This is what a full credit paper looks like. Read the problem, then compare
every box with your own.
The Problem
Kimberly is searching for a room to rent and found the payment plans offered by two
landlords, A and B. After how many months will renting with landlord A be cheaper than
landlord B?
Landlord A
Landlord B
Initial charge (one time)
$300
$100
Monthly charge
$450
$500
Orange = the charges you are given. Blue = landlord A. Red = landlord B. Purple = the tie. Green = the answer.
Box 1 — I know
half a point
Landlord A: $300 to start, then $450
each month.
Landlord B: $100 to start, then $500
each month.
Total cost = one-time charge + (monthly charge × number of months).
The one-time charge is paid once. It is never multiplied.
Box 2 — I need to find out
half a point
I need to find after how many months landlord A costs less than landlord B.
The answer is a number of months, not a number of dollars.
Box 3 — Steps
2 points · the biggest box
Step 1 — name the unknown
Let m = the number of months.
Step 2 — write each cost
A = 300 + 450m
B = 100 + 500m
Step 3 — turn "cheaper" into math
Cheaper means less than:
300 + 450m < 100 + 500m
Step 4 — solve it
Subtract 100 from both sides:
200 + 450m < 500m
Subtract 450m from both sides:
200 < 50m
Divide both sides by 50:
4 < m so m > 4
Step 5 — check the two months around the answer
This is the step that proves 4 is a tie and not the answer.
Landlord A
Landlord B
Who is cheaper?
m = 4
300 + 1,800 = $2,100
100 + 2,000 = $2,100
TIE — not cheaper
m = 5
300 + 2,250 = $2,550
100 + 2,500 = $2,600
A is cheaper
Box 4 — Answer
2 points
After 4 months. ✓ Landlord A is cheaper starting at month 5.
Why this earns all 5 points
Box 1 lists the four numbers with meanings. Not the story sentence about Kimberly.
And it says out loud that the one-time charge is paid once — which is the whole problem.
Box 2 keeps the words "how many months." Shortening it to "which landlord is cheaper"
is what made several students stop after one month.
Box 3 names the variable first, then builds both costs, then compares them —
the order Algebra II is asking for.
Box 3 checks m = 4 and m = 5. This is what separates a 2 from a 1 and a half.
It proves month 4 is a tie, so "cheaper" cannot start there.
Box 4 answers in months. A dollar amount in the answer box does not answer the
question.
Three mistakes to avoid
1. Do not multiply the one-time charge. (300 + 450) × 2 is wrong.
It is 300 + (450 × 2). The $300 is paid once, not every month.
2. Do not compare only the monthly charges. $450 is less than $500, but that does not
make A cheaper right away — A starts $200 behind.
3. Do not stop at one month. At month 1, B is cheaper ($600 vs $750). You have to keep
going until the lines cross.
How to remember it: A starts $200 behind and catches up $50 every month, so 200 ÷ 50 = 4 months to tie — then A wins.