Problem
Find an object's weight from balance
Four objects sit on three level balances, so each pair of pans weighs the same. Every balance becomes one equation. A and D together weigh 36 g. Find the weight of A, B, C and D.
Reasoning · Grade 3-2 Weighing with Balances
Your answer
Watch out Solving the second balance as written leaves three unknowns — taking the shared C off both pans simplifies it at once.
How to solve
Strategy Work Backwards — Each level balance gives one equation. Balance 2 has a C on BOTH pans, so the C's cancel and it tells us A in terms of B. Balance 1 ties B to D. With those two facts the total A + D = 36 pins down B (and hence A and D), and finally balance 3 gives C. We chain the balances backwards from the known total - a work-backwards / subproblem chain that needs only grade-3 add, subtract, double, and the unknown-factor idea.
1STEP 1
Turn each level balance into an equation
Level pans give one equation per balance.
B = D+6, C+2B = A+C, 2A+4 = C
A level scale is just an equals sign you can see.
3.OA.A.4Identify Subproblems2STEP 2
Cancel the matching C on balance 2 to get A = 2B
Removing C from both pans leaves A = two B.
C + 2B = A + C → 2B = A
Taking the same thing off both pans keeps a balance level - so the shared C simply disappears.
3.OA.A.4Work BackwardsLifting the same object C off both pans leaves the balance level and turns the second weighing into A equals two B.
Why?
A level scale stays level when the same amount is removed from both pans, so the shared object can simply be struck out.
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
Why?
Each pan holds its objects' weights added together, so removing one object subtracts exactly that object's weight from that side.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Write A and D using B, then use A + D = 36
Putting D = B − 6 into the 36 gives B = 14.
A = 2B, D = B-6, 2B + (B-6) = 36 → 3B = 42
Rewriting everything in terms of one object (B) collapses the puzzle to a single unknown-factor question.
3.OA.D.8Identify Subproblems4STEP 4
Solve for B, then A and D
So A = 28 and D = 8, summing to 36.
B = 42 ÷ 3 = 14, A = 2 × 14 = 28, D = 14-6 = 8
Once B is known, the other two weights fall out by one doubling and one subtraction.
3.OA.A.4Work Backwards5STEP 5
Find C from balance 3
The third balance then gives C = 60.
C = 2 × 28 + 4 = 60
The last balance was waiting for A; now we just plug it in.
3.NBT.A.2Identify SubproblemsAnswer
A 28 g, B 14 g, C 60 g, D 8 g
Check all three balances with A=28, B=14, C=60, D=8. Balance 1: B = 14 and D + 6 = 8 + 6 = 14 (level). Balance 2: C + 2B = 60 + 28 = 88 and A + C = 28 + 60 = 88 (level). Balance 3: 2A + 4 = 56 + 4 = 60 = C (level). And A + D = 28 + 8 = 36. Every condition holds and all weights are positive whole numbers, so the answer is reasonable.
Takeaway
When the SAME object sits on both pans, you can lift it off both sides and the scale stays level - that trick turns Balance 2 into the simple fact 'A is two B's', and the rest falls right out.
- Turn each level balance into an equation
- Cancel the matching C on balance 2 to get A = 2B
- Write A and D using B, then use A + D = 36
- Solve for B, then A and D
- Find C from balance 3