Population Comparison

Grade 7 · statistics · 100 practice problems · read aloud

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📊 Population Comparison

What is Population Comparison?

Population comparison is a way to compare the sizes of different groups using ratios and percents. It's useful for understanding real-world data, like comparing the number of students in different schools or the populations of different cities.

How to Solve Population Problems

  1. Identify the Populations: Write down the numbers you are comparing.
  2. Find the Ratio: Write the first population compared to the second (e.g., Population A : Population B).
  3. Simplify the Ratio: Divide both numbers by their greatest common factor.
  4. Find the Percent: Divide the smaller population by the larger one and multiply by 100 to see what percent one is of the other.

Visual Examples

Example 1: Town A has 12,000 people. Town B has 15,000 people. Compare them.

  1. Identify: 12,000 and 15,000.
  2. Ratio: 12,000 : 15,000
  3. Simplify: Divide both by 3,000 → 4 : 5. For every 4 people in Town A, there are 5 in Town B.
  4. Percent: (12,000 / 15,000) * 100 = 80%. Town A's population is 80% the size of Town B's.

Example 2: A school has 180 students in 7th grade and 220 in 8th grade.

  1. Identify: 180 and 220.
  2. Ratio: 180 : 220
  3. Simplify: Divide both by 20 → 9 : 11.
  4. Percent: (180 / 220) * 100 ≈ 82%. The 7th grade is about 82% the size of the 8th grade.

⚠️ Common Mistakes

  • Writing the Ratio Backwards: Always check which population is mentioned first in the problem.
  • Forgetting to Simplify: A ratio of 12000:15000 should always be simplified to 4:5.
  • Percent Confusion: Remember, the percent tells you what the smaller group is compared to the larger one.

💡 Tips & Tricks

  • Use the "part ÷ whole × 100" formula for percent.
  • To simplify ratios quickly, look for numbers that divide evenly into both (like 1,000 or 100).
  • Draw two bars to represent the populations visually—it helps you see the difference!

How to Practice

Look for populations to compare in your daily life! Compare the number of:

  • Students in your class vs. another class.
  • People in your city vs. a nearby city (look it up!).
  • Followers for two different sports teams on social media.

Calculate the ratio and percent for each. Practice makes perfect!

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 (3/4 + 1/2) × 24 = ?

Hint: First find a common denominator to add the fractions, then multiply the result by the whole number.

Show the answer

Answer: 30

  1. First, we need to calculate the expression inside the parentheses: 3/4 + 1/2.
  2. To add these fractions, we need a common denominator. The denominators are 4 and 2. The least common denominator is 4.
  3. Convert 1/2 to a fraction with denominator 4: 1/2 = (1 × 2)/(2 × 2) = 2/4.
  4. Now add: 3/4 + 2/4 = (3 + 2)/4 = 5/4.
  5. So the expression becomes: (5/4) × 24.
  6. Multiply 5/4 by 24. This is the same as: (5 × 24) / 4.
  7. First, 5 × 24 = 120.
  8. Now divide 120 by 4: 120 / 4 = 30. Final answer: 30

Let's solve step-by-step.

2 (-2)³ + 5 × (3 - 7) = ?

Hint: Remember to follow the order of operations and handle negative exponents carefully.

Show the answer

Answer: -28

  1. Evaluate the exponent: (-2)³ = (-2) × (-2) × (-2) = -8
  2. Calculate inside parentheses: (3 - 7) = -4
  3. Multiply: 5 × (-4) = -20
  4. Add the results: -8 + (-20) = -28

The answer is -28.

3 (-5)² + 3 × (-4) - 7 = ?

Hint: Remember the order of operations and how negative numbers behave with exponents and multiplication.

Show the answer

Answer: 6

  1. Calculate the exponent first: (-5)² = 25
  2. Perform the multiplication: 3 × (-4) = -12
  3. Rewrite the expression with the results: 25 + (-12) - 7
  4. Perform the addition and subtraction from left to right: 25 + (-12) = 13
  5. 13 - 7 = 6

The answer is 6.

4 (-2)³ + 5 × (3 - 7) ÷ 2 = ?

Hint: Remember to follow the order of operations (PEMDAS) and be careful with negative signs when working with exponents.

Show the answer

Answer: -18

  1. Start with the expression: (-2)³ + 5 × (3 - 7) ÷ 2
  2. Calculate the exponent: (-2)³ = -2 × -2 × -2 = 4 × -2 = -8
  3. Calculate inside the parentheses: (3 - 7) = -4
  4. Now we have: -8 + 5 × (-4) ÷ 2
  5. Perform multiplication and division from left to right: 5 × (-4) = -20
  6. Then: -20 ÷ 2 = -10
  7. Now we have: -8 + (-10) = -18
  8. The final answer is -18.

5 (-3)² × 4 - 18 ÷ (2 + 1) = ?

Hint: Remember to follow the order of operations: parentheses first, then exponents, then multiplication and division from left to right, and finally addition and subtraction.

Show the answer

Answer: 30

  1. Solve inside the parentheses: (2 + 1) = 3. The expression becomes (-3)² × 4 - 18 ÷ 3.
  2. Evaluate the exponent: (-3)² = 9. The expression becomes 9 × 4 - 18 ÷ 3.
  3. Perform multiplication and division from left to right: 9 × 4 = 36, and 18 ÷ 3 = 6. The expression becomes 36 - 6.
  4. Perform the subtraction: 36 - 6 = 30. The final answer is 30.

6 (-3)² × 4 + 18 ÷ (-2) - 5 = ?

Hint: Remember the order of operations and pay attention to negative signs when working with exponents and division.

Show the answer

Answer: 22

  1. Calculate the exponent first: (-3)² = 9
  2. Perform multiplication: 9 × 4 = 36
  3. Perform division: 18 ÷ (-2) = -9
  4. Now the expression is: 36 + (-9) - 5
  5. Add and subtract from left to right: 36 + (-9) = 27
  6. 27 - 5 = 22

The answer is 22.

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