Grade 9 · Unit 7: Percent, rate and per-capita

Track: A/B · Tools: calculator, spreadsheet or pandas · Lessons: 3 Datasets: census-districts

Big idea. Fair comparison needs rates. Per-capita and per-1,000 measures remove the effect of size; percentage points and percent change are not the same.

What you will be able to do

• I can compute shares, rates, ratios and per-capita values, and choose the scale (per 100, 1,000, 100,000).
• I can choose and state the right denominator, and combine rates correctly (weighted, not averaged).
• I can compute percent change, percentage-point change and an annual growth rate.

Words to know: numerator (अंश), denominator (हर), share, rate (दर), ratio (अनुपात), per capita (प्रतिव्यक्ति), weighted average, percentage point, growth rate. Full list: curriculum/framework/glossary.md.

Notebook: notebooks/rates.ipynb (or use a spreadsheet; every calculation is shown both ways).

Before you start

Grade 8 Unit 6 introduced counts vs rates and percentage points. This unit goes further: denominators, ratios, weighted rates from groups, and growth over time — the arithmetic behind most news stories about Nepal's economy, population and schools.

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Lesson 1: Share, percent and per 1,000

Hook (5 min). Which tells you more: "4,000 people from our district live abroad" or "4 in every 1,000 people from our district live abroad"? What would you need to know to compare with another district?

Explanation

Every rate is a fraction: numerator ÷ denominator. * The numerator is what you count (people abroad, girls enrolled, road deaths). * The denominator is the group the count comes from (all residents, all students, all people).

Then choose a scale so the number is easy to read:

Scale	Multiply by	Good for	Example
share	1	proportions, between 0 and 1	0.068
percent	100	common things	6.8%
per 1,000	1,000	births, deaths, migrants	68 per 1,000
per 100,000	100,000	rare things: road deaths, diseases	12 per 100,000

Ratios compare two groups instead of a part with a whole: * Sex ratio = males per 100 females. Nepal's Census 2021 reported a sex ratio of about 95.6: fewer males than females, partly because many men work abroad. * Pupil–teacher ratio = students per teacher. * Dependency ratio = (people aged 0–14 + 65+) per 100 people aged 15–64.

Per capita means per person: GDP per capita, electricity used per capita, rice produced per capita. It divides a total by the population.

Worked example (synthetic). People abroad per 1,000 residents:

District	Population	Absentee	Per 1,000
Terai-09	1,607,726	111,228	69.2
Terai-03	878,299	47,834	54.5
Hill-25	421,197	47,947	113.8
Hill-04	313,431	24,176	77.1
Mountain-01	107,373	7,065	65.8
Mountain-04	84,648	5,315	62.8
Mountain-03	70,970	4,415	62.2
Mountain-05	26,942	1,072	39.8

Hill-25 has about the same number of people abroad as Terai-03 (≈ 48,000) but more than twice the rate: about one person in nine.

Spreadsheet: =1000*K2/E2. pandas:

import pandas as pd
df = pd.read_csv("districts_synthetic.csv")
df["abroad_per_1000"] = 1000 * df["absentee_population"] / df["population"]
df["sex_ratio"] = 100 * df["male"] / df["female"]
df[["district", "abroad_per_1000", "sex_ratio"]].round(1).head()

Activity (20 min)

1. For the 8 districts above, rank by count and by rate. Which district moves the most?
2. Compute the sex ratio for three districts. What would a sex ratio of 100 mean?
3. Make: choose one more rate from the table (e.g. households per km², people per secondary school) and say what the numerator and denominator are.

Common mistakes

• Choosing a scale that gives awkward numbers (0.0000124 instead of 1.24 per 100,000).
• Mixing up a ratio (males per 100 females) with a share (males per 100 people).
• Forgetting to say the scale ("the rate is 69").

Check (5 min). 1. 12 of 200 as a percent? 2. 12 of 200 per 1,000? 3. Why per 1,000 for rare things?

Reflect (2 min). What surprised me? Who or what is missing from this data?

Lesson 2: Choosing a denominator

Hook (5 min). "Girls' enrolment is 50%." Fifty percent of what? All students? All girls of school age? All girls in the district? Each gives a different number — and answers a different question.

Explanation

The denominator defines the question. Three "girls" indicators that sound alike:

Indicator	Numerator	Denominator	Question it answers
Girls' share of enrolment	girls enrolled	all students enrolled	Are classrooms balanced?
Girls' enrolment rate	girls enrolled	girls of school age	Are girls going to school?
Girls per school	girls enrolled	number of schools	How crowded are schools?

