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In these passages, Ishmaelites and Midianites are referring to the same group of people, so there is no contradiction.
Some argue that the Bible is contradictory concerning whether the Ishmaelites or the Midianites purchased Joseph. Here are the relevant passages:
Genesis 37:28- Then Midianite traders passed by. And they drew Joseph up and lifted him out of the pit, and sold him to the Ishmaelites for twenty shekels of silver. They took Joseph to Egypt.
Genesis 39:1- Now Joseph had been brought down to Egypt, and Potiphar, an officer of Pharaoh, the captain of the guard, an Egyptian, had bought him from the Ishmaelites who had brought him down there.
Genesis 37:36- Meanwhile the Midianites had sold him in Egypt to Potiphar, an officer of Pharaoh, the captain of the guard.
In Scripture itself, we see the terms "Midianites" and "Ishmaelites" used interchangeably. Here are the relevant verses.
Judges 8:22-26- 22 Then the men of Israel said to Gideon, “Rule over us, you and your son and your grandson also, for you have saved us from the hand of Midian.” 23 Gideon said to them, “I will not rule over you, and my son will not rule over you; the Lord will rule over you.” 24 And Gideon said to them, “Let me make a request of you: every one of you give me the earrings from his spoil.” (For they had golden earrings, because they were Ishmaelites.) 25 And they answered, “We will willingly give them.” And they spread a cloak, and every man threw in it the earrings of his spoil. 26 And the weight of the golden earrings that he requested was 1,700 shekels of gold, besides the crescent ornaments and the pendants and the purple garments worn by the kings of Midian, and besides the collars that were around the necks of their camels.
To paraphrase, after Gideon delivered the Israelites from the Midianites, he requested the golden earrings from the Ishmaelites, whom they had just plundered. Ishmaelite here is being used synonymously with Midianite. To solidify this point, the narrative goes on to say that the earrings, crescent ornaments, pendants, and purple robes were all "worn by the kings of Midian."
In Harper's Bible Dictionary, Achtemeier says that the term "Ishmaelite" was synonomous with the term "Midianite."[1] The two terms probably referred to the same general group that descended from Abraham.
Ishmael was the son of Abraham through his handmaiden Hagar (Genesis 16), and Midian was another son of Abraham through his concubine Keturah. Both were sons of Abraham, and it is very possible that their descendents became intermingled with time.
A Midianite who resided in the area of the Ishmaelites could be referred to as an Ishmaelite, and an Ishmaelite who resided on the area of the Midianites could be referred to as a Midianite. The Bible does not record national boundaries between the two peoples, so it is very reasonable to believe that there was at least some mingling between the two groups of people.
Achtemeier goes on to say this:
The term ‘Midianite’ probably identified a confederation of tribes that roamed far beyond this ancestral homeland, a usage that explains the biblical references to Midianites in Sinai, Canaan, the Jordan Valley, Moab, and Transjordan’s eastern desert.[2]
Thus, it is likely that the Midianites mingled with other groups, such as the Ishmaelites, such that the two terms became interchangeable.
Here is what another commentary says:
The different names given to the traders...do not show that the account has been drawn from different legends, but that these tribes were often confounded, from the fact that they resembled one another so closely, not only in their common descent from Abraham (Gen 16:15and 25:2), but also in the similarity of their mode of life and their constant change of abode, that strangers could hardly distinguish them, especially when they appeared not as tribes but as Arabian merchants, such as they are here described as being (1996).[3]
Essentially, the Midianites and Ishmaelites were indistinguishable from one another because of where they lived, how they lived, and how they looked.
The reasonable explanation that Ishmaelites and Midianites referred to the same group of people eliminates the alleged contradiction between these passages in Genesis.
Some people have trouble accepting possible explanations to alleged contradictions in the Bible, especially ones that they consider a stretch. For example, Dan Barker, president of the Freedom From Religion Foundation, says this:
I have received numerous replies from Christians who think that these contradictions are either trivial or easily explained. Yet not a single "explanation" has been convincing.
The fact is that there are plausible explanations for every alleged contradiction in the Bible. That there is a plausible explanation for an alleged contradiction does not mean that it is definitely the correct explanation for the alleged contradiction.
However, as long as a possible explanation has been suggested, then it has been objectively demonstrated that there is no necessary contradiction regarding the Bible verses and passages brought up.
When people like Dan Barker say that they don't find a particular explanation for a contradiction "convincing," then that is merely their opinion. A plausible explanation has been suggested that eliminates the necessary alleged contradiction. They simply don't like it, which is not at all a relevant argument against the explanation.
To read more about solutions to Bible contradictions and difficulties, check out Norman Geisler's The Big Book of Bible Difficulties: Clear and Concise Answers from Genesis to Revelation. While we do not agree with some of Geisler's theology, particularly concerning his view of predestination, this book is still an excellent resource. It is thorough and filled with research.
Another book to check out is Tim Chaffey's Demolishing Supposed Bible Contradictions: Exploring Forty Alleged Contraditions, which also answers many alleged contradictions in the Bible.