An assortment of some of the trickier kinds of problems that students might see on the SHSAT. Things like… modulo division, mean & median, fractionally-filled jars, number theory, probabilities, and unit conversion – followed by a deeper exploration of word problems involving averages (aka: mean).

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Your ideas and suggestions are important to me! For example, maybe…

• It didn’t work on my device! (😢please give me actionable details)

• Would you like to see more questions? Less? Harder? You tell me 😄

• Do you want an assortment of categories within each sampler – not just one?

• How about the difficulty? Too easy, too hard, just right?

• Anything else?

This week’s theme: 🧧 Chinatown

My very first apartment in New York City was on Catherine Street, around the corner from Confucius Plaza. I absolutely loved this apartment, the neighborhood, and the people. Many New Yorkers think of Chinatown as the last authentic ethnic enclave in a City that’s becoming more and more gentrified, whitewashed, and replaced with a tourist-friendly Epcot center version of itself. I’ll just say that I adore the neighborhood, and always will – despite the rising cost of parking and dumplings. Below are a few interesting links:

• Washington Post Article about Bud Glick’s famous photo essay of Chinatown in the 80’s

• Guardian write-up of a family friend Freeman Wong – who appears in Bud Glick’s photo essay, along with his brother and dad

Deep dive: The Average of x is blah …

When my daughter would see the following kinds of questions, she would immediately get suspicious. Who wouldn’t? It has all the hallmarks of an SHSAT boobytrap, sometimes jam-packed with factual misdirection. Below are a few of my favorite variants:

• Rosa will take a total of 4 tests. She has already taken 2 of the tests and earned scores of 81 and 83. What is the least possible score Rosa can earn on the third test and still be able to finish the class with an average score of 85 on all 4 tests? (Assume that test scores can range from 0 to 100.)

• 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?

• The mean value of 8 numbers is 17. Three of these numbers (9, 11, and 20) are discarded. What is the mean of the 5 remaining numbers?

• The average (arithmetic mean) of 10 numbers is 5. The average of these 10 numbers and an eleventh number is 6. What is the eleventh number?

View the solutions

Tip: Demystify averages by treating them as sums of values (and ignore individual values)

Consider: I have 3 watermelons 🍉. Their average weight is 10 lbs. Multiple scenarios are possible, but for the sake of solving word problems, it is almost always more useful to think of the watermelons as a single group of things that weighs 30 lbs. The same is true of average game scores, average test scores, average heights of people, etc.

• Scenario 1

  • Watermelon 1 = 10 lbs

  • Watermelon 2 = 10 lbs

  • Watermelon 3 = 10 lbs

  • Total weight = 30 lbs

• Scenario 2

  • Watermelon 1 = 12 lbs

  • Watermelon 2 = 12 lbs

  • Watermelon 3 = 6 lbs

  • Total weight = 30 lbs

• Scenario 3

  • Watermelon 1 = 2 lbs

  • Watermelon 2 = 2 lbs

  • Watermelon 3 = 26 lbs

  • Total weight = 30 lbs

The key insight to Word Problems with Averages

The total value of an averaged-out group is the key fact that lets you infer other facts. Often times, if you don’t see this, you will waste your time doing other math that is not related to the task at hand.

Solutions to the examples from above

Rosa’s test average

• Rosa will take a total of 4 tests. She has already taken 2 of the tests and earned scores of 81 and 83. What is the least possible score Rosa can earn on the third test and still be able to finish the class with an average score of 85 on all 4 tests? (Assume that test scores can range from 0 to 100.)

  • An average score of 85 across 4 tests is the same as saying… the total value for all 4 tests MUST EQUAL 85 x 4

  • Rosa’s first 2 tests were 81 and 83 => giving a sum of 164

  • We don’t know what her 3rd test is, but we want to identify the lowest score she can get

  • Because of ☝🏽 we can safely assume it is OK to give Rosa the maximum score on her 4th test: 100

  • We can now write out an equation like so…

    • 81 + 83 + 100 + x = (4 * 85)

    • 264 + x = 340

    • x = 340 - 264

    • x = 76

    • The lowest score Rosa can get on her 3rd test = 76. It is impossible for her to go lower than this because it is impossible for her fourth test score to be above 100. Her 1st and 2nd test scores are already known.

Anita & tariq’s game scores

• 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?

  • First: remember that mean = average

  • How can we find out the mean score of 5 games if we don’t have 5 individual scores?

  • Easy… it doesn’t matter what the individual scores were. We know Anita’s 3-game average and Tariq’s 2-game average

  • Anita’s 3-game total = Anita’s 3-game average x 3… (140 x 3)

  • Tariq’s 2-game total = Tariq’s 2-game average x 2… (90 x 2)

  • The sum of 5 games = Anita’s 3-game total + Tariq’s 2-game total

  • We can now write out an equation like so…

    • Let x = the mean score of all 5 games

    • 5x = (140 x 3) + (90 x 2)

    • 5x = 420 + 180

    • 5x = 600

    • x = 120

mean value of 8 Numbers

• The mean value of 8 numbers is 17. Three of these numbers (9, 11, and 20) are discarded. What is the mean of the 5 remaining numbers?

  • The mean value (average) of 8 numbers is 17

  • The total value = 8 x 17 = 136

  • 3 of these numbers are: {9, 11, 20}

  • The sum of the 3 numbers = 9+11+20 = 40

  • We can now write out an equation like so…

    • Let x = the mean value of the 5 remaining numbers

    • x = ((the total value of all 8 numbers) - (the total value of 3 known numbers)) ÷ 5

    • x = (136 - 40) ÷ 5

    • x = 19.2

What is the 11th Number?

• The average (arithmetic mean) of 10 numbers is 5. The average of these 10 numbers and an eleventh number is 6. What is the eleventh number?

  • The average of 10 numbers = 5

  • The total value = 10 x 5 = 50

  • There is an eleventh number which, when added to the 10, brings the average up from 5 to 6

  • We can now write out an equation like so…

    • Let x = the eleventh number

    • (x + (the total value of 10 numbers)) ÷ 11 = 6

    • (x + 50) ÷ 11 = 6

    • x + 50 = 6 x 11

    • x + 50 = 66

    • x = 66 - 50

    • x = 16