Depth · one concept from first principles · step 2 of 4
Only the 3 repeats: subtract the non-repeating 58 from 583 and divide by 900 (one 9, two 0s).
Mia's misread fraction was in lowest terms, so 7/12 is exactly what she read, and its numerator 7 is the true numerator.
8.NS.A.1Work Backwards.
Calling the number , and end in the same lined-up tail of s, so subtracting them cancels the tail and leaves .
Each factor of moves every digit one place to the left, so carries the two digits and in front of the point and carries one more, .
In the two digits before the repeat have moved out, leaving only s after the point, and is shifted by exactly one more block of length , so its tail matches digit for digit.
Taking away from and away from keeps the two sides equal, and the matching tails cancel, leaving .
becomes by dividing top and bottom by , which only renames the fraction.
In only the 1 repeats, so use 90 as the denominator.
73 is prime, so 73/90 is in lowest terms and is exactly what Liam read; its denominator 90 is the true denominator.
8.NS.A.1Work BackwardsPut the correct numerator over the correct denominator. 7/90 is in lowest terms, and dividing gives 0.0777….
Only the 7 repeats, starting at the second decimal place.
7.NS.A.2Identify SubproblemsA wrong answer still hides a right part -- work backwards to find it.
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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 →