Play With Fractions: Five Acts and an Encore
Some problems can seem simple, yet can be challenging. The following are some examples involving fractions (not ratios).
Act 1: You are given the digits 1, 3, 4, 5, 6, 7...which you can use exactly once each. In the expression A/B + C/D, replace the letters A, B, C, D with one of the given digits (so only 4 digits can be used)...with the goal that you are finding a sum less than 1.
What is the largest possible sum less than 1?
Act 2: You are given the digits 1, 3, 4, 5, 6, 7...which you can use exactly once each. In the expression A/B  C/D, replace the letters A, B, C, D with one of the given digits (so only 4 digits can be used)...with the goal that you are finding a difference less than 1.
What is the largest possible difference less than 1?
Act 3: You are given the digits 1, 3, 4, 5, 6, 7...which you can use exactly once each. In the expression A/B x C/D, replace the letters A, B, C, D with one of the given digits (so only 4 digits can be used)...with the goal that you are finding a product less than 1.
What is the largest possible product less than 1?
Act 4: You are given the digits 1, 3, 4, 5, 6, 7...which you can use exactly once each. In the expression A/B ÷ C/D, replace the letters A, B, C, D with one of the given digits (so only 4 digits can be used)...with the goal that you are finding a quotient less than 1.
What is the largest possible quotient less than 1?
Act 5: You are given the digits 1, 3, 4, 5, 6, 7...which you can use exactly once each. In the expoential expression [A/B]^{C/D}, replace the letters A, B, C, D with one of the given digits (so only 4 digits can be used)...with the goal that you are finding a root or exponential result less than 1.
What is the largest possible root or exponential result less than 1?
Encore Act: What if you started with the initial numbers 1, 2, 3, ..., 99, 100. What would be your results in each Act?
Hint: Depending on your fractionsense, the best approach is to jump in with both feet and try some possibilities. In each act, finding a result less than 1 is much easier than finding the largest possible result less than 1.
Solution Commentary: As answers may vary, I will not share mine. But, I will note that the first time I gave Act 1 to students, their common result was: 3/5 + 7/6 = 10/11 < 1. Thus, the initial distiction that the play is with fractions, not ratios (and there is a difference!).
