AMC 10 · 2013 · #3
Grade 7 algebraPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
Let L be the low temperature; the high is 16 degrees higher, so the high is L+16.
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 is the pair's sum divided by 2, so setting that equal to 3 gives one equation in L.
The average of two numbers is just their total shared evenly between them.
The average of two numbers is just their total shared evenly between them.
▸ Why?
An average is a total shared over a count, so two numbers split their sum in half.
▸ Why?
Moving up from one end as far as you move down from the other keeps that sum, so the middle is fixed.
Simplify the equation
The top adds to 2L+16, and halving it leaves L+8, so the equation becomes L+8=3.
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
Subtract 8 from both sides: L=3-8, which drops below zero to -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