Open Middle®
Directions: Use the digits 0 through 9, without repeats, to solve the problem below.
Hint
Answer
Number of Unique Solutions: 338
1: 1/2 ÷ 3/4 = 6/9
2: 1/2 ÷ 3/9 = 6/4
3: 1/2 ÷ 6/4 = 3/9
4: 1/2 ÷ 6/9 = 3/4
5: 1/3 ÷ 2/4 = 6/9
6: 1/3 ÷ 2/9 = 6/4
7: 1/3 ÷ 4/8 = 6/9
8: 1/3 ÷ 4/9 = 6/8
9: 1/3 ÷ 6/4 = 2/9
10: 1/3 ÷ 6/8 = 4/9
11: 1/3 ÷ 6/9 = 2/4
12: 1/3 ÷ 6/9 = 4/8
13: 1/4 ÷ 3/8 = 6/9
14: 1/4 ÷ 3/9 = 6/8
15: 1/4 ÷ 6/8 = 3/9
16: 1/4 ÷ 6/9 = 3/8
17: 1/6 ÷ 2/4 = 3/9
18: 1/6 ÷ 2/9 = 3/4
19: 1/6 ÷ 3/4 = 2/9
20: 1/6 ÷ 3/8 = 4/9
21: 1/6 ÷ 3/9 = 2/4
22: 1/6 ÷ 3/9 = 4/8
23: 1/6 ÷ 4/8 = 3/9
24: 1/6 ÷ 4/9 = 3/8
25: 2/1 ÷ 4/3 = 9/6
26: 2/1 ÷ 4/6 = 9/3
27: 2/1 ÷ 4/8 = 36/9
28: 2/1 ÷ 4/9 = 36/8
29: 2/1 ÷ 9/3 = 4/6
30: 2/1 ÷ 9/6 = 4/3
31: 2/4 ÷ 1/3 = 9/6
32: 2/4 ÷ 1/6 = 9/3
33: 2/4 ÷ 1/8 = 36/9
34: 2/4 ÷ 1/9 = 36/8
35: 2/4 ÷ 9/3 = 1/6
36: 2/4 ÷ 9/6 = 1/3
37: 2/9 ÷ 1/3 = 4/6
38: 2/9 ÷ 1/6 = 4/3
39: 2/9 ÷ 4/3 = 1/6
40: 2/9 ÷ 4/6 = 1/3
41: 3/1 ÷ 2/5 = 60/8
42: 3/1 ÷ 2/8 = 60/5
43: 3/1 ÷ 4/2 = 9/6
44: 3/1 ÷ 4/6 = 9/2
45: 3/1 ÷ 6/5 = 20/8
46: 3/1 ÷ 6/8 = 20/5
47: 3/1 ÷ 8/4 = 9/6
48: 3/1 ÷ 8/6 = 9/4
49: 3/1 ÷ 9/2 = 4/6
50: 3/1 ÷ 9/4 = 8/6
51: 3/1 ÷ 9/6 = 4/2
52: 3/1 ÷ 9/6 = 8/4
53: 3/2 ÷ 1/5 = 60/8
54: 3/2 ÷ 1/8 = 60/5
55: 3/2 ÷ 6/5 = 10/8
56: 3/2 ÷ 6/7 = 14/8
57: 3/2 ÷ 6/8 = 10/5
58: 3/2 ÷ 6/8 = 14/7
59: 3/4 ÷ 1/2 = 9/6
60: 3/4 ÷ 1/6 = 9/2
61: 3/4 ÷ 9/2 = 1/6
62: 3/4 ÷ 9/6 = 1/2
63: 3/6 ÷ 1/5 = 20/8
64: 3/6 ÷ 1/8 = 20/5
65: 3/6 ÷ 2/5 = 10/8
66: 3/6 ÷ 2/7 = 14/8
67: 3/6 ÷ 2/8 = 10/5
68: 3/6 ÷ 2/8 = 14/7
69: 3/8 ÷ 1/4 = 9/6
70: 3/8 ÷ 1/6 = 9/4
71: 3/8 ÷ 9/4 = 1/6
72: 3/8 ÷ 9/6 = 1/4
73: 3/9 ÷ 1/2 = 4/6
74: 3/9 ÷ 1/4 = 8/6
