Scientific Comparison

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Answer Key & Explanations

Scientific Comparison · Grade 7 · Worksheet 1

1. Emma is comparing the populations of four cities, each written in scientific notation on a chart: City A: 2.5×10⁵, City B: 8.0×10⁴, City C: 1.5×10⁶, City D: 4.5×10⁵. Help Emma arrange the cities in order from smallest to largest population. Write your answer as a sequence of letters separated by commas (e.g., A, B, C, D). Answer: B, A, D, C Solution: List the numbers: City A = 2.5×10⁵, City B = 8.0×10⁴, City C = 1.5×10⁶, City D = 4.5×10⁵. Compare exponents: 10⁴ is the smallest exponent, then 10⁵, then 10⁶ is the largest.
   Full step-by-step solution

   Step 1: List the numbers: City A = 2.5×10⁵, City B = 8.0×10⁴, City C = 1.5×10⁶, City D = 4.5×10⁵. Step 2: Compare exponents: 10⁴ is the smallest exponent, then 10⁵, then 10⁶ is the largest. Step 3: City B has exponent 10⁴, so it is the smallest. Step 4: City A and City D both have exponent 10⁵. Compare coefficients: 2.5 < 4.5, so City A comes before City D. Step 5: City C has exponent 10⁶, so it is the largest. Step 6: Order from smallest to largest: B, A, D, C. The answer is B, A, D, C.

2. Emma is comparing the populations of three cities written in scientific notation on a bar graph. The bar for City X reaches a height representing 6.5 × 10⁵ people. The bar for City Y reaches a height representing 3.2 × 10⁶ people. The bar for City Z reaches a height representing 8.9 × 10⁴ people. Order the cities from smallest population to largest population. Answer: City Z, City X, City Y Solution: Write each population as a standard number to compare easily. City X: 6.5 × 10⁵ = 6.5 × 100,000 = 650,000 City Y: 3.2 × 10⁶ = 3.2 × 1,000,000 = 3,200,000 City Z: 8.9 × 10⁴ = 8.9 × 10,000 = 89,000 Order from smallest to largest.
   Full step-by-step solution

   Step 1: Write each population as a standard number to compare easily. City X: 6.5 × 10⁵ = 6.5 × 100,000 = 650,000 City Y: 3.2 × 10⁶ = 3.2 × 1,000,000 = 3,200,000 City Z: 8.9 × 10⁴ = 8.9 × 10,000 = 89,000 Step 2: Order from smallest to largest. 89,000 (City Z) < 650,000 (City X) < 3,200,000 (City Y) Step 3: Write the answer using the city names. Smallest to largest: City Z, City X, City Y. The answer is City Z, City X, City Y.

3. Order: 8.2×10⁷, 9.1×10⁶, 7.5×10⁸, 6.3×10⁷ Answer: 9.1×10⁶, 6.3×10⁷, 8.2×10⁷, 7.5×10⁸ Solution: List the numbers with their exponents: 8.2×10⁷ (exponent 7), 9.1×10⁶ (exponent 6), 7.5×10⁸ (exponent 8), 6.3×10⁷ (exponent 7).
   Full step-by-step solution

   Step 1: List the numbers with their exponents: 8.2×10⁷ (exponent 7), 9.1×10⁶ (exponent 6), 7.5×10⁸ (exponent 8), 6.3×10⁷ (exponent 7). Step 2: Order by exponents from smallest to largest: exponent 6 is smallest, then exponent 7 (two numbers), then exponent 8 is largest. Step 3: For the two numbers with exponent 7 (8.2×10⁷ and 6.3×10⁷), compare coefficients: 6.3 < 8.2, so 6.3×10⁷ comes before 8.2×10⁷. Step 4: Final order from least to greatest: 9.1×10⁶, 6.3×10⁷, 8.2×10⁷, 7.5×10⁸. The answer is 9.1×10⁶, 6.3×10⁷, 8.2×10⁷, 7.5×10⁸.

