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