75: 3/9 ÷ 1/6 = 4/2
76: 3/9 ÷ 1/6 = 8/4
77: 3/9 ÷ 4/2 = 1/6
78: 3/9 ÷ 4/6 = 1/2
79: 3/9 ÷ 8/4 = 1/6
80: 3/9 ÷ 8/6 = 1/4
81: 4/1 ÷ 3/5 = 60/9
82: 4/1 ÷ 3/6 = 72/9
83: 4/1 ÷ 3/9 = 60/5
84: 4/1 ÷ 3/9 = 72/6
85: 4/1 ÷ 6/5 = 30/9
86: 4/1 ÷ 6/9 = 30/5
87: 4/1 ÷ 8/3 = 9/6
88: 4/1 ÷ 8/6 = 27/9
89: 4/1 ÷ 8/6 = 9/3
90: 4/1 ÷ 8/9 = 27/6
91: 4/1 ÷ 9/3 = 8/6
92: 4/1 ÷ 9/6 = 8/3
93: 4/2 ÷ 3/1 = 6/9
94: 4/2 ÷ 3/9 = 6/1
95: 4/2 ÷ 6/1 = 3/9
96: 4/2 ÷ 6/9 = 3/1
97: 4/3 ÷ 1/6 = 72/9
98: 4/3 ÷ 1/9 = 60/5
99: 4/3 ÷ 1/9 = 72/6
100: 4/3 ÷ 2/1 = 6/9
101: 4/3 ÷ 2/9 = 6/1
102: 4/3 ÷ 6/1 = 2/9
103: 4/3 ÷ 6/5 = 10/9
104: 4/3 ÷ 6/9 = 10/5
105: 4/3 ÷ 6/9 = 2/1
106: 4/6 ÷ 1/9 = 30/5
107: 4/6 ÷ 2/1 = 3/9
108: 4/6 ÷ 2/9 = 3/1
109: 4/6 ÷ 3/1 = 2/9
110: 4/6 ÷ 3/5 = 10/9
111: 4/6 ÷ 3/9 = 10/5
112: 4/6 ÷ 3/9 = 2/1
113: 4/8 ÷ 1/3 = 9/6
114: 4/8 ÷ 1/6 = 27/9
115: 4/8 ÷ 1/6 = 9/3
116: 4/8 ÷ 1/9 = 27/6
117: 4/8 ÷ 9/3 = 1/6
118: 4/8 ÷ 9/6 = 1/3
119: 4/9 ÷ 1/3 = 8/6
120: 4/9 ÷ 1/6 = 8/3
121: 4/9 ÷ 8/3 = 1/6
122: 4/9 ÷ 8/6 = 1/3
123: 5/1 ÷ 2/3 = 60/8
124: 5/1 ÷ 2/8 = 60/3
125: 5/1 ÷ 3/4 = 60/9
126: 5/1 ÷ 3/9 = 60/4
127: 5/1 ÷ 6/3 = 20/8
128: 5/1 ÷ 6/4 = 30/9
129: 5/1 ÷ 6/8 = 20/3
130: 5/1 ÷ 6/9 = 30/4
131: 5/2 ÷ 1/3 = 60/8
132: 5/2 ÷ 1/8 = 60/3
133: 5/2 ÷ 3/6 = 40/8
134: 5/2 ÷ 3/8 = 40/6
135: 5/2 ÷ 3/8 = 60/9
136: 5/2 ÷ 3/9 = 60/8
137: 5/2 ÷ 4/6 = 30/8
138: 5/2 ÷ 4/8 = 30/6
139: 5/2 ÷ 6/3 = 10/8
140: 5/2 ÷ 6/8 = 10/3
141: 5/2 ÷ 6/8 = 30/9
142: 5/2 ÷ 6/9 = 30/8
143: 5/3 ÷ 1/4 = 60/9
144: 5/3 ÷ 2/8 = 40/6
145: 5/3 ÷ 2/8 = 60/9
146: 5/3 ÷ 4/8 = 20/6
147: 5/3 ÷ 6/4 = 10/9
148: 5/3 ÷ 6/8 = 20/9
149: 5/4 ÷ 2/6 = 30/8
150: 5/4 ÷ 2/8 = 30/6
151: 5/4 ÷ 3/6 = 20/8
