AMC 10 · 2006 · #4

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
percentagedecimal-arithmeticestimation identify-subproblems ↑ Prerequisites: percentage
📏 Short solution 💡 1 insight
Problem
Five items priced just under whole dollars are paid for with a twenty. Find which listed percentage is closest to the change.

Pick an answer.

(A)
5
(B)
10
(C)
15
(D)
20
(E)
25
How to solve
Strategy Identify Subproblems

The question stacks three separate jobs — total the prices, subtract to get the change, then express the change as a percent of $20 — so Tool #7 (Identify Subproblems) does them one at a time instead of in one messy expression. Tool #9 (Solve an Easier Related Problem) replaces each awkward price by the whole dollar just above it, turning five decimal additions into 8+5+3+2+1 with one small correction. Because the word "closest" is doing real work here, Tool #3 (Eliminate Possibilities) settles the comparison against the five listed percentages rather than assuming a rounded value lands on the right one, and Tool #11 (Work Backwards) re-derives the same choice from the other end by asking what total each listed percentage would require.

1STEP 1

Total the prices using whole dollars

Rounding up and correcting once makes the total easy.

7.99+4.99+2.99+1.99+0.99=(8+5+3+2+1)-5(0.01)=19-0.05=18.95
2STEP 2

Subtract to get the change

Subtracting gives the change as 1.05.

20.00-18.95=1.05
3STEP 3

Express the change as a percent

As a share of what was handed over that is 5.25 percent.

1.05/20=(1.05×5)/(20×5)=5.25/100=5.25%
4STEP 4

Test which choice is closest

It sits well below the midpoint to the next choice, so the nearest is clear.

|5.25-5|=0.25 ＜ 4.75=|5.25-10|, midpoint =(5+10)/2=7.5 ＞ 5.25
5STEP 5

Confirm from the other direction

Checking each choice as an amount confirms 5, choice (A).

5%→19.00,1019.00, 10%→18.00, 15%→17.00,2017.00, 20%→16.00, 25%→$15.00; |18.95-19.00|=0.05 → (A)
Answer
5
The spending eats almost the whole bill, so the change should be barely over a dollar, and one dollar out of twenty is exactly 5% — a hair above 5% is exactly what 5.25% is. The result is also sturdy: choice (A) beats choice (B) for any change under 1.50,thatis,foranytotalabove1.50, that is, for any total above18.50, and 18.95clearsthatlineby45cents.Soamisreadpenny,orevenamisreaddime,cannotchangetheanswer;whattheanswerreallyrestsonisthatthetotalisabove18.95 clears that line by 45 cents. So a misread penny, or even a misread dime, cannot change the answer; what the answer really rests on is that the total is above18.50, not on the exact 5-cent correction. Even the common slip of dividing by the amount spent instead of the bill, 1.05/18.95≈5.5%, still lands on (A).
💡Key takeaway

Prices ending in .99 are whole dollars minus pennies, and since one dollar is 5% of a twenty, a little over a dollar in change is a little over 5%.

  • Total the prices using whole dollars
  • Subtract to get the change
  • Express the change as a percent
  • Test which choice is closest
  • Confirm from the other direction

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