AMC 10 · 2013 · #1
Grade 7 algebraPick an answer.
There is one unknown, the low temperature, and every other quantity is described in terms of it, so Tool #4 (Introduce a Variable) names the low as L and lets the high be written as L+16. Tool #13 (Convert to Algebra) then turns the sentence “the average is 3” into a single equation to solve. Tool #3 (Eliminate Possibilities) is a fast backstop: only one listed low pairs with a high that is 16 larger and averages 3.
Name the low and write the high
One letter names both temperatures.
Once the low has a name, the high is just that name plus 16.
7.EE.B.4Introduce A VariableTurn the average into an equation
The average becomes one equation.
The average of two numbers is just their total shared evenly between them.
The average of two numbers is their total shared evenly between them.
▸ Why?
An average is a total divided by how many there are, so two numbers give the total halved.
▸ Why?
Two values sit symmetrically about their average, so the average is the low plus half the gap.
Simplify the equation
Simplifying leaves the low plus 8.
Halving the gap of 16 shows the average is exactly 8 warmer than the low.
7.EE.B.4Introduce A VariableSolve for the low temperature
Solving gives -5, choice (C).
Taking away more than you have pushes the number below zero.
7.NS.A.1Eliminate PossibilitiesThe average of two numbers sits exactly halfway between them, so the low is the average minus half the gap.
- Name the low and write the high
- Turn the average into an equation
- Simplify the equation
- Solve for the low temperature