152: 5/4 ÷ 3/8 = 20/6
153: 5/6 ÷ 1/4 = 30/9
154: 5/6 ÷ 1/8 = 20/3
155: 5/6 ÷ 2/8 = 10/3
156: 5/6 ÷ 2/8 = 30/9
157: 5/6 ÷ 3/4 = 10/9
158: 5/6 ÷ 3/8 = 20/9
159: 6/1 ÷ 3/4 = 72/9
160: 6/1 ÷ 3/9 = 72/4
161: 6/1 ÷ 4/2 = 9/3
162: 6/1 ÷ 4/3 = 9/2
163: 6/1 ÷ 8/3 = 9/4
164: 6/1 ÷ 8/4 = 27/9
165: 6/1 ÷ 8/4 = 9/3
166: 6/1 ÷ 8/9 = 27/4
167: 6/1 ÷ 9/2 = 4/3
168: 6/1 ÷ 9/3 = 4/2
169: 6/1 ÷ 9/3 = 8/4
170: 6/1 ÷ 9/4 = 8/3
171: 6/2 ÷ 3/5 = 40/8
172: 6/2 ÷ 3/8 = 40/5
173: 6/2 ÷ 4/5 = 30/8
174: 6/2 ÷ 4/8 = 30/5
175: 6/3 ÷ 1/4 = 72/9
176: 6/3 ÷ 1/9 = 72/4
177: 6/3 ÷ 2/5 = 40/8
178: 6/3 ÷ 2/8 = 40/5
179: 6/3 ÷ 4/5 = 20/8
180: 6/3 ÷ 4/8 = 20/5
181: 6/4 ÷ 1/2 = 9/3
182: 6/4 ÷ 1/3 = 9/2
183: 6/4 ÷ 2/5 = 30/8
184: 6/4 ÷ 2/8 = 30/5
185: 6/4 ÷ 3/5 = 20/8
186: 6/4 ÷ 3/8 = 20/5
187: 6/4 ÷ 9/2 = 1/3
188: 6/4 ÷ 9/3 = 1/2
189: 6/8 ÷ 1/3 = 9/4
190: 6/8 ÷ 1/4 = 27/9
191: 6/8 ÷ 1/4 = 9/3
192: 6/8 ÷ 1/9 = 27/4
193: 6/8 ÷ 9/3 = 1/4
194: 6/8 ÷ 9/4 = 1/3
195: 6/9 ÷ 1/2 = 4/3
196: 6/9 ÷ 1/3 = 4/2
197: 6/9 ÷ 1/3 = 8/4
198: 6/9 ÷ 1/4 = 8/3
199: 6/9 ÷ 4/2 = 1/3
200: 6/9 ÷ 4/3 = 1/2
201: 6/9 ÷ 8/3 = 1/4
202: 6/9 ÷ 8/4 = 1/3
203: 7/2 ÷ 4/8 = 63/9
204: 7/2 ÷ 4/9 = 63/8
205: 7/2 ÷ 6/3 = 14/8
206: 7/2 ÷ 6/8 = 14/3
207: 7/4 ÷ 2/8 = 63/9
208: 7/4 ÷ 2/9 = 63/8
209: 7/4 ÷ 6/8 = 21/9
210: 7/4 ÷ 6/9 = 21/8
211: 7/6 ÷ 2/8 = 14/3
212: 7/6 ÷ 4/8 = 21/9
213: 8/1 ÷ 2/3 = 60/5
214: 8/1 ÷ 2/5 = 60/3
215: 8/1 ÷ 4/2 = 36/9
216: 8/1 ÷ 4/9 = 36/2
217: 8/1 ÷ 6/3 = 20/5
218: 8/1 ÷ 6/5 = 20/3
219: 8/2 ÷ 1/3 = 60/5
220: 8/2 ÷ 1/5 = 60/3
221: 8/2 ÷ 3/5 = 40/6
222: 8/2 ÷ 3/5 = 60/9
223: 8/2 ÷ 3/6 = 40/5
224: 8/2 ÷ 3/9 = 60/5
225: 8/2 ÷ 4/5 = 30/6
226: 8/2 ÷ 4/6 = 30/5
