Depth · one concept from first principles · step 2 of 4
Use the four number cards , , , and , each exactly once, to build a multiplication of the form shown below.
Find the greatest possible product.
A ones digit is worth ten times the same digit in the tenths place, so 8 and 7 take the ones places.
Comparing decimals by place value is a grade-5 skill: bigger digits in higher places make a bigger number, so they belong in the ones places.
5.NBT.A.3Look For A PatternThe two largest cards have to go in the ones places.
A digit standing in the ones place is worth ten times the same digit standing in the tenths place.
A factor grows most when its most valuable place holds the biggest digit, and a bigger factor makes a bigger product.
Making a factor bigger means taking more of the same equal groups, so the product rises with it.
The leftover cards are 5 and 1, which go in the two tenths places. There are just two ways: 8.5 × 7.1 or 8.1 × 7.5.
With only two small cards left, a short complete list guarantees I do not miss the best option.
5.NBT.A.3Make A Systematic List85 x 71 = 6035, and one decimal place from each factor puts the point at 60.35.
Multiply as whole numbers, then count one decimal place from each factor (two total) to place the point.
5.NBT.B.7Guess And CheckMultiply 8.1 by 7.5. As whole numbers, 81 × 75 = 6075; with two decimal places that is 60.75.
Same decimal-point rule; 6075 hundredths is 60.75.
5.NBT.B.7Guess And Check60.75 beats 60.35, so pairing the smaller tenths digit with the larger ones digit is what wins.
Once both products are computed, the bigger one is the answer; comparing to hundredths is straightforward place-value work.
5.NBT.A.3Guess And CheckPut your biggest digits in the ones places, then test the last two ways to see which wins.
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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 →