4. 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.

5. 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.

6. A scientist is comparing the populations of two cities. City A has a population of 3.4 × 10^6 people, while City B has a population of 4.1 × 10^5 people. How many times larger is the population of City A compared to City B? Answer: 8.29 Solution: To find how many times larger City A's population is compared to City B, we need to divide the population of City A by the population of City B. Write down the populations. Population of City A = 3.4 × 10^6 Population of City B = 4.1 × 10^5 Set up the division.
   Full step-by-step solution

   To find how many times larger City A's population is compared to City B, we need to divide the population of City A by the population of City B. Step 1: Write down the populations. Population of City A = 3.4 × 10^6 Population of City B = 4.1 × 10^5 Step 2: Set up the division. Times larger = (Population of City A) / (Population of City B) = (3.4 × 10^6) / (4.1 × 10^5) Step 3: Separate the numbers and the powers of 10. We can rewrite the division as: (3.4 / 4.1) × (10^6 / 10^5) Step 4: Divide the numbers. 3.4 / 4.1 = 34 / 41 (multiplying numerator and denominator by 10 to make it easier) 34 ÷ 41 ≈ 0.829268... (we will round this later) Step 5: Divide the powers of 10. 10^6 / 10^5 = 10^(6 - 5) = 10^1 = 10 Step 6: Multiply the results from Step 4 and Step 5. 0.829268... × 10 = 8.29268... Step 7: Round the final answer to two decimal places, as is common for such comparisons. 8.29268... ≈ 8.29 Therefore, the population of City A is approximately 8.29 times larger than the population of City B.

7. Order: 7.2×10⁶, 4.8×10⁷, 9.3×10⁵, 6.1×10⁶ Answer: 9.3×10⁵, 6.1×10⁶, 7.2×10⁶, 4.8×10⁷ Solution: List the numbers: 7.2×10⁶, 4.8×10⁷, 9.3×10⁵, 6.1×10⁶ Compare exponents: 10⁵ (smallest), 10⁶ (two numbers), 10⁷ (largest) For the two numbers with exponent 10⁶: compare coefficients 6.1 and 7.2 → 6.1 < 7.2 Order from smallest to largest: 9.3×10⁵, 6.1×10⁶, 7.2×10⁶, 4.8×10⁷ The answer is 9.3×10⁵,…
   Full step-by-step solution

   Step 1: List the numbers: 7.2×10⁶, 4.8×10⁷, 9.3×10⁵, 6.1×10⁶ Step 2: Compare exponents: 10⁵ (smallest), 10⁶ (two numbers), 10⁷ (largest) Step 3: For the two numbers with exponent 10⁶: compare coefficients 6.1 and 7.2 → 6.1 < 7.2 Step 4: Order from smallest to largest: 9.3×10⁵, 6.1×10⁶, 7.2×10⁶, 4.8×10⁷ The answer is 9.3×10⁵, 6.1×10⁶, 7.2×10⁶, 4.8×10⁷.

8. (8.4 × 10⁷) ÷ (2.1 × 10⁴) = ? Answer: 4000 Solution: Separate the coefficients and exponents: (8.4 ÷ 2.1) × (10⁷ ÷ 10⁴) Divide the coefficients: 8.4 ÷ 2.1 = 4 Divide the exponents: 10⁷ ÷ 10⁴ = 10³ (subtract exponents: 7 - 4 = 3) Combine the results: 4 × 10³ = 4000 The answer is 4000.
   Full step-by-step solution

   Step 1: Separate the coefficients and exponents: (8.4 ÷ 2.1) × (10⁷ ÷ 10⁴) Step 2: Divide the coefficients: 8.4 ÷ 2.1 = 4 Step 3: Divide the exponents: 10⁷ ÷ 10⁴ = 10³ (subtract exponents: 7 - 4 = 3) Step 4: Combine the results: 4 × 10³ = 4000 The answer is 4000.