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Anita played 3 games

2019 Handbook

2019 Test A Math · Question 48 of 57

Question

Anita played 3 games and had a mean score of 140. Tariq played 2 games and had a mean score of 90. What was the mean of the scores for all 5 of these games?

  • A100
  • B115
  • C120
  • D230
Show the answer

Correct answer: C

The sum of the scores of Anita’s 3 games is 3(140)=4203(140) = 420. The sum of the scores of Tariq’s 2 games is 2(90)=1802(90) = 180. Add those two sums and divide by 5 to find the mean:

420+1805=6005=120\frac{420 + 180}{5} = \frac{600}{5} = 120

Question analysis

Anita averaged 140 points over 3 games and Tariq averaged 90 over 2 games. This mean, or average, question asks for the mean score of all 5 games combined. The two means stand for different numbers of games, so students have to think about the totals behind them.

NYS standards
  • 6.SP.5c range, center, and overall pattern
  • 6.SP.3 measures of center and variation
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
At grade 7 Means are grade 6 content. Combining two groups of different sizes requires turning each mean back into a total, a step that grade 7 test items often demand and many students skip.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
3 steps
Reading level
Question and choices: about grade 1, 30 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The quick move, averaging 140 and 90, gives a listed choice, so the question rewards students who stop to ask what each mean stands for. Anita's three games carry more weight than Tariq's two, which pushes the combined mean above the 115 midpoint toward 140. Turning both means into totals and dividing by all 5 games is only one step longer than the shortcut.

  1. Turning words into math moderate Understanding the question

    Each mean stands for a total that must be rebuilt first: 140 over 3 games is 420 points and 90 over 2 games is 180, before the 600 is shared among 5 games. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  2. Something you must already know minor Working it out

    The rule that a mean times its count gives the total is never stated, and it is what shows that two means over different numbers of games cannot simply be averaged. (Anderson & Krathwohl, 2001)

  3. A very close wrong answer minor Choosing the answer

    115 and 120 both lie between the two means, so a reasonableness check passes both; only the weighting, 3 games at 140 against 2 at 90, lifts the true mean above the midpoint. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Multiply each player's mean by the games that player played to get two point totals, add them, and divide by all 5 games.

Traps

  • Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
  • Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Mean (average)
mean=sum of the scoresnumber of scores\text{mean} = \frac{\text{sum of the scores}}{\text{number of scores}}
The mean, or average, shares a total evenly. The sum of the scores is all the scores added up. The number of scores is how many games there are. Dividing tells what each game would score if all the games scored the same.
Total from a mean
sum of the scores=mean×number of scores\text{sum of the scores} = \text{mean} \times \text{number of scores}
This is the mean formula turned around. If you know the mean and how many scores there are, multiply them to get the total. It works because the mean is the same share given to every game.

  1. Sort out the question

    Anita played 3 games, and her mean score was 140. Tariq played 2 games, and his mean score was 90. The mean is the average: add the scores and divide by how many there are. You need the mean of all 5 games together. You don’t know any single score, but you can find each player’s total.

  2. Find Anita’s total

    Use the mean formula turned around. Anita’s mean is 140, over 3 games.

    sum of the scores=mean×number of scores\text{sum of the scores} = \text{mean} \times \text{number of scores}

    sum of the scores=140×3=420\text{sum of the scores} = 140 \times 3 = 420

    Anita scored 420 points in her 3 games.

  3. Find Tariq’s total

    Do the same for Tariq. His mean is 90, over 2 games.

    sum of the scores=90×2=180\text{sum of the scores} = 90 \times 2 = 180

    Tariq scored 180 points in his 2 games.

  4. Find the mean of all 5 games

    Add the two totals. Then divide by all 5 games.

    mean=sum of the scoresnumber of scores\text{mean} = \frac{\text{sum of the scores}}{\text{number of scores}}

    mean=420+1805=6005=120\text{mean} = \frac{420 + 180}{5} = \frac{600}{5} = 120

    The mean of all 5 games is 120 points.

    Watch out: don’t just average the two means. (140+90)÷2=115(140 + 90) \div 2 = 115, choice B, counts Tariq’s 2 games as much as Anita’s 3. Adding the means, 140+90=230140 + 90 = 230, forgets to divide. That is choice D.

    So the answer is C, 120.

Check your answer

The mean must be between 90 and 140. It should be closer to 140, because Anita played more games. 120 is. Also, Anita’s 3 games average 20 points above 120, which is 60 points in all. Tariq’s 2 games average 30 points below 120, also 60 in all. The two balance, so 120 is the middle.

More questions like this

The original DOE question

Question 105 as printed on page 105 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2019–2020, page 105, question 105. Open page 105 of the handbook (PDF)