227: 8/2 ÷ 4/7 = 63/9
228: 8/2 ÷ 4/9 = 63/7
229: 8/2 ÷ 6/3 = 10/5
230: 8/2 ÷ 6/3 = 14/7
231: 8/2 ÷ 6/5 = 10/3
232: 8/2 ÷ 6/5 = 30/9
233: 8/2 ÷ 6/7 = 14/3
234: 8/2 ÷ 6/9 = 30/5
235: 8/3 ÷ 2/6 = 40/5
236: 8/3 ÷ 2/9 = 60/5
237: 8/3 ÷ 4/1 = 6/9
238: 8/3 ÷ 4/6 = 20/5
239: 8/3 ÷ 4/9 = 6/1
240: 8/3 ÷ 6/1 = 4/9
241: 8/3 ÷ 6/5 = 20/9
242: 8/3 ÷ 6/9 = 20/5
243: 8/3 ÷ 6/9 = 4/1
244: 8/4 ÷ 1/2 = 36/9
245: 8/4 ÷ 1/9 = 36/2
246: 8/4 ÷ 2/5 = 30/6
247: 8/4 ÷ 2/6 = 30/5
248: 8/4 ÷ 2/7 = 63/9
249: 8/4 ÷ 2/9 = 63/7
250: 8/4 ÷ 3/1 = 6/9
251: 8/4 ÷ 3/5 = 20/6
252: 8/4 ÷ 3/6 = 20/5
253: 8/4 ÷ 3/9 = 6/1
254: 8/4 ÷ 6/1 = 3/9
255: 8/4 ÷ 6/7 = 21/9
256: 8/4 ÷ 6/9 = 21/7
257: 8/4 ÷ 6/9 = 3/1
258: 8/6 ÷ 1/3 = 20/5
259: 8/6 ÷ 2/3 = 10/5
260: 8/6 ÷ 2/3 = 14/7
261: 8/6 ÷ 2/7 = 14/3
262: 8/6 ÷ 2/9 = 30/5
263: 8/6 ÷ 3/1 = 4/9
264: 8/6 ÷ 3/5 = 20/9
265: 8/6 ÷ 3/9 = 20/5
266: 8/6 ÷ 3/9 = 4/1
267: 8/6 ÷ 4/1 = 3/9
268: 8/6 ÷ 4/7 = 21/9
269: 8/6 ÷ 4/9 = 21/7
270: 8/6 ÷ 4/9 = 3/1
271: 9/1 ÷ 3/4 = 60/5
272: 9/1 ÷ 3/4 = 72/6
273: 9/1 ÷ 3/5 = 60/4
274: 9/1 ÷ 3/6 = 72/4
275: 9/1 ÷ 4/2 = 36/8
276: 9/1 ÷ 4/8 = 36/2
277: 9/1 ÷ 6/4 = 30/5
278: 9/1 ÷ 6/5 = 30/4
279: 9/1 ÷ 8/4 = 27/6
280: 9/1 ÷ 8/6 = 27/4
281: 9/2 ÷ 3/1 = 6/4
282: 9/2 ÷ 3/4 = 6/1
283: 9/2 ÷ 3/5 = 60/8
284: 9/2 ÷ 3/8 = 60/5
285: 9/2 ÷ 4/7 = 63/8
286: 9/2 ÷ 4/8 = 63/7
287: 9/2 ÷ 6/1 = 3/4
288: 9/2 ÷ 6/4 = 3/1
289: 9/2 ÷ 6/5 = 30/8
290: 9/2 ÷ 6/8 = 30/5
291: 9/3 ÷ 1/4 = 60/5
292: 9/3 ÷ 1/4 = 72/6
293: 9/3 ÷ 1/5 = 60/4
294: 9/3 ÷ 1/6 = 72/4
295: 9/3 ÷ 2/1 = 6/4
296: 9/3 ÷ 2/4 = 6/1
297: 9/3 ÷ 2/5 = 60/8
298: 9/3 ÷ 2/8 = 60/5
299: 9/3 ÷ 4/1 = 6/8
300: 9/3 ÷ 4/8 = 6/1
301: 9/3 ÷ 6/1 = 2/4
302: 9/3 ÷ 6/1 = 4/8
