Scientific Comparison

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Scientific Comparison · Grade 7 · Worksheet 3

1. Olivia is studying the distances of three planets from the Sun for a science project. She records the distances in scientific notation: - Mercury: 5.79 × 10⁷ km - Venus: 1.08 × 10⁸ km - Earth: 1.50 × 10⁸ km Olivia creates a visual diagram showing the planets on a number line. She places a point for each distance on the line, which is marked in increments of 1 × 10⁷ km. Based on the diagram, which planet is closest to the Sun, and what is the correct order of the planets from closest to farthest? Answer: Mercury, Venus, Earth Solution: Write each distance in standard form to compare easily. - Mercury: 5.79 × 10⁷ = 57,900,000 km - Venus: 1.08 × 10⁸ = 108,000,000 km - Earth: 1.50 × 10⁸ = 150,000,000 km Compare the numbers.
   Full step-by-step solution

   Step 1: Write each distance in standard form to compare easily. - Mercury: 5.79 × 10⁷ = 57,900,000 km - Venus: 1.08 × 10⁸ = 108,000,000 km - Earth: 1.50 × 10⁸ = 150,000,000 km Step 2: Compare the numbers. - 57,900,000 is less than 108,000,000, which is less than 150,000,000. Step 3: Order from closest (smallest distance) to farthest (largest distance). Mercury (57,900,000 km) < Venus (108,000,000 km) < Earth (150,000,000 km) Final answer: Mercury, Venus, Earth

2. Compare: 9.8 × 10⁷ __ 1.2 × 10⁸ Answer: < Solution: Identify the exponents. 9.8 × 10⁷ has an exponent of 7. 1.2 × 10⁸ has an exponent of 8.
   Full step-by-step solution

   Step 1: Identify the exponents. 9.8 × 10⁷ has an exponent of 7. 1.2 × 10⁸ has an exponent of 8. Step 2: Since 8 > 7, the number with exponent 8 is larger, regardless of the coefficients. Step 3: Therefore, 9.8 × 10⁷ < 1.2 × 10⁸. The answer is <.

3. Charlotte is looking at a diagram of a number line that shows the distances of four stars from Earth, each written in scientific notation. The positions on the number line are labeled as follows: Star P is at 7.2 × 10² light-years, Star Q is at 1.7 × 10² light-years, Star R is at 2.7 × 10³ light-years, and Star S is at 4.2 × 10³ light-years. Arrange the stars in order from the closest to Earth (smallest distance) to the farthest from Earth (largest distance). Write your answer as a sequence of letters separated by commas (e.g., P, Q, R, S). Answer: Q, P, R, S Solution: Write all distances in scientific notation: Star P = 7.2 × 10², Star Q = 1.7 × 10², Star R = 2.7 × 10³, Star S = 4.2 × 10³. Compare the exponents. Stars P and Q have exponent 2 (hundreds).
   Full step-by-step solution

   Step 1: Write all distances in scientific notation: Star P = 7.2 × 10², Star Q = 1.7 × 10², Star R = 2.7 × 10³, Star S = 4.2 × 10³. Step 2: Compare the exponents. Stars P and Q have exponent 2 (hundreds). Stars R and S have exponent 3 (thousands). Since 2 < 3, Stars P and Q are smaller (closer) than Stars R and S. Step 3: Compare Stars P and Q, which both have exponent 2. Compare coefficients: 1.7 < 7.2, so Star Q (1.7 × 10²) is smaller than Star P (7.2 × 10²). Step 4: Compare Stars R and S, which both have exponent 3. Compare coefficients: 2.7 < 4.2, so Star R (2.7 × 10³) is smaller than Star S (4.2 × 10³). Step 5: Order from smallest to largest: Q (1.7 × 10²), P (7.2 × 10²), R (2.7 × 10³), S (4.2 × 10³). The answer is Q, P, R, S.

4. Aroha is comparing the distances of two exoplanets from Earth for her astronomy project. Exoplanet Kepler-22b is approximately 6.2 × 10^18 meters away, while Exoplanet Proxima Centauri b is approximately 4.0 × 10^17 meters away. How many times farther from Earth is Kepler-22b compared to Proxima Centauri b? Answer: 15.5 Solution: Write the division expression: (6.2 × 10^18) ÷ (4.0 × 10^17). Divide the coefficients: 6.2 ÷ 4.0 = 1.55. Subtract the exponents: 18 - 17 = 1.
   Full step-by-step solution

   Step 1: Write the division expression: (6.2 × 10^18) ÷ (4.0 × 10^17). Step 2: Divide the coefficients: 6.2 ÷ 4.0 = 1.55. Step 3: Subtract the exponents: 18 - 17 = 1. Step 4: Combine: 1.55 × 10^1 = 15.5. Step 5: Kepler-22b is 15.5 times farther from Earth than Proxima Centauri b.

5. Compare: 9.1×10⁶ __ 8.6×10⁷ Answer: < Solution: Identify the exponents. 9.1×10⁶ has an exponent of 6. 8.6×10⁷ has an exponent of 7.
   Full step-by-step solution

   Step 1: Identify the exponents. 9.1×10⁶ has an exponent of 6. 8.6×10⁷ has an exponent of 7. Step 2: Since 7 > 6, the number with the larger exponent is greater. So 8.6×10⁷ is greater than 9.1×10⁶. Step 3: Therefore, 9.1×10⁶ < 8.6×10⁷. The answer is <.

6. A research lab is studying bacterial growth. One strain doubles every hour, starting with 8.0 × 10³ bacteria. Another strain triples every hour, starting with 3.0 × 10² bacteria. After 4 hours, which strain has more bacteria, and by approximately how many times more? Answer: The doubling strain has about 2.96 times more bacteria. Solution: Exponential growth means a quantity multiplies by a constant factor each time period.
   Full step-by-step solution

   Exponential growth means a quantity multiplies by a constant factor each time period. To compare two exponentially growing quantities, calculate their values after the same number of periods using exponent rules, then find their ratio. This concept appears in population studies, finance, and natural phenomena.

7. Compare: 7.2 × 10⁷ __ 8.1 × 10⁶ Answer: > Solution: Write both numbers in standard form to compare. 7.2 × 10⁷ = 72,000,000 8.1 × 10⁶ = 8,100,000 Compare the values: 72,000,000 > 8,100,000 Therefore, 7.2 × 10⁷ > 8.1 × 10⁶ The answer is >.
   Full step-by-step solution

   Step 1: Write both numbers in standard form to compare. 7.2 × 10⁷ = 72,000,000 8.1 × 10⁶ = 8,100,000 Step 2: Compare the values: 72,000,000 > 8,100,000 Step 3: Therefore, 7.2 × 10⁷ > 8.1 × 10⁶ The answer is >.