Scientific Comparison

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Answer Key & Explanations

Scientific Comparison · Grade 7 · Worksheet 2

1. Olivia is studying the masses of four planets in a distant solar system, each written in scientific notation on a chart: - Planet A: 9.87 × 10²³ kg - Planet B: 1.23 × 10²⁴ kg - Planet C: 5.71 × 10²³ kg - Planet D: 3.45 × 10²⁴ kg Arrange the planets in order from the smallest mass to the largest mass. Then, identify which planet has the greatest mass. Answer: Planet C, Planet A, Planet B, Planet D; Planet D has the greatest mass. Solution: Planet A: 9.87 × 10²³ (exponent 23) Planet B: 1.23 × 10²⁴ (exponent 24) Planet C: 5.71 × 10²³ (exponent 23) Planet D: 3.45 × 10²⁴ (exponent 24) Compare exponents first.
   Full step-by-step solution

   Step 1: List the numbers with their exponents: Planet A: 9.87 × 10²³ (exponent 23) Planet B: 1.23 × 10²⁴ (exponent 24) Planet C: 5.71 × 10²³ (exponent 23) Planet D: 3.45 × 10²⁴ (exponent 24) Step 2: Compare exponents first. Planets with exponent 24 (B and D) are larger than those with exponent 23 (A and C). Step 3: Among exponent 23 planets (A and C), compare the decimal parts: 5.71 < 9.87, so Planet C (5.71 × 10²³) is smaller than Planet A (9.87 × 10²³). Step 4: Among exponent 24 planets (B and D), compare decimal parts: 1.23 < 3.45, so Planet B (1.23 × 10²⁴) is smaller than Planet D (3.45 × 10²⁴). Step 5: Order from smallest to largest: Planet C (5.71 × 10²³), Planet A (9.87 × 10²³), Planet B (1.23 × 10²⁴), Planet D (3.45 × 10²⁴). The greatest mass is Planet D.

2. Liam is comparing the storage capacities of two new smartphones. The Galaxy X has 5.6 × 10^11 bytes of storage, while the Tech Pro has 8.0 × 10^10 bytes. How many times more storage does the Galaxy X have compared to the Tech Pro? Answer: 7 Solution: Write the ratio of Galaxy X storage to Tech Pro storage: (5.6 × 10^11) ÷ (8.0 × 10^10) Divide the coefficients: 5.6 ÷ 8.0 = 0.7 Subtract the exponents: 11 - 10 = 1 Combine the results: 0.7 × 10^1 = 7 The Galaxy X has 7 times more storage than the Tech Pro.
   Full step-by-step solution

   Step 1: Write the ratio of Galaxy X storage to Tech Pro storage: (5.6 × 10^11) ÷ (8.0 × 10^10) Step 2: Divide the coefficients: 5.6 ÷ 8.0 = 0.7 Step 3: Subtract the exponents: 11 - 10 = 1 Step 4: Combine the results: 0.7 × 10^1 = 7 Step 5: The Galaxy X has 7 times more storage than the Tech Pro. The answer is 7.

3. Compare: 9.2×10⁷ __ 1.3×10⁸ Answer: < Solution: Identify the exponents. 9.2×10⁷ has an exponent of 7, and 1.3×10⁸ has an exponent of 8. Since 8 > 7, the number with exponent 8 is larger, regardless of the coefficients.
   Full step-by-step solution

   Step 1: Identify the exponents. 9.2×10⁷ has an exponent of 7, and 1.3×10⁸ has an exponent of 8. Step 2: Since 8 > 7, the number with exponent 8 is larger, regardless of the coefficients. Step 3: Therefore, 9.2×10⁷ is less than 1.3×10⁸. Step 4: The correct symbol is <. The answer is <.

4. Compare: 6.8 × 10⁶ __ 4.2 × 10⁷ Answer: < Solution: Identify the exponents. For 6.8 × 10⁶, the exponent is 6. For 4.2 × 10⁷, the exponent is 7.
   Full step-by-step solution

   Step 1: Identify the exponents. For 6.8 × 10⁶, the exponent is 6. For 4.2 × 10⁷, the exponent is 7. Step 2: Compare the exponents: 6 < 7. Step 3: Since the exponent of 10 is larger for 4.2 × 10⁷, this number is greater. Step 4: Therefore, 6.8 × 10⁶ < 4.2 × 10⁷. The answer is <.

5. Compare: 6.5 × 10⁵ __ 8.0 × 10⁴ Answer: > Solution: Write both numbers in standard form for comparison. 6.5 × 10⁵ = 6.5 × 100000 = 650,000 8.0 × 10⁴ = 8.0 × 10000 = 80,000 Compare the standard forms: 650,000 and 80,000. 650,000 is greater than 80,000.
   Full step-by-step solution

   Step 1: Write both numbers in standard form for comparison. 6.5 × 10⁵ = 6.5 × 100000 = 650,000 8.0 × 10⁴ = 8.0 × 10000 = 80,000 Step 2: Compare the standard forms: 650,000 and 80,000. 650,000 is greater than 80,000. Step 3: Therefore, 6.5 × 10⁵ > 8.0 × 10⁴. The answer is >.

