AMC 10 · 2012 · #6
Grade 8 arithmeticPick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Two unknown numbers are tied together by two facts: a product and a reciprocal relationship. Naming each number with a letter lets me turn both facts into equations, then combine them into one equation in a single variable and solve it.
Name the two numbers
Call the two positive numbers a and b, so each fact can be written as an equation instead of words.
A letter is just a placeholder for a number you don't know yet.
6.EE.B.6Introduce A VariableTurn the two facts into equations
"The product is 9" becomes ab = 9; "one reciprocal is 4 times the other" becomes 1/a = 4·(1/b).
Each English sentence maps directly onto one equation.
6.EE.A.2Convert To AlgebraSimplify the reciprocal equation
Cross-multiplying 1/a = 4/b gives b = 4a: one number is 4 times the other.
A smaller number has a larger reciprocal, so the one with the bigger reciprocal is the smaller number.
A smaller number has a larger reciprocal, so the bigger reciprocal marks the smaller number.
▸ Why?
A reciprocal undoes multiplying, so making the number bigger makes what returns it to one smaller.
▸ Why?
That flip reverses the order consistently, so comparing reciprocals compares the numbers backwards.
Substitute and solve for a
Replacing b in ab = 9 gives 4a² = 9, so a² = 9/4, and a is positive, so a = 3/2.
Substituting collapses two unknowns into one, and taking the positive square root undoes the squaring.
8.EE.A.2Introduce A VariableFind b and add
Then b = 4a = 6, and the sum is 3/2 + 12/2 = 15/2, choice (D).
Write the whole number as halves so both terms share a denominator before adding.
5.NF.A.1Introduce A VariableTurn each sentence into an equation, use one fact to replace a letter, and you shrink two unknowns down to one you can solve.
- Name the two numbers
- Turn the two facts into equations
- Simplify the reciprocal equation
- Substitute and solve for a
- Find b and add