AMC 10 · 2007 · #3
Grade 6 algebraPick an answer.
The smaller integer is the one unknown that controls everything else, so Tool #4 (Introduce a Variable) names it s and writes the larger one as s+2 (the next odd number is always 2 bigger). Tool #13 (Convert to Algebra) then turns the sentence "the larger is three times the smaller" into an equation, which pins down s exactly instead of guessing.
Name the two integers
Consecutive odd numbers differ by two.
Odd numbers step by 2, so the neighbour above s is s+2.
Odd numbers step by two, so the next odd number above one is that number plus two.
▸ Why?
Adding one to an odd number makes it even, so another one is needed to reach the next odd number.
▸ Why?
The odd numbers climb by the same fixed step every time, so one letter and one step describe both.
Turn the words into an equation
The sentence becomes a one-step equation.
"Is" means the two sides name the same number, so set them equal.
6.EE.B.6Convert To AlgebraSolve for the smaller integer
Solving gives the smaller as 1.
The gap between the two numbers is both 2 and two copies of the smaller, so the smaller must be 1.
6.EE.B.7Introduce A VariableAdd the two integers
Adding gives 4, choice (A).
Once both integers are known, the sum is just adding them.
6.EE.A.2Convert To AlgebraCall the smaller number s, write the next odd number as s+2, turn the sentence into an equation, and the numbers fall out.
- Name the two integers
- Turn the words into an equation
- Solve for the smaller integer
- Add the two integers