AMC 10 · 2011 · #1
Easy mode Grade 5Michelle's phone bill for one month is made of three parts:
- a flat $20 to start,
- 5 cents for every text she sends,
- 10 cents for every minute she talks past 30 hours.
In January she sent 100 texts and talked for 30.5 hours. What was her total bill?
Pick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A phone bill has three parts: a flat $20 base, a charge of 5 cents for every text sent, and a charge of 10 cents for every minute of talking beyond 30 hours. Michelle sent 100 texts and talked for 30.5 hours. Find her total bill.
Givens: Base charge: $20 per month; Text charge: 5 cents per text sent; Overage charge: 10 cents per minute used over 30 hours; Michelle sent 100 texts; Michelle talked for 30.5 hours
Unknowns: The total amount Michelle pays for January
Understand
Restated: A phone bill has three parts: a flat $20 base, a charge of 5 cents for every text sent, and a charge of 10 cents for every minute of talking beyond 30 hours. Michelle sent 100 texts and talked for 30.5 hours. Find her total bill.
Givens: Base charge: $20 per month; Text charge: 5 cents per text sent; Overage charge: 10 cents per minute used over 30 hours; Michelle sent 100 texts; Michelle talked for 30.5 hours
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units
The bill is a sum of three independent pieces, so I compute each piece by itself and add them. The pieces mix units (cents, dollars, hours, minutes), so I convert carefully as I go to keep everything in dollars.
Execute — Answer: D
4.OA.A.3 Step 1 Split the bill into three parts
- The total is the base fee plus the texting cost plus the extra-minutes cost.
- I will find each part separately and then add them.
💡 A messy bill becomes easy once you break it into pieces you can price one at a time.
5.NBT.B.7 Step 2 Cost of the 100 texts
Each text costs 5 cents, and $5$ cents is $\$0.05$. For 100 texts, multiply: $100 \times \$0.05 = \$5.00$.
💡 Cost per item times number of items gives the total cost for that item.
4.MD.A.1 Step 3 How many extra minutes
- Only time past 30 hours is charged.
- She talked $30.5$ hours, so the extra is $30.5 - 30 = 0.5$ hour.
- Half an hour is $30$ minutes.
💡 Charges start at the cutoff, so subtract the free amount first, then switch to the unit that gets billed.
5.NBT.B.7 Step 4 Cost of the extra minutes
Each extra minute costs 10 cents, and $10$ cents is $\$0.10$. For 30 minutes, multiply: $30 \times \$0.10 = \$3.00$.
💡 Same idea as the texts: rate times amount gives the cost of that piece.
5.NBT.B.7 Step 5 Add the three parts
- Add the base, the texting cost, and the extra-minutes cost: $\$20 + \$5 + \$3 = \$28$.
- Michelle paid $\$28.00$, which is choice (D).
💡 Once every piece is priced in dollars, the total is just their sum.
4.OA.A.3 The total is the base fee plus the texting cost plus the extra-minutes cost. I w 5.NBT.B.7 Each text costs 5 cents, and $5$ cents is $\$0.05$. For 100 texts, multiply: $10 4.MD.A.1 Only time past 30 hours is charged. She talked $30.5$ hours, so the extra is $30 5.NBT.B.7 Each extra minute costs 10 cents, and $10$ cents is $\$0.10$. For 30 minutes, mu 5.NBT.B.7 Add the base, the texting cost, and the extra-minutes cost: $\$20 + \$5 + \$3 = Review
Reasonableness: The base alone is $20, and two small usage charges of $5 and $3 push the bill up by $8, landing at $28. That is a modest, sensible total for one month, and it matches choice (D). A common slip is charging all 30.5 hours or forgetting to turn 0.5 hour into 30 minutes; guarding against those keeps the extra-minutes cost at exactly $3.
Alternative: Work entirely in cents to avoid decimals: base $= 2000$ cents, texts $= 100 \times 5 = 500$ cents, minutes $= 30 \times 10 = 300$ cents. The sum $2000 + 500 + 300 = 2800$ cents $= \$28.00$, the same answer.
CCSS standards used (min grade 5)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Breaking the bill into a base part, a texting part, and an extra-minutes part, then combining them.)4.MD.A.1Know relative sizes of measurement units and convert larger to smaller units (Converting the 0.5 hour of overage into 30 minutes so it can be billed per minute.)5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths (Multiplying the cent rates in dollars and adding the three dollar amounts to the total.)
⭐ Price each part of the bill on its own, remember to bill only the time past the cutoff, then add everything up.
⭐ Price each part of the bill on its own, remember to bill only the time past the cutoff, then add everything up.
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