Scientific Comparison

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Answer Key & Explanations

Scientific Comparison · Grade 7 · Worksheet 2

1. (7.2 × 10⁸) × (3.0 × 10⁻⁴) ÷ (1.2 × 10³) = ? Answer: 180 Solution: Multiply the first two numbers: (7.2 × 3.0) × (10⁸ × 10⁻⁴) = 21.6 × 10⁴ Now divide by the third number: (21.6 ÷ 1.2) × (10⁴ ÷ 10³) = 18 × 10¹ Convert to standard form: 18 × 10 = 180 The answer is 180.
   Full step-by-step solution

   Step 1: Multiply the first two numbers: (7.2 × 3.0) × (10⁸ × 10⁻⁴) = 21.6 × 10⁴ Step 2: Now divide by the third number: (21.6 ÷ 1.2) × (10⁴ ÷ 10³) = 18 × 10¹ Step 3: Convert to standard form: 18 × 10 = 180 The answer is 180.

2. Liam is comparing the energy consumption of two different countries for his environmental science project. Country X uses 7.2 × 10^15 kilowatt-hours per year, while Country Y uses 4.8 × 10^14 kilowatt-hours per year. How many times greater is Country X's energy consumption compared to Country Y's? Answer: 15 Solution: Write the division expression: (7.2 × 10^15) ÷ (4.8 × 10^14) Separate the coefficients and powers of 10: (7.2 ÷ 4.8) × (10^15 ÷ 10^14) Divide the coefficients: 7.2 ÷ 4.8 = 1.5 Divide the powers of 10: 10^15 ÷ 10^14 = 10^(15-14) = 10^1 = 10 Multiply the results: 1.5 × 10 = 15 Country X's energy…
   Full step-by-step solution

   Step 1: Write the division expression: (7.2 × 10^15) ÷ (4.8 × 10^14) Step 2: Separate the coefficients and powers of 10: (7.2 ÷ 4.8) × (10^15 ÷ 10^14) Step 3: Divide the coefficients: 7.2 ÷ 4.8 = 1.5 Step 4: Divide the powers of 10: 10^15 ÷ 10^14 = 10^(15-14) = 10^1 = 10 Step 5: Multiply the results: 1.5 × 10 = 15 Country X's energy consumption is 15 times greater than Country Y's.

3. Compare: 8.7 × 10⁶ __ 9.2 × 10⁵ Answer: > Solution: Write both numbers in standard form to compare easily. 8.7 × 10⁶ = 8.7 × 1,000,000 = 8,700,000 9.2 × 10⁵ = 9.2 × 100,000 = 920,000 Compare the standard forms: 8,700,000 and 920,000. 8,700,000 is greater than 920,000.
   Full step-by-step solution

   Step 1: Write both numbers in standard form to compare easily. 8.7 × 10⁶ = 8.7 × 1,000,000 = 8,700,000 9.2 × 10⁵ = 9.2 × 100,000 = 920,000 Step 2: Compare the standard forms: 8,700,000 and 920,000. 8,700,000 is greater than 920,000. Step 3: Therefore, 8.7 × 10⁶ > 9.2 × 10⁵. The answer is >.

4. Liam is comparing the storage capacities of two new smartphones. The Galaxy Xtreme has 5.6 × 10^11 bytes of storage, while the TechMaster Pro has 8.0 × 10^10 bytes. How many times more storage does the Galaxy Xtreme have compared to the TechMaster Pro? Answer: 7 Solution: Write the division expression: (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 Convert to standard form: 0.7 × 10 = 7 The Galaxy Xtreme has 7 times more storage than the TechMaster Pro.
   Full step-by-step solution

   Step 1: Write the division expression: (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 Step 5: Convert to standard form: 0.7 × 10 = 7 Step 6: The Galaxy Xtreme has 7 times more storage than the TechMaster Pro.

5. 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 <.

6. A scientist is comparing the masses of two celestial objects. The first object has a mass of 4.2 × 10²¹ kilograms, and the second object has a mass of 6.8 × 10¹⁹ kilograms. Which object is heavier, and by approximately how many times greater is its mass? Answer: The first object is heavier, and its mass is approximately 62 times greater. Solution: Let’s compare the two masses step by step. First object mass: \( 4.2 \times 10^{21} \) kg Second object mass: \( 6.8 \times 10^{19} \) kg Since both are in scientific notation, we can compare the powers of 10 first.
   Full step-by-step solution

   Let’s compare the two masses step by step. --- **Step 1: Write the masses clearly** First object mass: \( 4.2 \times 10^{21} \) kg Second object mass: \( 6.8 \times 10^{19} \) kg --- **Step 2: Compare exponents** Since both are in scientific notation, we can compare the powers of 10 first. \( 10^{21} \) is larger than \( 10^{19} \), so the first object is heavier. --- **Step 3: Find how many times heavier** We divide the larger mass by the smaller mass: \[ \frac{4.2 \times 10^{21}}{6.8 \times 10^{19}} \] --- **Step 4: Separate the numbers and powers of 10** \[ = \frac{4.2}{6.8} \times \frac{10^{21}}{10^{19}} \] --- **Step 5: Simplify powers of 10** \[ \frac{10^{21}}{10^{19}} = 10^{21-19} = 10^{2} = 100 \] So we have: \[ \frac{4.2}{6.8} \times 100 \] --- **Step 6: Simplify 4.2 / 6.8** \[ \frac{4.2}{6.8} = \frac{42}{68} = \frac{21}{34} \] --- **Step 7: Approximate the fraction** \( 21/34 \) is approximately \( 0.6176 \). --- **Step 8: Multiply by 100** \[ 0.6176 \times 100 \approx 61.76 \] --- **Step 9: Round to a whole number for “approximately”** About 62 times. --- **Final Answer:** The first object is heavier, and its mass is approximately 62 times greater.

7. 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.

8. A research lab is studying bacteria growth. They observe that one type of bacteria doubles its population every hour. If the lab starts with 5.2 × 10³ bacteria in a petri dish, how many bacteria will be present after 4 hours? Express your answer in scientific notation. Answer: 8.32 × 10⁴ Solution: The bacteria population doubles every hour. Initial population = \( 5.2 \times 10^3 \) bacteria. We want the population after 4 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The bacteria population doubles every hour. Initial population = \( 5.2 \times 10^3 \) bacteria. We want the population after 4 hours. --- **Step 2: Set up the doubling formula** After \( t \) hours, population = initial × \( 2^t \). So after 4 hours: \[ \text{Final} = (5.2 \times 10^3) \times 2^4 \] --- **Step 3: Calculate \( 2^4 \)** \[ 2^4 = 16 \] So: \[ \text{Final} = (5.2 \times 10^3) \times 16 \] --- **Step 4: Multiply 5.2 by 16** \[ 5.2 \times 16 = 83.2 \] So: \[ \text{Final} = 83.2 \times 10^3 \] --- **Step 5: Convert to proper scientific notation** \( 83.2 \times 10^3 = 8.32 \times 10^1 \times 10^3 \) \[ = 8.32 \times 10^{1+3} = 8.32 \times 10^4 \] --- **Step 6: Final answer** \[ \boxed{8.32 \times 10^4} \]