303: 9/3 ÷ 6/4 = 10/5
304: 9/3 ÷ 6/4 = 2/1
305: 9/3 ÷ 6/5 = 10/4
306: 9/3 ÷ 6/5 = 20/8
307: 9/3 ÷ 6/8 = 20/5
308: 9/3 ÷ 6/8 = 4/1
309: 9/4 ÷ 1/2 = 36/8
310: 9/4 ÷ 1/8 = 36/2
311: 9/4 ÷ 2/7 = 63/8
312: 9/4 ÷ 2/8 = 63/7
313: 9/4 ÷ 3/1 = 6/8
314: 9/4 ÷ 3/8 = 6/1
315: 9/4 ÷ 6/1 = 3/8
316: 9/4 ÷ 6/7 = 21/8
317: 9/4 ÷ 6/8 = 21/7
318: 9/4 ÷ 6/8 = 3/1
319: 9/6 ÷ 1/4 = 30/5
320: 9/6 ÷ 1/5 = 30/4
321: 9/6 ÷ 2/1 = 3/4
322: 9/6 ÷ 2/4 = 3/1
323: 9/6 ÷ 2/5 = 30/8
324: 9/6 ÷ 2/8 = 30/5
325: 9/6 ÷ 3/1 = 2/4
326: 9/6 ÷ 3/1 = 4/8
327: 9/6 ÷ 3/4 = 10/5
328: 9/6 ÷ 3/4 = 2/1
329: 9/6 ÷ 3/5 = 10/4
330: 9/6 ÷ 3/5 = 20/8
331: 9/6 ÷ 3/8 = 20/5
332: 9/6 ÷ 3/8 = 4/1
333: 9/6 ÷ 4/1 = 3/8
334: 9/6 ÷ 4/7 = 21/8
335: 9/6 ÷ 4/8 = 21/7
336: 9/6 ÷ 4/8 = 3/1
337: 9/8 ÷ 1/4 = 27/6
338: 9/8 ÷ 1/6 = 27/4
Source: Shaun Errichiello
I think these are all the possibilities, listed here in form a/b divided by c/d, but where a and d can be switched and/or b and c can be switched:
5/2 divided by 4/6 = 30/8
5/2 divided by 3/8 = 40/6 [as above]
7/2 divided by 4/9 = 63/8
Would this a possible solution also? Or would it be invalid?
9/3 divided by 1/6 equals 72/4
I think 9/3 divided by 1/6 would be 72/3. I think it might be cool to have a class of solutions like this, where there is 1 repeated digit. In this case, the 1.
9/3 divided by 1/6 would actually be 54/3 so it wouldn’t work since the “3” is repeated.
“9/3 divided by 1/6 equals 72/4” was original query
9/3 divided by 1/6 reduces to 54/3 hence 18
72/4 reduces to 18
Hence 18=18
I suppose the question is of repeats in original equation or no repeats in all the equations as you simplify and reduce.