A district could have a 50% girls' share (balanced classrooms) while only 60% of girls of school age are enrolled (and maybe only 60% of boys, too). Our synthetic schools table lets us compute the first and third, but not the second: it has no "children of school age" column. Knowing what you cannot compute is part of the skill.

Combining rates: sum first, then divide (weighted). Suppose:

District	Population	Literacy
A	1,000,000	80%
B	50,000	50%

• Unweighted average of the two rates: (80 + 50) ÷ 2 = 65%.
• Weighted (true rate for all people): literate people = 800,000 + 25,000 = 825,000; total = 1,050,000 → 825,000 ÷ 1,050,000 ≈ 78.6%.

The unweighted average treats the small district as if it were as big as the large one. Unless you really want the "average district", sum the numerators and the denominators, then divide:

schools = pd.read_csv("schools_synthetic.csv")
m = df.merge(schools, on="district_id")
belt = m.groupby("ecological_belt")[["enrolment_girls", "enrolment_boys"]].sum()
belt["girls_share_pct"] = 100 * belt["enrolment_girls"] / (belt["enrolment_girls"] + belt["enrolment_boys"])
belt.round(1)

In a spreadsheet: =SUMIF(belt,"Hill",girls) / (SUMIF(belt,"Hill",girls) + SUMIF(belt,"Hill",boys)).

Denominators can mislead too. * Road deaths per vehicle fall when vehicles increase fast, even if deaths per person rise. * A hospital's death rate per patient is high if it treats the sickest patients. * "Per capita" uses the population living there — census, projection, or the absent population included? Check.

Activity (20 min)

1. For each indicator, write the numerator and the correct denominator: (a) pupil–teacher ratio; (b) share of households with electricity; (c) infant mortality rate (hint: per 1,000 live births); (d) GDP per capita.
2. Compute girls' share of enrolment for each belt the weighted way, and also as the mean of district shares. Are they different? Why only a little here?
3. Explain in two sentences why the synthetic table cannot give the girls' enrolment rate.

Common mistakes

• Averaging percentages across groups of different sizes.
• A denominator that does not contain the numerator (girls enrolled ÷ total population).
• Not stating the denominator in the chart title or caption.

Check (5 min). 1. Girls' share of enrolment divides by… 2. Why can two 'girls' percentages differ? 3. Always state the denominator in…

Reflect (2 min). What surprised me? Who or what is missing from this data?

Lesson 3: Percent change and percentage points

Hook (5 min). Girls' share of enrolment grew from 48% to 50%. Headline A: "up 2%". Headline B: "up 4.2%". Is the change big or small? Which headline is honest?

Explanation

Two kinds of change: * Absolute change = new − old. For percentages it is measured in percentage points: 50% − 48% = 2 percentage points. * Relative (percent) change = (new − old) ÷ old × 100 = 2 ÷ 48 × 100 ≈ 4.2%.

Headline A is confusing ("2%" of what?); Headline B is technically right but sounds bigger. The honest version gives both values: "Girls' share rose from 48% to 50% (2 percentage points)."

Growth over several years. Nepal's population was 26,494,504 in the 2011 census and 29,164,578 in 2021. * Total change: 29,164,578 − 26,494,504 = 2,670,074 people. * Percent change: 2,670,074 ÷ 26,494,504 ≈ 10.1% over the decade. * Is that 1.01% per year? Not quite: growth compounds (each year's growth builds on the previous year's total). The average annual growth rate is

(new ÷ old)^(1 ÷ years) − 1 = (1.1008)^(1/10) − 1 ≈ 0.96% per year.

NSO reports 0.92%, because the two censuses were about 10.4 years apart (June 2011 to November 2021) and NSO uses a slightly different (exponential) formula. Small details like the exact dates matter when you check official numbers.

old, new, years = 26_494_504, 29_164_578, 10
growth = (new / old) ** (1 / years) - 1
print(f"{100 * (new - old) / old:.1f}% over the decade, {100 * growth:.2f}% per year")

Index numbers compare many years with one base year set to 100: index = value ÷ base value × 100. Prices are usually shown this way: the consumer price index (CPI) published by Nepal Rastra Bank. An index of 125 means prices are 25% higher than in the base year.

