Problem
Deduce missing digits in a multiplication
A three-digit number A74 (hundreds digit A, tens 7, ones 4) times a single digit B equals 6992. Find A and B.
OperationsBase-ten numbers
Your answer
Watch out Don't pick a candidate without checking -- assuming B = 3 works falls apart once you see 6992 divided by 3 isn't a whole number.
How to solve
Strategy Make a Systematic List — Start at the ones place: 4 times B must end in 2, which only a couple of digits do. Test each candidate B by dividing 6992 to see if it gives a clean A74 form.
1STEP 1
Use the ones digit to limit B
The product's ones digit is 2, and 4 x B must end in 2 -- checking the options shows B is 3 or 8.
4 × 3 = 12, 4 × 8 = 32
The ones digit of a product depends only on the ones digits of the factors.
3.OA.C.7Make A Systematic ListThe ones digit of A74 times B is the same as the ones digit of 4 times B.
Why?
A74 is A hundreds, 7 tens, and 4 ones added together, so B times A74 equals B times the hundreds plus B times the 7 tens plus B times 4.
Why?
A74 is exactly its hundreds amount, its tens amount, and its ones amount stacked back together.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Multiplying that stacked-up sum by B is the same as multiplying each part by B and then adding the pieces.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
B times the hundreds and B times the 7 tens are whole tens, so they put a 0 in the ones place and only B times 4 reaches the ones place.
Why?
A count of whole tens is complete ten-bundles with no leftover ones, so its ones place holds 0.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Adding those whole tens drops zero ones into the ones place, so that place stays whatever 4 times B made it.
🧱Additive identity (zero)Adding zero adds nothing, so the number stays exactly the same.
2STEP 2
Test B = 3, then B = 8
Divide 6992 by each candidate -- only B = 8 gives a whole number, 874, matching A74 with A = 8.
6992 ÷ 8 = 874
Dividing the product by B should rebuild the original number with 7 in the tens and 4 in the ones place.
3.OA.A.4Guess And Check3STEP 3
Confirm
Check 874 x 8 = 6992 -- it matches, so A = 8 and B = 8.
874 × 8 = 6400 + 560 + 32 = 6992
Multiplying back must return the original 6992.
3.OA.C.7Guess And CheckAnswer
A = 8, B = 8
874 × 8 = 6992
874 x 8 = 6992 matches exactly, the tens digit is 7 and ones digit is 4 as required, so the solution fits all conditions.
Takeaway
Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!
- Use the ones digit to limit B
- Test B = 3, then B = 8
- Confirm
Where next?
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