></code> from both of our original equations.</p>
+ </div>
+ <p>Simplifying the right side of this equation, we get: <code><var>personVar(1)</var> = <var>C</var><var>personVar(1)</var> - <var>C * A</var></code>.</p>
+ <p>Solving for <code><var>personVar(1)</var></code>, we get: <code><var>C - 1</var> <var>personVar(1)</var> = <var>A * C</var></code>.</p>
+ <p><code><var>personVar(1)</var> = <var>A * C / (C - 1)</var></code>.</p>
+ </div>
+ </div>
+
+ <div id="solve-younger-2" data-type="solve-older-2">
+ <div class="question">
+ <p><span class="first"><var>person(1)</var> is <var>C</var> times as old as
+ <var>person(2)</var></span> and <span class="second">is also <var>A</var>
+ years older than <var>person(2)</var>.</span></p>
+
+ <p>How old is <var>person(2)</var>?</p>
+ </div>
+ <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>person(1)</var> and <var>person(2)</var>.</p>
+ <p>Let <var>person(1)</var>'s current age be <code><var>personVar(1)</var></code> and <var>person(2)</var>'s current age be <code><var>personVar(2)</var></code>.</p>
+ <div>
+ <div>
+ <p><code class="hint_blue"><var>personVar(1)</var> = <var>C</var><var>personVar(2)</var></code></p>
+ </div>
+ <div>
+ <p><code class="hint_red"><var>personVar(1)</var> = <var>personVar(2)</var> + <var>A</var></code></p>
+ </div>
+ <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>personVar(2)</var></code>, and both of our equations have <code><var>personVar(1)</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>
+ <div>
+ <p><code>0 =</code> <code class="hint_blue"><var>C</var><var>personVar(2)</var></code> <code>-</code> <code class="hint_red"> (<var>personVar(2)</var> + <var>A</var>)</code></p>
+ </div>
+ <p>which combines the information about <code><var>personVar(2)</var></code> from both of our original equations.</p>
+ </div>
+ <p>Solving for <code><var>personVar(2)</var></code>, we get: <code><var>C - 1</var> <var>personVar(2)</var> = <var>A</var></code>.</p>
+ <p><code><var>personVar(2)</var> = <var>A / (C - 1)</var></code>.</p>
+ </div>
+ </div>
+
+ <div id="solve-older-3">
+ <div class="vars" data-ensure="C - A !== A && A * B * (C - 1) < 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>
+
+ <div class="question">
+ <p><span class="first"><var>person(1)</var> is <var>A</var> times as old as <var>person(2)</var>.</span> <span class="second"><var>Cardinal(B)</var> years ago, <var>person(1)</var> was <var>C</var> times as old as <var>person(2)</var>.</span></p>
+
+ <p>How old is <var>person(1)</var> now?</p>
+ </div>
+ <div class="solution"><var>A * 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>person(1)</var> and <var>person(2)</var>.</p>
+ <p>Let <var>person(1)</var>'s current age be <code><var>personVar(1)</var></code> and <var>person(2)</var>'s current age be <code><var>personVar(2)</var></code>.</p>
+ <div>
+ <p>The information in the first sentence can be expressed in the following equation:</p>
+ <div>
+ <p><code class="hint_blue"><var>personVar(1)</var> = <var>A</var><var>personVar(2)</var></code></p>
+ </div>
+ <div class="graphie">
+ $(".first").addClass("hint_blue");
+ </div>
+ </div>
+ <p><var>Cardinal(B)</var> years ago, <var>person(1)</var> was <code><var>personVar(1)</var> - <var>B</var></code> years old, and <var>person(2)</var> was <code><var>personVar(2)</var> - <var>B</var></code> years old.</p>
+ <div>
+ <p>The information in the second sentence can be expressed in the following equation:</p>
+ <div>
+ <p><code class="hint_red"><var>personVar(1)</var> - <var>B</var> = <var>C</var>(<var>personVar(2)</var> - <var>B</var>)</code></p>
+ </div>
+ <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>personVar(1)</var></code>, it might be easiest to solve our first equation for <code><var>personVar(2)</var></code> and substitute it into our second equation.</p>
+ <div>
+ <p>Solving our first equation for <code><var>personVar(2)</var></code>, we get: <code class="hint_blue"><var>personVar(2)</var> = <var>personVar(1)</var> / <var>A</var></code>. Substituting this into our second equation, we get: </p>
+ <div>
+ <p><code class="hint_red"><var>personVar(1)</var> - <var>B</var> = <var>C</var>(</code> <code class="hint_blue">(<var>personVar(1)</var> / <var>A</var>)</code> <code class="hint_red">- <var>B</var>)</code></p>
+ </div>
+ <p>which combines the information about <code><var>personVar(1)</var></code> from both of our original equations.</p>
+ </div>
+ <p>Simplifying the right side of this equation, we get: <code><var>personVar(1)</var> - <var>B</var> = <var>fractionReduce(C, A)</var> <var>personVar(1)</var> - <var>C * B</var></code>.</p>
+ <p>Solving for <code><var>personVar(1)</var></code>, we get: <code><var>fractionReduce(C - A, A)</var> <var>personVar(1)</var> = <var>B * (C - 1)</var></code>.</p>
+ <p><code><var>personVar(1)</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">
+ <div class="question">
+ <p><span class="first"><var>person(1)</var> is <var>A</var> times as old as <var>person(2)</var>.</span> <span class="second"><var>Cardinal(B)</var> years ago, <var>person(1)</var> was <var>C</var> times as old as <var>person(2)</var>.</span></p>
+
+ <p>How old is <var>person(2)</var> now?</p>
+ </div>
+ <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>person(1)</var> and <var>person(2)</var>.</p>
+ <p>Let <var>person(1)</var>'s current age be <code><var>personVar(1)</var></code> and <var>person(2)</var>'s current age be <code><var>personVar(2)</var></code>.</p>
+ <div>
+ <p>The information in the first sentence can be expressed in the following equation:</p>
+ <div>
+ <p><code class="hint_blue"><var>personVar(1)</var> = <var>A</var><var>personVar(2)</var></code></p>
+ </div>
+ <div class="graphie">
+ $(".first").addClass("hint_blue");
+ </div>
+ </div>
+ <p><var>Cardinal(B)</var> years ago, <var>person(1)</var> was <code><var>personVar(1)</var> - <var>B</var></code> years old, and <var>person(2)</var> was <code><var>personVar(2)</var> - <var>B</var></code> years old.</p>
+ <div>
+ <p>The information in the second sentence can be expressed in the following equation:</p>
+ <div>
+ <p><code class="hint_red"><var>personVar(1)</var> - <var>B</var> = <var>C</var>(<var>personVar(2)</var> - <var>B</var>)</code></p>
+ </div>
+ <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>personVar(2)</var></code>, it might be easiest to use our first equation for <code><var>personVar(1)</var></code> and substitute it into our second equation.</p>
+ <div>
+ <p>Our first equation is: <code class="hint_blue"><var>personVar(1)</var> = <var>A</var><var>personVar(2)</var></code>. Substituting this into our second equation, we get:</p>
+ <div>
+ <p><code class="hint_blue"><var>A</var><var>personVar(2)</var></code> <code class="hint_red">-</code> <code class="hint_red"><var>B</var> = <var>C</var>(<var>personVar(2)</var> - <var>B</var>)</code></p>
+ </div>
+ <p>which combines the information about <code><var>personVar(2)</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>personVar(2)</var> - <var>B</var> = <var>C</var> <var>personVar(2)</var> - <var>B * C</var></code>.</p>
+ <p>Solving for <code><var>personVar(2)</var></code>, we get: <code><var>C - A</var> <var>personVar(2)</var> = <var>B * (C - 1)</var>.</code>
+ <p><code><var>personVar(2)</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 <= 60">
+ <var id="A">randRange(3, 20)</var>
+ <var id="B">randRange(7, 24) * (A - 1)</var>
+ </div>
+
+ <div class="question">
+ <p>In <var>B</var> years, <var>person(1)</var> will be <var>A</var> times as old as <var>he(1)</var> is right now.</p>
+
+ <p>How old is <var>he(1)</var> right now?</p>
+ </div>
+ <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>person(1)</var>'s age.</p>
+ <p>Let <var>person(1)</var>'s age be <code><var>personVar(1)</var></code>.</p>
+ <p>In <var>B</var> years, <var>he(1)</var> will be <code><var>personVar(1)</var> + <var>B</var></code> years old.</p>
+ <p>At that time, <var>he(1)</var> will also be <code><var>A</var> <var>personVar(1)</var></code> years old.</p>
+ <div>
+ <p>Writing this information as an equation, we get:</p>
+ <div>
+ <p><code><var>personVar(1)</var> + <var>B</var> = <var>A</var> <var>personVar(1)</var></code></p>
+ </div>
+ </div>
+ <p>Solving for <code><var>personVar(1)</var></code>, we get: <code><var>A - 1</var> <var>personVar(1)</var> = <var>B</var></code>.</p>
+ <p><code><var>personVar(1)</var> = <var>B / (A - 1)</var></code>.</p>
+ </div>
+ </div>
+
+ <div id="solve-single-5" data-weight="2">
+ <div class="vars" data-ensure="A <= 80 && B >= 2 && (A - B * C) > (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>
+
+ <div class="question">
+ <p><var>person(1)</var> is <var>A</var> years old and <var>person(2)</var> is <var>B</var> years old.</p>
+
+ <p>How many years will it take until <var>person(1)</var> is only <var>C</var> times as old as <var>person(2)</var>?</p>
+ </div>
+ <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>person(1)</var> will be <code><var>A</var> + y</code> years old and <var>person(2)</var> will be <code><var>B</var> + y</code> years old.</p>
+ <p>At that time, <var>person(1)</var> will be <var>C</var> times as old as <var>person(2)</var>.</p>
+ <div>
+ <p>Writing this information as an equation, we get:</p>
+ <div>
+ <p><code><var>A</var> + y = <var>C</var> (<var>B</var> + y)</code></p>
+ </div>
+ </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>
+</body>
+</html>
+

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