AMC 10 · 2016 · #14
Grade 8 algebraPick an answer.
The question asks for a smallest possible value, which is a minimization problem, so the Extreme Principle leads. To use it, name the first term and ratio as variables, turn the two facts into algebra, and write S as a single expression in r. Then S is smallest exactly when one denominator quantity is largest, and that maximum is easy to find.
Write the sum with a and r
Two letters describe the whole series.
Naming the two numbers that define the series lets every fact become an equation.
6.EE.B.6Introduce A VariableUse the second term to cut a variable
The second term removes one of them.
One given fact removes one unknown, so the whole problem collapses to a single variable.
7.EE.A.2Convert To AlgebraSmallest S means largest denominator
The smallest total needs the largest denominator.
Pushing on the bottom of a fraction the opposite way pushes the whole fraction toward its extreme.
The smallest sum comes from the largest denominator, because the fraction moves the opposite way.
▸ Why?
Dividing by a larger positive number is multiplying by a smaller reciprocal, so the value falls.
▸ Why?
Because that trade never turns around, the extreme of the sum sits at the extreme of the denominator.
Maximize the denominator
Completing the square gives 4, choice (E).
Rewriting as a square minus a constant makes the biggest value obvious, because a square can only add or stay at zero.
7.EE.A.2Convert To AlgebraTo make a fraction as small as possible, make its bottom as big as possible.
- Write the sum with a and r
- Use the second term to cut a variable
- Smallest S means largest denominator
- Maximize the denominator