Competition · AMC preparation · step 4 of 4

AMC 8 · 2010 · #3

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

percentagegraph-readingfraction-arithmetic identify-subproblems ↑ Prerequisites: percentagemulti-digit-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A bar graph shows the price of 5 gallons of gasoline for each of the first 10 months of the year. Reading the bars, the highest price is $17 (Month 1) and the lowest price is $10 (Month 3). By what percent does the highest price exceed the lowest price?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

100

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The problem has a simple shape — read two numbers off the graph, then compute a percent increase — so the work splits cleanly into Tool #7 subproblems: (1) extract the maximum and minimum from the bar chart, (2) compute the percent increase using (high - low) / low × 100%. Tool #5 (Look for a Pattern) is what makes the second step quick: $7 on a base of$ 10 is the familiar pattern 7/10 = 0.70 = 70%, so no long division is needed.

1STEP 1

Read the highest and lowest

Scan all ten bars: Month 1 is the tallest at the $17 level, Month 3 the shortest at the $10 gridline.

highest =

17, lowest =

2STEP 2

Subtract the two prices

The gap between the two prices is $7 — how much the high price clears the low.

17 -

10 = $7

3STEP 3

Set up the percent ratio

"Exceeds by what percent" means divide the gap by the lowest price — the base — then scale to a percent.

percent more = (

17 -

10)/

10 × 100% =

7/$10 × 100%

Choosing the lowest price as the base of the percent — not the highest — is the key Grade 6 ratio-reasoning move.

By what percent the highest price is more than the lowest is found by taking the extra dollars the highest price has above the lowest, dividing that extra by the lowest price, and reading the result as a count per hundred.

▸ Why?

The words "more than the lowest" point to the extra amount the highest price carries above the lowest, so we first need that extra amount.

▸ Why?

The highest price is the lowest price with some extra added on top, so the extra is exactly the highest price minus the lowest price.

▸ Why?

The highest price splits cleanly into the lowest-price part and the leftover part, with nothing missing and nothing counted twice, so those two parts add back to the highest price.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

To pull the extra back out of that sum, we undo the adding by subtracting the lowest price from the highest price.

Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.

▸ Why?

Calling that extra "some percent of the lowest" means the extra equals that percent, read as parts of a hundred, times the lowest price — so to name the percent we must know what "parts of a hundred" means and then undo that multiplying.

▸ Why?

Saying the extra is p percent of the lowest means p out of every hundred equal parts of the lowest price, which is the same as (p/100) times the lowest price.

A percent is parts per hundredp% means p out of every hundred, so p% of a quantity is (p/100) times it.

▸ Why?

Since the extra equals the percent-fraction multiplied by the lowest price, we recover the percent-fraction by dividing the extra by the lowest price, because dividing undoes that multiplying.

Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.

6.RP.A.3Look For A Pattern

4STEP 4

Compute the percent

The fraction $\frac{7}{10}$ is exactly 0.7, so the high price is 70% more than the low.

7/10 × 100% = 70% → (C)

Answer

If the high price were double the low price, the increase would be 100%; doubling $10 would mean a high of $20. Since $17 is below $20 but well above $10, the answer must be less than 100% but more than 50%. That rules out (A) and (E) and leaves (B) 62, (C) 70, (D) 89. Our calculation

\frac{7}{10}

= 70% lands on (C), which sits in the expected window.

💡Key takeaway

This AMC 8 problem only needs Grade 6 percent reasoning — divide the gap by the smaller number — that you already know!

Read the highest and lowest
Subtract the two prices
Set up the percent ratio
Compute the percent

A parent dashboard for the family lives at sensimlab.com.

Read the highest and lowest

Subtract the two prices

Set up the percent ratio

Compute the percent

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