Place big digits high for largest product

3.OA.C.73.NBT.A.3 · take · grade 3

Archetype: Build the Largest or Smallest Value from Digit Cards · step in a 7-type progression

Using the number cards $4$ , $7$ , $9$ , $2$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $4$ , $7$ , $9$ , $2$ into the four blanks, using each card exactly once.

Show solution

Understand

Place the four cards 4, 7, 9, 2 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 4, 7, 9, 2, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Digits in the tens place count for the most, so the two biggest cards (9 and 7) should be tens digits. Then test the few ways to place 4 and 2 in the ones places and pick the largest product.

Execute

#6 Guess and Check 3.NBT.A.3

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 9 and 7, in the tens places. That leaves 4 and 2 for the ones places.

9\square \times 7\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (4) to the factor with the larger tens digit (9), so it is multiplied by more. Compare 94 x 72 with 92 x 74.

94 \times 72 = 6768,\quad 92 \times 74 = 6808

Only a few arrangements are possible, so listing and comparing them is quick and safe.

#6 Guess and Check 3.OA.C.7

Of the candidates, 92 x 74 gives the biggest result, 6808.

92 \times 74 = 92 \times 70 + 92 \times 4 = 6440 + 368 = 6808

Break 74 into 70 and 4, multiply each part, then add.

Answer: 6808 (from 92 x 74)

Review

Both factors are in the 70s-90s, so the product should be roughly 90 x 75 = 6750. The answer 6808 sits right in that range, so it is reasonable.

Make a systematic list of every two-digit x two-digit arrangement of 4, 7, 9, 2; the maximum among all of them is 6808, confirming the place-value shortcut.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡 Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!