Scientific Comparison

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Answer Key & Explanations

Scientific Comparison · Grade 7 · Worksheet 1

1. Noah is comparing the masses of two large asteroids for his astronomy project. Asteroid Pallas has a mass of approximately 2.07 × 10^20 kilograms, while Asteroid Vesta has a mass of approximately 2.59 × 10^19 kilograms. How many times greater is the mass of Asteroid Pallas compared to Asteroid Vesta? Answer: 8 Solution: Write the comparison as a division: (2.07 × 10^20) ÷ (2.59 × 10^19). Divide the coefficients: 2.07 ÷ 2.59 = 0.799... This is approximately 0.8.
   Full step-by-step solution

   Step 1: Write the comparison as a division: (2.07 × 10^20) ÷ (2.59 × 10^19). Step 2: Divide the coefficients: 2.07 ÷ 2.59 = 0.799... This is approximately 0.8. Step 3: Subtract the exponents: 20 - 19 = 1. Step 4: Combine the results: 0.8 × 10^1 = 8. Step 5: The mass of Asteroid Pallas is 8 times greater than the mass of Asteroid Vesta. The answer is 8.

2. Tane is examining a visual chart that shows the diameters of three planets in kilometers, each written in scientific notation. The chart displays: - Planet P: 1.49 × 10⁸ km - Planet Q: 9.12 × 10⁹ km - Planet R: 3.56 × 10⁷ km If Tane arranges the planets in order from the smallest diameter to the largest diameter, what is the correct order? Write your answer as a sequence of letters separated by commas (e.g., P, Q, R). Answer: R, P, Q Solution: Write the diameters: Planet P = 1.49 × 10⁸, Planet Q = 9.12 × 10⁹, Planet R = 3.56 × 10⁷. Compare the exponents: Planet Q has exponent 9, Planet P has exponent 8, Planet R has exponent 7.
   Full step-by-step solution

   Step 1: Write the diameters: Planet P = 1.49 × 10⁸, Planet Q = 9.12 × 10⁹, Planet R = 3.56 × 10⁷. Step 2: Compare the exponents: Planet Q has exponent 9, Planet P has exponent 8, Planet R has exponent 7. Step 3: The smallest exponent is 7, so Planet R (3.56 × 10⁷) is the smallest diameter. Step 4: Next smallest exponent is 8, so Planet P (1.49 × 10⁸) is the middle diameter. Step 5: The largest exponent is 9, so Planet Q (9.12 × 10⁹) is the largest diameter. Step 6: Order from smallest to largest: R, P, Q. The answer is R, P, Q.

3. Mason is studying a diagram of the solar system that shows the distances from the Sun to various planets. The distance from the Sun to Mercury is shown as 5.79 × 10⁷ km, and the distance from the Sun to Venus is shown as 1.08 × 10⁸ km. On the diagram, a third planet, Mars, is labeled with a distance of 2.28 × 10⁷ km from the Sun. Arrange the three planets (Mercury, Venus, and Mars) in order from the shortest distance to the longest distance from the Sun. Answer: Mars, Mercury, Venus Solution: Write all distances in scientific notation: Mercury = 5.79 × 10⁷, Venus = 1.08 × 10⁸, Mars = 2.28 × 10⁷. Compare the exponents: Venus has exponent 8, while Mercury and Mars have exponent 7.
   Full step-by-step solution

   Step 1: Write all distances in scientific notation: Mercury = 5.79 × 10⁷, Venus = 1.08 × 10⁸, Mars = 2.28 × 10⁷. Step 2: Compare the exponents: Venus has exponent 8, while Mercury and Mars have exponent 7. Since 8 > 7, Venus is the largest distance. Step 3: Compare Mercury and Mars, which both have exponent 7. Compare coefficients: 2.28 < 5.79, so Mars (2.28 × 10⁷) is smaller than Mercury (5.79 × 10⁷). Step 4: Order from shortest to longest: Mars (2.28 × 10⁷), Mercury (5.79 × 10⁷), Venus (1.08 × 10⁸). The answer is Mars, Mercury, Venus.

4. (3.2 × 10⁴) ÷ (8 × 10²) = ? Answer: 40 Solution: (3.2 × 10⁴) ÷ (8 × 10²) We can separate the decimal part and the power-of-ten part: = (3.2 ÷ 8) × (10⁴ ÷ 10²) 3.2 ÷ 8 = 0.4 0.4 × (10⁴ ÷ 10²) 10⁴ ÷ 10² = 10^(4 - 2) = 10² = 100 0.4 × 100 0.4 × 100 = 40 Final answer: 40
   Full step-by-step solution

   Let's solve step by step. We have: (3.2 × 10⁴) ÷ (8 × 10²) **Step 1: Break into parts** We can separate the decimal part and the power-of-ten part: = (3.2 ÷ 8) × (10⁴ ÷ 10²) **Step 2: Divide the decimal numbers** 3.2 ÷ 8 = 0.4 So now we have: 0.4 × (10⁴ ÷ 10²) **Step 3: Divide the powers of ten** 10⁴ ÷ 10² = 10^(4 - 2) = 10² = 100 So now we have: 0.4 × 100 **Step 4: Multiply** 0.4 × 100 = 40 **Final answer:** 40

5. Compare: 5.7 × 10⁷ __ 9.3 × 10⁶ Answer: > Solution: Identify the exponents. For 5.7 × 10⁷, the exponent is 7. For 9.3 × 10⁶, the exponent is 6.
   Full step-by-step solution

   Step 1: Identify the exponents. For 5.7 × 10⁷, the exponent is 7. For 9.3 × 10⁶, the exponent is 6. Step 2: Since 7 > 6, the number with the larger exponent is greater. So 5.7 × 10⁷ > 9.3 × 10⁶. Step 3: To verify, convert to standard form: 5.7 × 10⁷ = 57,000,000 and 9.3 × 10⁶ = 9,300,000. Clearly 57,000,000 > 9,300,000. The answer is >.

6. Compare: 7.2×10⁷ __ 8.7×10⁶ Answer: > Solution: Write both numbers in standard form to compare. 7.2×10⁷ = 72,000,000. 8.7×10⁶ = 8,700,000.
   Full step-by-step solution

   Step 1: Write both numbers in standard form to compare. 7.2×10⁷ = 72,000,000. 8.7×10⁶ = 8,700,000. Step 2: Compare the values: 72,000,000 is greater than 8,700,000. Step 3: Therefore, 7.2×10⁷ > 8.7×10⁶. The answer is >.

7. Isabella is an astronomer comparing the distances of two newly discovered asteroids from Earth. Asteroid X-27 is approximately 2.7 × 10^7 kilometers away, while Asteroid Y-42 is approximately 8.1 × 10^6 kilometers away. How many times farther from Earth is Asteroid X-27 compared to Asteroid Y-42? Answer: 3.333... or 10/3 Solution: Write the division to compare the distances: (2.7 × 10^7) ÷ (8.1 × 10^6). Divide the coefficients: 2.7 ÷ 8.1 = 0.333... Subtract the exponents: 7 - 6 = 1.
   Full step-by-step solution

   Step 1: Write the division to compare the distances: (2.7 × 10^7) ÷ (8.1 × 10^6). Step 2: Divide the coefficients: 2.7 ÷ 8.1 = 0.333... Step 3: Subtract the exponents: 7 - 6 = 1. Step 4: Combine the results: 0.333... × 10^1 = 0.333... × 10 = 3.333... Step 5: The distance is 3.333... or 10/3 times greater. The answer is 3.333... or 10/3.