Problem
Dividend equals divisor times quotient
Two numbers A and B add up to 10, and they make the equation 8 divided by A equal to B divided by 3. We must find each of A and B.
Operations
Your answer
Watch out Don't stop at a pair that just sums to 10 — you must also check that it makes 8 divided by A equal to B divided by 3.
How to solve
Strategy Guess and Check — Because A and B are small whole numbers that sum to 10, we can list the few sensible pairs and check which one keeps both sides of 8 divided by A equal to B divided by 3 equal. Guess and check fits this bounded set perfectly.
1STEP 1
List the pairs that sum to 10
Since 8 ÷ A must come out even, A is a factor of 8 — the sum-to-10 pairs are (1,9), (2,8), (4,6), (8,2).
(A,B) = (1,9), (2,8), (4,6), (8,2)
Grade 3 sense: dividing 8 evenly only works for the factors of 8, so only a few A values are worth trying.
3.OA.A.4Make A Systematic List2STEP 2
Test each pair in the equation
Testing each pair, only A = 4, B = 6 makes 8 ÷ A equal B ÷ 3 (both give 2).
8 ÷ 4 = 2, 6 ÷ 3 = 2
Grade 3 division: the equation balances only when both quotients come out to the same whole number, 2.
3.OA.B.6Guess And CheckPutting A = 4 and B = 6 into 8 \div A = B \div 3 keeps the two sides equal.
Why?
Each side works out to the same number, 2, so the equation balances.
Why?
The left side 8 \div 4 is 2, because dividing asks what number times 4 makes 8, and 2 \times 4 = 8.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The right side 6 \div 3 is 2, because dividing asks what number times 3 makes 6, and 2 \times 3 = 6.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Since the left side is 2 and 2 is the right side, the left side must be equal to the right side.
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
Answer
A = 4, B = 6
Check both conditions: A + B = 4 + 6 = 10 (correct), and 8 divided by 4 equals 2 while 6 divided by 3 equals 2, so the two sides match. Both requirements hold.
Takeaway
Try the few pairs that add to 10 and make 8 divide nicely, and only 4 and 6 keep both sides equal!
- List the pairs that sum to 10
- Test each pair in the equation
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested