Depth · one concept from first principles · step 2 of 4
Both sides of an equation can be multiplied or divided by the same number
The two expressions below give the same result. Find the value of . (Neither nor is .)
Expressions
Your answer
How to solve
Strategy Work Backwards — Simplify each side into a single multiple of its unknown. Then the equality is one clean sentence relating the two, and the ratio falls out without ever needing the individual values.
1STEP 1
Simplify the first expression
Dividing by 2 and then multiplying by 6 is the same as multiplying by 6 ÷ 2.
○ ÷ 2 × 6 = ○ × 3
The first side is just three times ○.
6.EE.A.3Convert To Algebra2STEP 2
Simplify the second expression
Dividing by 3 twice is dividing by 9 once.
△ ÷ 3 ÷ 3 = △ ÷ 9
The second side is △ split into nine.
6.EE.A.3Convert To Algebra3STEP 3
Set the two sides equal
The problem says the results match, so write that as one equation and clear the division by multiplying both sides by 9.
○ × 3 = △ ÷ 9 → △ = ○ × 27
△ is 27 times ○.
6.EE.B.5Convert To Algebrais times .
Why?
The two expressions are said to give the same result, so whatever the first is worth, the second is worth exactly that.
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
Why?
Three times equalling split into nine becomes once both sides are multiplied by nine, because multiplying by nine undoes dividing by nine.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
4STEP 4
Read off the ratio
If △ is 27 copies of ○, then ○ is one twenty-seventh of △, whatever the actual numbers are.
○ ÷ △ = 1/27
The ratio is 1/27.
6.EE.B.6Work BackwardsAnswer
Try a case: with ○ = 1 the first side is 3, and △ must be 27 because 27 ÷ 9 = 3. Then 1 ÷ 27 = 1/27.
Takeaway
Tidy each side on its own first -- an equation is much easier to read once both sides are as simple as they can be.
- Simplify the first expression
- Simplify the second expression
- Set the two sides equal
- Read off the ratio
Where next?
Another one like thissuggested
Step back and solidifysuggested
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Multiplicative Ratio with Integrality Constraints. Why the shapes repeat →