Traps: * A 50% fall followed by a 50% rise ends 25% lower (100 → 50 → 75). * Percent changes on small bases look huge: 2 → 6 cases is "+200%". * Comparing "percent change" of two different-sized starting points: a district growing from 10,000 to 12,000 (+20%) added fewer people than one growing from 1,000,000 to 1,050,000 (+5%).

Activity (20 min)

1. For each pair compute percentage points and percent change, and write a fair headline: 48% → 50%; 80% → 84%; 60% → 45%; 5% → 10%; 92% → 96%.
2. A district grew from 400,000 to 460,000 people in 10 years. Compute the decade percent change and the average annual growth rate.
3. Rice cost Rs 80/kg in a base year (index 100). Two years later it is Rs 92. What is the index? What percent rise?

Common mistakes

• "Percent" when you mean "percentage points".
• Dividing a decade's percent change by 10 for an annual rate (close for small rates, wrong for large ones).
• Starting from the new value instead of the old one.

Check (5 min). 1. 48% → 50%: points? 2. Relative change? 3. Fair headline wording?

Reflect (2 min). What surprised me? Who or what is missing from this data?

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Unit summary

• Rate = numerator ÷ denominator × scale (100, 1,000, 100,000). Ratios compare two groups (sex ratio = males per 100 females). Per capita = per person.
• The denominator defines the question; say it in every title and caption; know what you cannot compute.
• Combine groups by summing numerators and denominators, not averaging rates.
• Percentage points (absolute) vs percent change (relative); give old and new values.
• Annual growth = (new ÷ old)^(1/years) − 1; index = value ÷ base × 100.

Practice exercises

1. ★ 35 of 1,400 students failed. Give it as a share, a percent and per 1,000.
2. ★ A school has 840 students and 28 teachers. Pupil–teacher ratio?
3. ★ A village has 1,180 males and 1,250 females. Sex ratio?
4. ★★ Three classes have pass rates 90% (of 20), 70% (of 40), 50% (of 10). What is the school's overall pass rate? Compare with the simple average of the three rates.
5. ★★ For each, name the numerator and denominator: (a) literacy rate; (b) remittances as % of GDP; (c) hospital beds per 10,000 people; (d) students per computer.
6. ★★ Literacy rose from 65.9% (2011) to 76.2% (2021) in Nepal (census, population aged 5+). Give the change in percentage points and as percent change. Write an honest one-sentence headline.
7. ★★ A district's population went from 250,000 to 300,000 in 10 years. Find the percent change and the average annual growth rate.
8. ★★★ "Road deaths per 10,000 vehicles fell by half in ten years, so roads are safer." Explain why this might be false.
9. ★★★ Using the synthetic table, compute the sex ratio by belt (sum males ÷ sum females × 100). Which belt has the lowest? Suggest a reason that would apply to real data.
10. ★★★ Prices rose 10% one year and 10% the next. Did they rise 20% in total? Compute exactly.

Exercises with real data

R1. Remittances (RD-01). From the World Bank, download "Personal remittances, received (% of GDP)" (BX.TRF.PWKR.DT.GD.ZS) for Nepal and three neighbours. Which country depends most on remittances? For Nepal, find the change in percentage points between 2000 and the latest year. Why is "% of GDP" a better comparison than the amount in US dollars?

R2. Census growth by province (RD-13). From the census results portal, get 2011 and 2021 population for the 7 provinces. Compute decade percent change and average annual growth rate for each. Which province grew fastest? Which might have lost people? Rank by growth rate, not by count. Offline fallback: invent a 2011 column for 5 synthetic districts by multiplying by 0.9, then compute growth.

R3. Prices and an index (RD-08). Choose one vegetable in the Kalimati data. Take its mean price in 2014 as index 100 and compute an index for each year to 2020. Which year was most expensive? Compare with a second vegetable.

Unit project: Rate detective

Follow Problem · Plan · Data · Analysis · Conclusion. Find a news story about Nepal that uses a number (population, migration, prices, enrolment, health). Find the source data, recompute the number with the right numerator, denominator and scale, check percent vs percentage points, and write a "rate detective" report: what the article got right, what it got wrong or left out, and a corrected headline. Full brief and rubric: project.md.

If you use an AI assistant, follow the AI-use policy (explain, don't answer; disclose; be ready for an oral check).

Going further

• Our World in Data, "Per capita" and "growth rate" explainers; many charts let you switch between total and per capita — try it.
• Nepal Rastra Bank's "Current Macroeconomic and Financial Situation" reports (https://www.nrb.org.np): practise reading percent changes in real tables.