Quiz 10 · Numbers in Social Studies — Ancient America to the Singularity

Who made this?

1. Who made it?: The drawing is by Felipe Guaman Poma de Ayala, an Andean nobleman and Quechua speaker, whose book you met in Quiz 6 and in Interlude I. The man he drew is named in the caption: Condor Chaua, chief accountant and treasurer, “the one who keeps the quipu of Tawantinsuyu.” The board at his feet, a yupana in Quechua (yupay means “to count”), was made and used by Andean accountants; who first made one, and when, is not recorded. Boards of stone and clay that match its shape have been dug up in Ecuador and Peru since 1869.
2. When, and where?: The drawing was finished in Peru in 1615, about eighty years after the Spanish conquest; it shows an office of the Inca state as it worked before 1532. Guaman Poma drew the quipu without knots and the board without counters: the tools, not a count.
3. Why, and for whom?: For the king of Spain. The book is a long letter to Philip III on how the Inca state had run and how the colony was failing, with a drawing on nearly every other page so that a busy king would still see. This page sits in the chapter on Inca administration: the accountants counted the people of the state in tens, hundreds, thousands, and ten thousands, and the counts decided who owed labor and how much.
4. Who has it now, and how did it get there?: The Royal Danish Library in Copenhagen, where it has been since at least 1785; how it got there is not recorded, and Interlude I tells what is known. The page you see is from the facsimile printed in Paris in 1936, which traced the drawings cleanly; the library’s own photographs of the manuscript show the ink bleeding through from the other side.
5. Whose voice is missing here, and where can we hear it?: The accountants’. Guaman Poma’s one sentence on the board says they counted “by tables, from a hundred thousand down to one”; he does not say how the board was used, and no one else who saw one wrote it down. For more than a hundred years scholars have proposed ways the rows and dots might have worked, and none is agreed. The quipu’s numbers are better understood: in 1912 Leland Locke showed that the knots record numbers by tens, and in the 1970s the Aschers showed that many quipus carry a cord that holds the total of the others. What the numbers counted, the cords rarely say.

For your notebook: He drew an empty board and a quipu with no knots: the tools, not the count. What count of your own life would you want kept, and who should hold the cord?

Sources: Royal Danish Library, digital edition of the Nueva corónica (kb.dk), pages 360 and 361; Nueva corónica y buen gobierno, facsimile edition (Paris: Institut d’Ethnologie, 1936); Locke, The Ancient Quipu (1923); Ascher and Ascher, Code of the Quipu (1981); on the yupana finds and the unsettled readings, Chris Staecker, “The Unconquered Mathematics of the Inca” (Fairfield University, 2025).

First, the idea

Social studies runs on counts. A count is how many: 1,459,648 people. A count is the kind of evidence Quiz 9 asked for, something a stranger could check; but a count by itself is hard to compare, because places and groups are different sizes. So the test also asks for four ways of putting counts side by side.

A percent is a share out of a hundred: 28.2 percent means 28.2 of every 100. Always ask percent of what, the question Quiz 5 taught; 28 percent of Chiapas and 28 percent of the country are different numbers of people. A rate is a share out of some other round number, usually because the share is small: 3 births per 1,000 people, 12 speakers of no Spanish per 100 speakers. Per capita means per person: a city’s total divided by its people, so a big city and a small one can be compared. And when a question asks what is typical, the test has three answers: the mean, which adds everything and divides by how many; the median, the middle value once the numbers are put in order; and the mode, the value that appears most often.

Look at one row of a hundred seats. A stadium in Chiapas holds more people than one in Yucatán, so counting green shirts across the whole crowd tells you which crowd is bigger, not which crowd is greener. A percent walks you to any one row of a hundred seats and counts the green shirts there. In that row the two stadiums can be compared, whatever their size. That is all a percent is: the count in one row of a hundred.

Two habits for the test. Before you compute, decide which question is being asked: how many (a count) or how common (a percent, a rate, per capita). And when one value is far from the others, prefer the median to the mean, because a single large number drags the mean toward itself and away from what is typical. The on-screen calculator is there for the arithmetic; the judgment about which number to compute is yours.

Worked example — thinking out loud

The Inca state kept its counts on cords. An accountant like the one in the masthead tied knots in a hanging cord, and the height of a knot on the cord told its place: knots near the top were hundreds, knots in the middle were tens, and a long knot at the bottom, with as many turns as the number, was the ones. I have three cords to read.

Three cords, drawn for this example after the way Leland Locke read Inca knots in 1912: the knots nearest the main cord are hundreds, the middle group is tens, and the bottom knot is a long knot whose turns count the ones. On many real quipus the cord that holds a total is tied so that it hangs the other way; here all three hang down so you can read them alike.

Drawn for this example. The reading rule follows L. Leland Locke, The Ancient Quipu (1923), and Marcia and Robert Ascher, Code of the Quipu (1981).

Cord 1: one knot in the hundreds, three in the tens, a long knot with two turns. That is 132. Cord 2: four, one, and seven turns: 417. Cord 3: five, four, nine: 549. My first pull is to stop there, three numbers read, done. But 132 plus 417 is 549. Cord 3 is not a third count; it is the total of the first two, and on a real quipu it would often be tied to hang the other way so the reader would know. The cord tells you what it counts if you look at how it is tied.

Now the social studies question, which is never just “read the knots.” Suppose cord 1 counted households in one village and cord 2 in the next, with cord 3 the district total. What share of the district’s households were in the second village? That is a percent of what: 417 out of 549. On the calculator, 417 ÷ 549 = 0.7595, which is about 76 percent. The first village is the other 24 percent. And if the question were how many households per village on average, that is a mean: 549 ÷ 2, about 275, a number that describes neither village, which is the mean’s way. Read the count, then ask which comparison the question wants.

