Dividend equals divisor times quotient

3.OA.B.63.OA.A.4 · take · grade 3

Archetype: Division as the Inverse of Multiplication · step in a 4-type progression

The sum of $A$ and $B$ is $10$ . Find the value of $A$ and the value of $B$ .

$8 \div A = B \div 3$

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Understand

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.

Givens

A plus B equals 10.
8 divided by A equals B divided by 3.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

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

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.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 8 nicely only for certain values, focus on pairs where A is a number 8 can be divided by: A could be 1, 2, 4, or 8, with B being 10 minus A. That gives the pairs (A=1,B=9), (A=2,B=8), (A=4,B=6), (A=8,B=2).

(A,B) \in \{(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.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (4,6): 8 divided by 4 is 2, and 6 divided by 3 is 2, so both sides equal 2. The other pairs fail: (1,9) gives 8 vs 3, (2,8) gives 4 vs not-whole, (8,2) gives 1 vs not-whole. Only A = 4, B = 6 works.

8 \div 4 = 2 \quad \text{and} \quad 6 \div 3 = 2

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 2.

Answer: A = 4, B = 6

Review

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.

Rewrite 8 divided by A equals B divided by 3 as a cross-product: 8 times 3 equals A times B, so A times B = 24. Find two numbers that add to 10 and multiply to 24, which are 4 and 6; since A must divide 8, A = 4 and B = 6.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 8 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 10 and make 8 divide nicely, and only 4 and 6 keep both sides equal!