This repository has been archived by the owner on May 11, 2021. It is now read-only.
/
age_word_problems.html
355 lines (329 loc) · 24.7 KB
/
age_word_problems.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
<!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="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 friends.|<var>person(1)</var> and <var>person(2)</var> first met 3 years ago.|}
<span class="second"><var>CardinalThrough20(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>(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>CardinalThrough20(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> <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>CardinalThrough20(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>CardinalThrough20(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>CardinalThrough20(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>CardinalThrough20(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>CardinalThrough20(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>CardinalThrough20(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><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 data-if="isMale(1)">In <var>B</var> years, <var>person(1)</var> will be <var>A</var> times as old as he is right now.</p><p data-else="">In <var>B</var> years, <var>person(1)</var> will be <var>A</var> times as old as she is right now.</p>
<p data-if="isMale(1)">How old is he right now?</p><p data-else="">How old is she 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 data-if="isMale(1)">In <var>B</var> years, he will be <code><var>personVar(1)</var> + <var>B</var></code> years old.</p><p data-else="">In <var>B</var> years, she will be <code><var>personVar(1)</var> + <var>B</var></code> years old.</p>
<p data-if="isMale(1)">At that time, he will also be <code><var>A</var> <var>personVar(1)</var></code> years old.</p><p data-else="">At that time, she 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>