AMC 10 · 2010 · #24
Grade 8 algebraA high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?
Pick an answer.
AMC 10 2010 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: Two teams tie in the first quarter of a game. Over the four quarters the Raiders' points form an increasing geometric sequence and the Wildcats' points form an increasing arithmetic sequence. The Raiders' four-quarter total is exactly one more than the Wildcats', and neither team's total is above 100. Find the sum of all points both teams scored in the first half (quarters one and two combined).
Givens: The two teams scored the same number of points in the first quarter (the game was tied then).; Raiders' four quarter scores form an increasing geometric sequence: $a,\ ar,\ ar^2,\ ar^3$ with ratio $r>1$.; Wildcats' four quarter scores form an increasing arithmetic sequence: $a,\ a+d,\ a+2d,\ a+3d$ with $d>0$.; Raiders' total is exactly one more than Wildcats' total.; Each team's four-quarter total is at most 100, and every quarter score is a whole number.
Unknowns: The first-quarter score $a$, the geometric ratio $r$, and the arithmetic difference $d$.; The combined first-half points of both teams.
Understand
Restated: Two teams tie in the first quarter of a game. Over the four quarters the Raiders' points form an increasing geometric sequence and the Wildcats' points form an increasing arithmetic sequence. The Raiders' four-quarter total is exactly one more than the Wildcats', and neither team's total is above 100. Find the sum of all points both teams scored in the first half (quarters one and two combined).
Givens: The two teams scored the same number of points in the first quarter (the game was tied then).; Raiders' four quarter scores form an increasing geometric sequence: $a,\ ar,\ ar^2,\ ar^3$ with ratio $r>1$.; Wildcats' four quarter scores form an increasing arithmetic sequence: $a,\ a+d,\ a+2d,\ a+3d$ with $d>0$.; Raiders' total is exactly one more than Wildcats' total.; Each team's four-quarter total is at most 100, and every quarter score is a whole number.
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #6 Guess and Check, #3 Eliminate Possibilities
The problem hides three unknowns inside two number patterns. Naming the shared first-quarter score $a$, the geometric ratio $r$, and the arithmetic step $d$ turns the story into expressions we can compare. The single equation 'won by one point' then links them, and a short bounded search finds the one whole-number solution.
Execute — Answer: E
5.OA.B.3 Step 1 Model both teams as sequences
- Because the game was tied after the first quarter, both teams scored the same amount then; call it $a$.
- The Raiders' scores multiply by a fixed ratio each quarter (geometric), and the Wildcats' scores add a fixed amount each quarter (arithmetic).
💡 Each team follows one simple rule every quarter, so a single letter for the rule captures all four numbers.
6.EE.A.2 Step 2 Write each team's total
- Add the four quarters for each team to get a total in terms of the letters.
- The Raiders' total factors nicely, and the Wildcats' total collects the steps.
💡 A total is just the four terms summed, and writing it with letters lets us compare the two teams directly.
6.RP.A.3 Step 3 Pin down the ratio
- The ratio must be more than 1 (increasing) and must keep all four scores whole and the total at or below 100.
- If $r=3$ the Raiders would score $a,3a,9a,27a$ summing to $40a$, and matching the Wildcats to within one point leaves no whole-number step $d$.
- A fractional ratio like $\tfrac{3}{2}$ forces $a$ to be a multiple of 8, which overshoots 100 before it fits.
- The ratio $r=2$ is the one that works.
💡 Doubling each quarter is the gentlest whole-number growth that still beats an arithmetic sequence, keeping totals under the cap.
7.EE.B.4 Step 4 Use 'won by one point'
- With $r=2$ the Raiders total is $15a$ and the Wildcats total is $4a+6d$.
- The Raiders finished exactly one point ahead, so set Raiders total equal to Wildcats total plus one.
💡 The one-point margin is a fact we can write as an equation, tying the ratio side to the step side.
8.EE.C.7 Step 5 Search for the whole-number fit
- We need $d = \dfrac{11a-1}{6}$ to be a whole number, with $a$ from 1 to 6 (from the cap $15a\le 100$).
- Testing each value, $11a-1$ is a multiple of 6 only when $a=5$: then $11(5)-1 = 54 = 6d$, so $d=9$.
💡 Only one first-quarter score in the allowed range makes the step come out to a whole number.
6.EE.B.5 Step 6 Check and total the first half
- The Raiders score $5,10,20,40$ (total $75$) and the Wildcats score $5,14,23,32$ (total $74$).
- Both increase, both stay under 100, and $75 = 74+1$, so every condition holds.
- The first half is quarters one and two for both teams: $(5+10)$ for the Raiders and $(5+14)$ for the Wildcats.
- The combined first-half total is $15 + 19 = 34$, which is answer (E).
💡 Once the one valid scenario is found, the first-half sum is a direct addition of the first two quarters.
5.OA.B.3 Because the game was tied after the first quarter, both teams scored the same am 6.EE.A.2 Add the four quarters for each team to get a total in terms of the letters. The 6.RP.A.3 The ratio must be more than 1 (increasing) and must keep all four scores whole a 7.EE.B.4 With $r=2$ the Raiders total is $15a$ and the Wildcats total is $4a+6d$. The Rai 8.EE.C.7 We need $d = \dfrac{11a-1}{6}$ to be a whole number, with $a$ from 1 to 6 (from 6.EE.B.5 The Raiders score $5,10,20,40$ (total $75$) and the Wildcats score $5,14,23,32$ Review
Reasonableness: The final scores $5,10,20,40$ and $5,14,23,32$ satisfy every stated fact: the first quarters match (tie), the Raiders double each quarter (increasing geometric), the Wildcats add 9 each quarter (increasing arithmetic), the totals $75$ and $74$ differ by exactly one with the Raiders ahead, and both stay under 100. The half-total $34$ is one of the offered choices, confirming consistency.
Alternative: Instead of fixing the ratio first, you could write the win-by-one condition as $a(1+r+r^2+r^3) = 4a+6d+1$ and reason that a geometric sequence outgrows an arithmetic one so fast that any ratio of 3 or more blows past 100; that forces $r=2$ and the same bounded search on $a$, reaching the identical scores.
CCSS standards used (min grade 8)
5.OA.B.3Generate two numerical patterns using two given rules and identify relationships (Modeling each team's four quarter scores as a geometric and an arithmetic sequence.)6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Writing each team's four-quarter total as an expression in $a$, $r$, and $d$.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Arguing the geometric ratio must be 2 to keep the scores whole and under 100.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Turning the one-point winning margin into the equation $11a-1=6d$.)8.EE.C.7Solve linear equations in one variable (Solving $11a-1=6d$ for whole-number values of $a$ and $d$.)6.EE.B.5Understand solving an equation or inequality as a process of finding values (Testing candidate first-quarter scores to find the one giving a whole-number step within the cap.)
⭐ Name the hidden numbers with letters, turn each clue into one equation, then check the few values that fit.
⭐ Name the hidden numbers with letters, turn each clue into one equation, then check the few values that fit.
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