Scientific Comparison

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Answer Key & Explanations

Scientific Comparison · Grade 7 · Worksheet 3

1. Compare: 9.1 × 10⁷ __ 3.7 × 10⁸ Answer: < Solution: Identify the exponents. For 9.1 × 10⁷, the exponent is 7. For 3.7 × 10⁸, the exponent is 8.
   Full step-by-step solution

   Step 1: Identify the exponents. For 9.1 × 10⁷, the exponent is 7. For 3.7 × 10⁸, the exponent is 8. Step 2: Compare the exponents. Since 7 < 8, the number with exponent 8 is larger. Step 3: Therefore, 9.1 × 10⁷ is less than 3.7 × 10⁸. Step 4: Write the comparison: 9.1 × 10⁷ < 3.7 × 10⁸. The answer is <.

2. 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.

3. A shipping company transports two types of cargo. Container A carries 7.5 × 10^6 kilograms of goods, while Container B carries 1.25 × 10^5 kilograms of goods. How many times heavier is Container A than Container B? Answer: 60 Solution: Write the division expression: (7.5 × 10^6) ÷ (1.25 × 10^5) Divide the coefficients: 7.5 ÷ 1.25 = 6 Divide the powers of 10: 10^6 ÷ 10^5 = 10^(6-5) = 10^1 = 10 Multiply the results: 6 × 10 = 60 Container A is 60 times heavier than Container B.
   Full step-by-step solution

   Step 1: Write the division expression: (7.5 × 10^6) ÷ (1.25 × 10^5) Step 2: Divide the coefficients: 7.5 ÷ 1.25 = 6 Step 3: Divide the powers of 10: 10^6 ÷ 10^5 = 10^(6-5) = 10^1 = 10 Step 4: Multiply the results: 6 × 10 = 60 Step 5: Container A is 60 times heavier than Container B. The answer is 60.

4. 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 >.

5. Order: 6.4×10⁶, 8.2×10⁵, 3.1×10⁷ Answer: 8.2×10⁵ < 6.4×10⁶ < 3.1×10⁷ Solution: List the numbers: 6.4×10⁶, 8.2×10⁵, 3.1×10⁷ Compare exponents: 8.2×10⁵ has exponent 5, 6.4×10⁶ has exponent 6, 3.1×10⁷ has exponent 7.
   Full step-by-step solution

   Step 1: List the numbers: 6.4×10⁶, 8.2×10⁵, 3.1×10⁷ Step 2: Compare exponents: 8.2×10⁵ has exponent 5, 6.4×10⁶ has exponent 6, 3.1×10⁷ has exponent 7. Step 3: Since 5 < 6 < 7, the order from smallest to largest is: 8.2×10⁵, then 6.4×10⁶, then 3.1×10⁷. Step 4: Write the final order: 8.2×10⁵ < 6.4×10⁶ < 3.1×10⁷ The answer is 8.2×10⁵ < 6.4×10⁶ < 3.1×10⁷.

6. A scientist is comparing the masses of two distant exoplanets. Planet Alpha has a mass of 4.2 × 10²⁴ kilograms, while Planet Beta has a mass of 6.8 × 10²³ kilograms. Liam, an astronomy student, needs to determine which planet is more massive and by approximately how many times. Can you help him with this comparison? Answer: Planet Alpha is more massive, by about 6.2 times Solution: To compare the masses of Planet Alpha and Planet Beta, we need to divide the mass of Planet Alpha by the mass of Planet Beta. Write down the masses. Mass of Planet Alpha = 4.2 × 10²⁴ kg Mass of Planet Beta = 6.8 × 10²³ kg Set up the division to find how many times more massive Alpha is than Beta.
   Full step-by-step solution

   To compare the masses of Planet Alpha and Planet Beta, we need to divide the mass of Planet Alpha by the mass of Planet Beta. Step 1: Write down the masses. Mass of Planet Alpha = 4.2 × 10²⁴ kg Mass of Planet Beta = 6.8 × 10²³ kg Step 2: Set up the division to find how many times more massive Alpha is than Beta. Ratio = (Mass of Alpha) / (Mass of Beta) Ratio = (4.2 × 10²⁴) / (6.8 × 10²³) Step 3: Separate the numbers and the powers of 10. Ratio = (4.2 / 6.8) × (10²⁴ / 10²³) Step 4: Calculate the division of the powers of 10. When dividing powers with the same base, subtract the exponents. 10²⁴ / 10²³ = 10^(24 - 23) = 10¹ = 10 Step 5: Calculate the division of the decimal numbers. 4.2 / 6.8 To make this easier, we can think of it as 42 / 68. Dividing both numerator and denominator by 2 gives 21 / 34. Now, calculate 21 ÷ 34. 21 ÷ 34 ≈ 0.6176 Step 6: Combine the two results. Ratio ≈ 0.6176 × 10 Step 7: Simplify the expression. 0.6176 × 10 = 6.176 Step 8: Round to a reasonable number of significant figures. Both original numbers (4.2 and 6.8) have two significant figures, so we round our answer to two significant figures. 6.176 rounds to 6.2 Conclusion: Planet Alpha is more massive than Planet Beta. The mass of Planet Alpha is approximately 6.2 times the mass of Planet Beta.

7. Mason is an astronomer comparing the masses of two distant exoplanets. Planet Kepler-442b has a mass of 8.4 × 10^24 kilograms, while Planet Kepler-186f has a mass of 2.8 × 10^23 kilograms. How many times more massive is Kepler-442b than Kepler-186f? Answer: 30 Solution: Write the division expression: (8.4 × 10^24) ÷ (2.8 × 10^23) Divide the coefficients: 8.4 ÷ 2.8 = 3 Subtract the exponents: 24 - 23 = 1 Combine the results: 3 × 10^1 = 30 Kepler-442b is 30 times more massive than Kepler-186f.
   Full step-by-step solution

   Step 1: Write the division expression: (8.4 × 10^24) ÷ (2.8 × 10^23) Step 2: Divide the coefficients: 8.4 ÷ 2.8 = 3 Step 3: Subtract the exponents: 24 - 23 = 1 Step 4: Combine the results: 3 × 10^1 = 30 Step 5: Kepler-442b is 30 times more massive than Kepler-186f. The answer is 30.

8. Mason is an astronomer studying two distant stars. Star Alpha has a diameter of 4.2 × 10^7 kilometers, and Star Beta has a diameter of 7.0 × 10^6 kilometers. How many times larger is the diameter of Star Alpha compared to Star Beta? Answer: 6 Solution: Write the division expression: (4.2 × 10^7) ÷ (7.0 × 10^6) Divide the coefficients: 4.2 ÷ 7.0 = 0.6 Subtract the exponents: 7 - 6 = 1 Combine the results: 0.6 × 10^1 = 0.6 × 10 = 6 Star Alpha's diameter is 6 times larger than Star Beta's diameter.
   Full step-by-step solution

   Step 1: Write the division expression: (4.2 × 10^7) ÷ (7.0 × 10^6) Step 2: Divide the coefficients: 4.2 ÷ 7.0 = 0.6 Step 3: Subtract the exponents: 7 - 6 = 1 Step 4: Combine the results: 0.6 × 10^1 = 0.6 × 10 = 6 Step 5: Star Alpha's diameter is 6 times larger than Star Beta's diameter. The answer is 6.