Depth · one concept from first principles · step 2 of 4
Subtracting -2 adds 2, and the quotient of the two positives is positive; dividing the larger by the smaller makes it bigger.
The other order, 2/3 ÷ 4/5 + 2 = 5/6 + 2 = 17/6, is smaller, since 17/6 ＜ 3 ＜ 16/5.
7.NS.A.2Eliminate PossibilitiesThe greatest value comes from putting in the last box, so that it is subtracted, and dividing the larger positive number by the smaller one.
Subtracting is the same as adding .
Adding to gives , so adding undoes an added ; taking away undoes the same thing, so the two moves must be equal.
Any placement with inside the quotient ends up below , so it cannot be the greatest.
A quotient with one negative number in it is negative, because division undoes multiplication and a negative number of positive groups is negative.
Taking a positive number away from a negative quotient pushes it further below , and anything below is below every positive result.
Of the two positive quotients, is more than while is less than , so after adding one result is above and the other below it.
Put -2 in the quotient. A large-sized negative quotient comes from -2 on top.
Since 19/5 ＞ 19/6, the number -19/5 is the smaller one.
7.NS.A.2Make A Systematic ListWith -2 in the second box the quotient is small in size.
Both lie between -2 and -1, so neither beats -19/5 or 16/5.
7.NS.A.2Make A Systematic ListGreatest 16/5, least -19/5.
The sum is -3/5.
7.NS.A.1Make A Systematic ListDecide the signs first; then only a couple of placements are worth computing.
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The same shape shows up later on the AMC, as Branching State Enumeration, Casework by Position / Vertex Type. Why the shapes repeat →