Reasoning · Grade 4-1 Multiplication

Problem

Order products by where digits are placed

Six expressions are given, each a two-digit times a two-digit. Each uses the digits 3, 4, 6 and 8 exactly once. Compare them without actually multiplying. List the letters from the largest product down.
Your answer
How to solve
Strategy Make a Systematic List — The tens digits carry almost all of the size of a two-digit number, so I first sort the six expressions into groups by which two digits were sent to the tens places. That group order settles most of the ranking in one move. Only two ties are left, each between expressions built from the same pair of tens digits, and each tie can be broken with a one-line comparison that is much easier than a full multiplication.
1STEP 1

See what actually changes from one expression to the next

All that changes is where each digit sits.

2STEP 2

Sort the six expressions by their pair of tens digits

Grouping by the tens pair separates the sizes.

80 × 60 = 4800, 80 × 40 = 3200, 60 × 40 = 2400, 40 × 30 = 1200
3STEP 3

Check that the groups really cannot overlap

The groups' ranges never overlap.

{C,E} ＞ 4900 ＞ A ＞ 3300 ＞ B ＞ 2800 ＞ {D,F}
4STEP 4

Break the tie between C and E

Between C and E the leftovers make C larger.

83 × 64 = 83 × 63 + 83, 84 × 63 = 83 × 63 + 63
5STEP 5

Break the tie between D and F the same way

The same trick makes D larger than F.

46 × 38 = 46 × 36 + 2 × 46, 48 × 36 = 46 × 36 + 2 × 36
6STEP 6

Write the letters in order

The order is C, E, A, B, D, F.

5312 ＞ 5292 ＞ 3818 ＞ 3024 ＞ 1748 ＞ 1728
Answer
C, E, A, B, D, F
5312, 5292, 3818, 3024, 1748, 1728
Each product is a two-digit number times a two-digit number, so every one should land between about 1000 and about 6000, and the computed values 5312, 5292, 3818, 3024, 1748 and 1728 all do. The ordering also matches a general habit worth noticing: when the sum of the two factors is fixed, the closer together the factors are the bigger the product. C has factors 83 and 64, which differ by 19, while E has 84 and 63, which differ by 21 and the same sum of 147, so C should win, and it does.
Takeaway

Big digits belong in the tens places, and when two products are almost the same, compare the one extra row instead of multiplying everything out.

  • See what actually changes from one expression to the next
  • Sort the six expressions by their pair of tens digits
  • Check that the groups really cannot overlap
  • Break the tie between C and E
  • Break the tie between D and F the same way
  • Write the letters in order