My student found this solution:
9/2 divided by 4/7 equals 63/8
5/4 divided by 2/6 = 30/8
5/3 divided by 2/8 = 40/6
7/2 divided by 4/9 = 63/8
9/4 divided 2/7 = 63/8
5 / 2 divide 3/8 =40 / 6
6/4 divided by 2/5 equal 30/8
3/8 ÷ 1/4 = 9/6
Without leading zeros, there are 54 unique solutions. You can quadruple that number, because swapping the numerator of the first fraction with the denominator of the second fraction, or swapping the denominator of the first fraction with the numerator of the second fraction produces another solution:
2/1 ÷ 4/8 = 36/9
2/1 ÷ 4/9 = 36/8
3/1 ÷ 2/5 = 60/8
3/1 ÷ 2/8 = 60/5
3/1 ÷ 6/5 = 20/8
3/1 ÷ 6/8 = 20/5
3/2 ÷ 6/5 = 10/8
3/2 ÷ 6/7 = 14/8
3/2 ÷ 6/8 = 10/5
3/2 ÷ 6/8 = 14/7
4/1 ÷ 3/5 = 60/9
4/1 ÷ 3/6 = 72/9
4/1 ÷ 3/9 = 60/5
4/1 ÷ 3/9 = 72/6
4/1 ÷ 6/5 = 30/9
4/1 ÷ 6/9 = 30/5
4/1 ÷ 8/6 = 27/9
4/1 ÷ 8/9 = 27/6
4/3 ÷ 6/5 = 10/9
4/3 ÷ 6/9 = 10/5
5/1 ÷ 2/8 = 60/3
5/1 ÷ 3/9 = 60/4
5/1 ÷ 6/8 = 20/3
5/1 ÷ 6/9 = 30/4
5/2 ÷ 3/6 = 40/8
5/2 ÷ 3/8 = 40/6
5/2 ÷ 3/8 = 60/9
5/2 ÷ 3/9 = 60/8
5/2 ÷ 4/6 = 30/8
5/2 ÷ 4/8 = 30/6
5/2 ÷ 6/8 = 10/3
5/2 ÷ 6/8 = 30/9
5/2 ÷ 6/9 = 30/8
5/3 ÷ 4/6 = 20/8
5/3 ÷ 4/8 = 20/6
5/3 ÷ 6/8 = 20/9
5/3 ÷ 6/9 = 10/4
5/3 ÷ 6/9 = 20/8
6/1 ÷ 3/9 = 72/4
6/1 ÷ 8/9 = 27/4
6/2 ÷ 3/8 = 40/5
6/2 ÷ 4/8 = 30/5
6/3 ÷ 4/8 = 20/5
7/2 ÷ 4/8 = 63/9
7/2 ÷ 4/9 = 63/8
7/2 ÷ 6/8 = 14/3
7/4 ÷ 6/8 = 21/9
7/4 ÷ 6/9 = 21/8
8/1 ÷ 4/9 = 36/2
8/2 ÷ 3/9 = 60/5
8/2 ÷ 4/9 = 63/7
8/2 ÷ 6/9 = 30/5
8/3 ÷ 6/9 = 20/5
8/4 ÷ 6/9 = 21/7
With a leading zero in the numerator of the last fraction, there are another 2 unique solutions (1/3 ÷ 4/8 = 06/9 and 1/2 ÷ 3/4 = 06/9), where you can not only do swaps like described above (multiplying the number of solutions by 4), but also:
If a/b ÷ c/d = e/f is a solution, then so are
c/d ÷ f/e = a/b
a/b ÷ c/f = e/d
a/b ÷ e/d = c/f
a/b ÷ e/f = c/d
a/c ÷ e/d = b/f
a/c ÷ e/f = b/d
b/d ÷ f/c = a/e and
b/d ÷ f/e = a/c,
multiplying the number of solutions by 9. Also, if a/b ÷ c/d = e/f is a solution, so is b/a ÷ d/c = f/e (reciprocals of all fractions), multiplying the number of solutions again by 2, for a total of 2 * 4 * 9 * 2 = 144 solutions with a leading zero.
In total, there’s not just 338 solutions like the solution section claims, but 54*4 + 144 = 360. It’s missing the following 22 solutions:
4/3 ÷ 1/5 = 60/9
4/6 ÷ 1/5 = 30/9
5/3 ÷ 1/9 = 60/4
5/3 ÷ 2/6 = 40/8
5/3 ÷ 2/9 = 60/8
5/3 ÷ 4/6 = 20/8
5/3 ÷ 6/9 = 10/4
5/3 ÷ 6/9 = 20/8
5/6 ÷ 1/3 = 20/8
5/6 ÷ 1/9 = 30/4
5/6 ÷ 2/3 = 10/8
5/6 ÷ 2/9 = 30/8
5/6 ÷ 3/9 = 10/4
5/6 ÷ 3/9 = 20/8
7/6 ÷ 2/3 = 14/8
7/6 ÷ 4/9 = 21/8
8/3 ÷ 2/5 = 40/6
8/3 ÷ 2/5 = 60/9
8/3 ÷ 4/5 = 20/6
8/6 ÷ 1/5 = 20/3
8/6 ÷ 2/7 = 14/3
8/6 ÷ 2/9 = 30/5
2/5 divided by 8/3 equals 6/40