Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

Reverse the steps to recover the original fraction

Someone started with a mystery fraction, changed it in three steps (take 4 away from the denominator, add 3 to the numerator, then reduce by dividing top and bottom by 4), and ended with 6/7. We need to figure out what fraction they started with.
Fractions
Your answer
How to solve
Strategy Work Backwards — We are told the end result (6/7) and the exact sequence of steps that produced it, and we want the start. That is the classic 'work backwards' situation: replay each step in reverse order using the opposite operation. Breaking the three steps into separate reverse subproblems keeps it organized.
1STEP 1

Undo the reducing (multiply back by 4)

Reducing divided top and bottom by 4, so multiplying them back by 4 gives 24/28.

6/7 = (6 × 4)/(7 × 4) = 24/28
2STEP 2

Undo adding 3 to the numerator (subtract 3)

Undoing the 3 added to the numerator: 24 - 3 = 21, leaving 21/28.

24 - 3 = 21 → 21/28
3STEP 3

Undo subtracting 4 from the denominator (add 4)

Undoing the 4 taken off the denominator: 28 + 4 = 32, giving 21/32.

28 + 4 = 32 → 21/32
Answer
2132\frac{21}{32}
Check by replaying forward from 21/32: subtract 4 from the denominator gives 21/28; add 3 to the numerator gives 24/28; divide top and bottom by 4 gives 6/7. That matches the given result exactly, so 21/32 is correct.
Takeaway

To find a starting number, replay the steps backwards and use each operation's opposite.

  • Undo the reducing (multiply back by 4)
  • Undo adding 3 to the numerator (subtract 3)
  • Undo subtracting 4 from the denominator (add 4)
Where next?
Level upsuggested
Reverse decimal operations to recover the startUndo a wrong decimal operation to recover the start
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Skip-count backward to the startRecover the unknown, then recompute
7 reasoning problems grow out of this type — the bridge toward competition →

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▶ Practice — 12 problems

The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Multiplicative Ratio with Integrality Constraints. Why the shapes repeat →