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Practice sheet · Government Path · Lens 1: Math

Votes & budgets

Practice for The Math of Government: voting systems, seat counts, budgets, per capita figures, taxes, polls and boundaries (pp.8–16), plus Algebra A1–A2 (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

Majority = more than half; plurality = the most votes. Per capita = total ÷ people. Marginal rate = tax on the next dollar; average rate = total tax ÷ income.

Margin of error (95%, random sample) ≈ 1.96 × √(0.25 ÷ n) × 100 = 98 ÷ √n points (p.15). Property tax: 1 mill = $1 per $1,000 of assessed value. Every place, party and number here is made up.

Warm-up

One step

Votes, shares and taxes.
1

Plurality

In a made-up riding, Fir gets 420 votes, Pine 380 and Spruce 300. What share did Fir win? Is that a majority?

2

Per person

A made-up town spends $96 million a year and has 40,000 people. How much is that per person?

3

GST

Alberta has 5% GST and no PST. What does a $60 item cost with tax?

4

Mills

A house is assessed at $350,000. The made-up tax rate is 6.5 mills. What is the property tax?

Core

Two or three steps

Show which total each percent is “of”.
5

Ranked ballot

200 voters rank Fir, Pine and Spruce. First choices: Fir 80, Pine 70, Spruce 50. Spruce is dropped; 35 of its ballots go to Pine and 15 to Fir. Who wins, with what share?

6

Seats by share

A made-up council has 8 seats. Parties win 47%, 32% and 21%. Find each party’s exact share of 8 seats, then give out seats by largest remainder.

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Government Path · Unizon · PracticeVotes, budgets & polls
7

Pie chart

A $600 million budget spends $90 million on roads. What percent is that, and what angle on a pie chart?

8

Per capita trap

Spending rises from $500M to $520M while population grows from 200,000 to 210,000. What happens to spending per person, in dollars and percent?

9

Brackets

Made-up brackets: 10% on the first $50,000, 20% on income above that. Find the tax on $85,000, the average rate and the marginal rate.

10

Poll

A poll of 600 people has Party A at 42% and Party B at 38%. Find the margin of error. Can you say A is really ahead?

Level Up

Equations

A1–A2 (pp.17–18).
11

Work back

With the brackets from 9, someone paid $11,000 tax. Solve 5,000 + 0.2(I − 50,000) = 11,000.

12

Undo tax

A receipt says $84.00 including 5% GST. Solve 1.05p = 84. A $400,000 home paid $2,600: what mill rate is that?

13

Sample size

A pollster wants a ±4-point margin. Solve 98 ÷ √n = 4. How many people should they ask?

14

Draw the lines

35 voters (21 Teal, 14 Gold) are split into 7 districts of 5. What is the most districts Gold could win? The most Teal could win?

Make your own problem

Your turn to be the election analyst

  1. Invent a small election: candidates, voters and a voting system.
  2. Write a question about shares, seats or a poll’s margin of error.
  3. Solve it, then try a different voting system. Does the winner change?

My problem, and my worked answer:

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Government Path · Unizon · PracticeVotes, budgets & polls

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.

Answer key

Worked solutions. Every town, party, tax rate and poll is fictional; this sheet does not favour any real party.

Warm-up
1
≈ 38.2% · no (plurality only) 420 ÷ 1,100 ≈ 0.382. A majority needs over 550.
2
$2,400 96,000,000 ÷ 40,000 = 2,400.
3
$63.00 60 × 1.05 = 63.
4
$2,275 6.5 mills = $6.50 per $1,000: 350 × 6.50 = 2,275.
Core
5
Pine · 105 of 200 = 52.5% Fir 80 + 15 = 95; Pine 70 + 35 = 105, a majority. Fir led first choices but lost.
6
3.76, 2.56, 1.68 → 4, 2, 2 Whole parts give 3 + 2 + 1 = 6. The two biggest remainders (0.76, 0.68) get the last 2 seats.
7
15% · 54° 90 ÷ 600 = 0.15; 0.15 × 360 = 54.
8
$2,500 → ≈ $2,476.19 (≈ −0.95%) Total spending went up 4%, but per person it went down, because population grew 5%.
9
$12,000 · ≈ 14.1% · 20% 10% × 50,000 = 5,000; 20% × 35,000 = 7,000. 12,000 ÷ 85,000 ≈ 14.1%. The next dollar is taxed at 20%.
10
±4.0 points · no 98 ÷ √600 ≈ 4.0. A could be 38–46 and B 34–42: the ranges overlap, so it’s too close to call.
Level Up
11
$80,000 0.2(I − 50,000) = 6,000 → I − 50,000 = 30,000.
12
$80.00 · 6.5 mills p = 84 ÷ 1.05 = 80. 2,600 ÷ 400 = 6.5 per $1,000.
13
601 people √n = 24.5 → n = 600.25. Round up so the margin is no bigger than 4.
14
Gold up to 4 · Teal up to 7 Gold: 3 voters in 4 districts (12) and the other 2 elsewhere. Teal: 3 Teal + 2 Gold in all 7. Same votes, very different seats (p.16).
Make your own: say what each percent is “of”, round sample sizes up, and remember every place and party here is made up.
Government Practice · Government Path · UnizonAnswer key