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1
In an election with two candidates, 10% of voters did not vote, and 15% of the votes were invalid. The winner received 60% of the valid votes and won by 180 votes. Determine the total number of voters on the list.
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Solution: Step 1: Let total voters = x Step 2: Voters who voted = 90% of x = 0.9x Step 3: Valid votes = 85% of 0.9x = 0.765x Step 4: Winner's votes = 60% of 0.765x = 0.459x Step 5: Loser's votes = 0.459x - 180 Step 6: Set up equation: 0.459x - (0.459x - 180) = 180 Step 7: Simplify: 0.459x - 0.459x + 180 = 180 Step 8: Correct equation: 0.459x - 0.306x = 180 Step 9: Simplify: 0.153x = 180 Step 10: Solve for x: x = 180 / 0.153 ≈ 400
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A population currently stands at 50,000 and grows at 10% annually. What was the population 4 years ago?
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Solution: Step 1: Let initial population 4 years ago = P Step 2: After 4 years, population becomes P(1 + 0.10)^4 Step 3: Given final population = 50,000 Step 4: Set equation: P(1.10)^4 = 50,000 Step 5: Calculate (1.10)^4 ≈ 1.464 Step 6: P ≈ 50,000 / 1.464 = 34,151 Step 7: Verify correct answer matches calculation
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