6. Compare: 4.5×10⁶ __ 8.5×10⁵ Answer: > Solution: Write both numbers in standard form for comparison. 4.5×10⁶ = 4.5 × 1,000,000 = 4,500,000. 8.5×10⁵ = 8.5 × 100,000 = 850,000.
   Full step-by-step solution

   Step 1: Write both numbers in standard form for comparison. 4.5×10⁶ = 4.5 × 1,000,000 = 4,500,000. 8.5×10⁵ = 8.5 × 100,000 = 850,000. Step 2: Compare the exponents first. The first number has exponent 6, the second has exponent 5. Since 6 > 5, the first number is larger. Step 3: Even though 4.5 is smaller than 8.5, the exponent 6 makes 4.5×10⁶ much larger than 8.5×10⁵. Step 4: Therefore, 4.5×10⁶ > 8.5×10⁵. The answer is >.

7. A research team is studying bacteria growth. They observe that one bacterial colony has 3.8 × 10^7 cells, while another colony has 4.2 × 10^6 cells. Which colony has more bacterial cells, and how many times larger is it? Answer: The first colony is larger by approximately 9 times Solution: Identify the number of cells in each colony. First colony: 3.8 × 10^7 Second colony: 4.2 × 10^6 Compare the exponents of 10. 10^7 means 10 million, 10^6 means 1 million.
   Full step-by-step solution

   Step 1: Identify the number of cells in each colony. First colony: 3.8 × 10^7 Second colony: 4.2 × 10^6 Step 2: Compare the exponents of 10. 10^7 means 10 million, 10^6 means 1 million. Since 10^7 is 10 times larger than 10^6, the first colony is likely larger. Step 3: To find how many times larger the first colony is, divide the first colony's cell count by the second colony's cell count. (3.8 × 10^7) / (4.2 × 10^6) Step 4: Separate the numbers and the powers of 10. = (3.8 / 4.2) × (10^7 / 10^6) Step 5: Simplify each part. 3.8 / 4.2 = 38 / 42 = 19 / 21 ≈ 0.90476 10^7 / 10^6 = 10^(7 - 6) = 10^1 = 10 Step 6: Multiply the results. 0.90476 × 10 ≈ 9.0476 Step 7: Round to a reasonable value for comparison. 9.0476 is approximately 9. Conclusion: The first colony has more bacterial cells, and it is about 9 times larger than the second colony.

8. Emma is comparing the sizes of three planets as shown in a diagram. The diagram shows three circles labeled with their approximate diameters in scientific notation: Planet A: 1.27×10⁴ km, Planet B: 6.79×10³ km, and Planet C: 4.88×10⁴ km. Arrange the planets in order from smallest diameter to largest diameter. Answer: Planet B, Planet A, Planet C Solution: Write each diameter in standard form for comparison. Planet A: 1.27×10⁴ = 1.27 × 10000 = 12700 km Planet B: 6.79×10³ = 6.79 × 1000 = 6790 km Planet C: 4.88×10⁴ = 4.88 × 10000 = 48800 km Compare the numbers: 6790 < 12700 < 48800.
   Full step-by-step solution

   Step 1: Write each diameter in standard form for comparison. Planet A: 1.27×10⁴ = 1.27 × 10000 = 12700 km Planet B: 6.79×10³ = 6.79 × 1000 = 6790 km Planet C: 4.88×10⁴ = 4.88 × 10000 = 48800 km Step 2: Compare the numbers: 6790 < 12700 < 48800. Step 3: Order from smallest to largest: Planet B (6790 km), Planet A (12700 km), Planet C (48800 km). The answer is Planet B, Planet A, Planet C.

9. Order: 9.1×10⁶, 8.6×10⁷, 1.6×10⁶, 6.1×10⁷ Answer: 1.6×10⁶, 9.1×10⁶, 6.1×10⁷, 8.6×10⁷ Solution: List the numbers: 9.1×10⁶, 8.6×10⁷, 1.6×10⁶, 6.1×10⁷ Compare exponents first. Numbers with exponent 10⁶ are smaller than those with 10⁷.
   Full step-by-step solution

   Step 1: List the numbers: 9.1×10⁶, 8.6×10⁷, 1.6×10⁶, 6.1×10⁷ Step 2: Compare exponents first. Numbers with exponent 10⁶ are smaller than those with 10⁷. Step 3: Group by exponent: - 10⁶ group: 9.1×10⁶, 1.6×10⁶ - 10⁷ group: 8.6×10⁷, 6.1×10⁷ Step 4: Within the 10⁶ group, compare coefficients: 1.6 < 9.1, so 1.6×10⁶ < 9.1×10⁶ Step 5: Within the 10⁷ group, compare coefficients: 6.1 < 8.6, so 6.1×10⁷ < 8.6×10⁷ Step 6: Combine all in order from smallest to largest: 1.6×10⁶, 9.1×10⁶, 6.1×10⁷, 8.6×10⁷ The answer is 1.6×10⁶, 9.1×10⁶, 6.1×10⁷, 8.6×10⁷.