The first thing we need to do is to express both fractions in terms of a common denominator.
1⅜ is 8 eighths plus 3 eighths making 11 eighths, which we can write 11/8.
The common denominator of 5 and 8 is 40, which we get by just multiplying 5 and 8.
To make 11/8=?/40, we multiply top and bottom by 5 so 11/8=55/40.
To make 2/5=?/40, we multiply top and bottom by 8 so 2/5=16/40.
We can ignore the denominator now and just use partitive division on the numerators 55 and 16.
We split 55 into 2 groups. The way to do this is to make 55 into something which we more easily divide, so we know that 55=54+1 and we know we can split 54 into two groups, with 27 in each. And we know that we can split 1 into 2 groups by putting ½ into each so we have two groups with 27+½ in each group, that’s 27½.
We now split each of the two groups into two again. So we need to split 27½ into 2. Well, we go for the nearest even whole number which is smaller than 27½. That means 26 with 1½ left over. We know half of 26 is 13, but what’s half of 1½? Well, half of 1 is ½ and half of a half is ¼. And ½+¼=¾. So half of 27½ is 13¾. We now have the original 55 split into 4 groups, each with 13¾ in a group.
If we split each of the 4 groups in half we will have split 55 into 8 groups. So, again we see that 13¾ is 12+1¾, and half of 12=6, and since 1¾ is 7 quarters, or 7/4, half of this gives us ⅞. Therefore, we have 6⅞ in each of the 8 groups. We have to go one step further and halve each of the 8 groups so that we get 16 groups altogether, then we will have split 55 into 16 groups.
Split 6⅞ into two halves and we get 3⁷⁄₁₆. That’s it! We have divided 55 into 16 groups, each containing 3⁷⁄₁₆, so the result of dividing 1⅜ by ⅖ is 3⁷⁄₁₆.