Part A · The skill by itself

1. A community board in Queens has 50 members. Twelve of them are under 40. What percent of the board is under 40?

12 percent 24 percent 38 percent 50 percent

2. Two library branches. Branch A had 8,000 visits last month and serves a neighborhood of 20,000 people. Branch B had 12,000 visits and serves 60,000 people. Which branch had more visits per person?

Branch B, because 12,000 is more than 8,000 Branch A: 8,000 ÷ 20,000 is 0.4 visits per person, against 0.2 for Branch B They are equal It cannot be told without knowing the branches’ sizes

3. Five apartments on one Bronx block rent for $1,600, $1,700, $1,700, $1,900, and $4,100 a month. The mean is $2,200 and the median is $1,700. Why is the median the better description of a typical rent on this block?

Because the median is always better Because one rent of $4,100 pulls the mean up toward itself; four of the five apartments rent for less than the mean Because the mean was computed wrong Because $1,700 appears twice, which makes it the mean

Part B · The skill in the wild

A short passage and one table, from Mexico’s 2020 census. Read the table’s title and its two number columns before the questions: the first column counts people, the second gives a share. The on-screen calculator is fair game.

Counting speakers · Mexico, 2020

1Every ten years Mexico counts its people, and the 2020 count asked everyone aged three and older which language they speak. 2The count found 7,364,645 people who speak one of the country’s 68 Indigenous languages, which is 6.1 percent of everyone aged three and older. 3Of every 100 of those speakers, 12 speak no Spanish. 4Four states hold about half of all the speakers in the country, and the table shows them.

People aged 3 and older who speak an Indigenous language, 2020
State	Speakers	Share of the state’s people aged 3+
Chiapas	1,459,648	28.2%
Oaxaca	1,221,555	31.2%
Yucatán	525,092	23.7%
Guerrero	515,487	15.5%
All of Mexico	7,364,645	6.1%

Passage written for this quiz. Numbers: INEGI, Censo de Población y Vivienda 2020 (Mexico’s census office), basic tables and the press release of August 8, 2022. Quiz 5 used the same census, by language; this table is by state.

4. According to the table, which state has the most speakers, and which has the highest share?

Chiapas has both the most speakers and the highest share Oaxaca has both the most speakers and the highest share Chiapas has the most speakers; Oaxaca has the highest share Yucatán has the most speakers; Guerrero has the highest share

5. About what percent of all the Indigenous-language speakers in Mexico live in Chiapas?

About 14 percent About 20 percent About 28 percent About 50 percent

6. Written as a rate per 1,000 people, Yucatán’s share is about

24 speakers per 1,000 people 237 speakers per 1,000 people 2,370 speakers per 1,000 people 525 speakers per 1,000 people

7. Guerrero’s 515,487 speakers are 15.5 percent of its people aged three and older. About how many people aged three and older live in Guerrero?

About 80,000 About 800,000 About 3.3 million About 33 million

8. Complete the sentence so that it matches the table and the lesson.

Chiapas has more speakers than Oaxaca but a lower share, which tells you that Chiapas [has more people in total / counted its people wrong / has fewer people in total]; to compare how common these languages are in the two states, use the [count of speakers / share / mean of the two numbers].

The key — and why the tempting answers tempt

1. B. 12 out of 50. Fifty is half of a hundred, so double both: 24 out of 100, 24 percent. On the calculator, 12 ÷ 50 = 0.24. A is the count offered as the percent, the most common trap on the whole test: 12 is how many, not how many per hundred. C is 12 + 24 + 2, the number a tired guesser reaches for; D is half.

2. B. Per person means divide by the people: 8,000 ÷ 20,000 = 0.4, and 12,000 ÷ 60,000 = 0.2. Branch A is used twice as heavily for its size. A answers “how many” when the question asked “how common,” the count in place of the rate. D is the hedge: the sizes are in the question, in people, which is the only size that matters here.

3. B. Put the five rents in order and the middle one is $1,700. The mean adds the $4,100 in and shares it out across all five, so it lands above four of the five rents, and describes none of them. A is a rule where the lesson gave a reason; sometimes the mean is the right number (a landlord’s total income, for one). D confuses the mode, the value that appears most, with the mean; here the mode and the median happen to agree, which they often do not.

4. C. Read down each column separately. Speakers: Chiapas, 1,459,648, is the largest count. Share: Oaxaca, 31.2 percent, is the largest share. A and B each assume the two columns must agree. They need not, and the table is on the page to show you that they do not: the biggest count and the biggest share can sit in different rows.

5. B. Percent of what: Chiapas’s speakers out of Mexico’s speakers, 1,459,648 ÷ 7,364,645 = 0.198, about 20 percent. C is the trap: 28.2 percent is on the page, but it is Chiapas’s speakers out of Chiapas’s people, a different “of what.” D is what the passage says about all four states together, not this one.

6. B. 23.7 per hundred is 237 per thousand: multiply by ten when the base goes from 100 to 1,000. A moves the decimal the wrong way; C moves it too far. D is the count of speakers in thousands, which is not a rate at all.

7. C. If 515,487 is 15.5 percent of the whole, the whole is 515,487 ÷ 0.155, which the calculator gives as about 3,326,000. Check it the other way: 15.5 percent of 3.3 million is about 512,000, close enough. B multiplies 515,487 by 0.155 instead of dividing, and gets a number smaller than the speakers, which cannot be: the whole must be bigger than the part. When the answer is smaller than what you started with, you went the wrong way.

8. has more people in total; share. More speakers from a smaller share can only mean a bigger crowd to take the share from. And “how common” is the share’s question; the count answers “how many.” “Counted its people wrong” invents a fault to explain a difference that is no fault; and the mean of a count and a percent is a number with no meaning at all.

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