Problem
Dividing a smaller number by a bigger one shrinks the quotient
Four number cards each go in one place to make one proper fraction divided by another. We need the smallest quotient possible.
Number systemExpressions
Your answer
How to solve
Strategy Make a Systematic List — Rewriting the division as one fraction shows which two cards end up on top and which two underneath. Then the goal is plain -- small numbers above, large ones below -- and the proper-fraction rule cuts the choices down to a handful worth checking.
1STEP 1
Rewrite the division as one fraction
Dividing by a fraction flips it, so the second fraction's denominator moves to the top and its numerator to the bottom.
2/5 ÷ 3/4 = (2 × 4)/(5 × 3)
Two cards end up above the line and two below.
6.NS.A.1Convert To AlgebraThe division of two fractions is one fraction with two cards on top and two underneath.
Why?
Dividing by a fraction asks how many of it fit into the first, and that is the same as multiplying by the fraction turned over.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Multiplying two fractions multiplies the tops together and the bottoms together, because cutting each of b parts into d pieces makes b times d pieces in all.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Decide what makes a quotient small
A fraction is smallest when the top is small and the bottom is large, so put the small cards above the line.
Small cards up, large cards down.
6.EE.A.2Guess And Check3STEP 3
Work through the ways of pairing the cards
Both fractions must be proper, so each pair's smaller card is its numerator. The three ways of splitting the cards into two pairs give six arrangements, and the smallest from each pairing is 5/6, 5/6 and 8/15.
2/5 ÷ 3/4 = 8/15
8/15 is smaller than 5/6.
6.NS.A.1Make A Systematic ListAnswer
Compare as decimals: 8/15 ≈ 0.53 against 5/6 ≈ 0.83, so 8/15 really is the smallest.
Takeaway
Rewrite a fraction division as one fraction first -- then you can see which card lands where.
- Rewrite the division as one fraction
- Decide what makes a quotient small
- Work through the ways of pairing the cards