Grade 8 Math: Verify True Statements About Scientific Notation Ratios

Question

Select all of the true statements: 1. $1 \times 10^{-4}$ is 0.05 times as much as $2 \times 10^{-3}$. 2. $3 \times 10^{5}$ is 60 times as much as $5 \times 10^{4}$. 3. $1 \times 10^{-1}$ is 2,500 times as much as $4 \times 10^{-5}$. 4. $2 \times 10^{-4}$ is 4 times as much as $5 \times 10^{-3}$.

Accepted Answer

Statements 1, 3 are true; Statements 2, 4 are false

Answer

Step 1: For each statement, calculate the ratio of the first number to the second number to verify the relationship. Step 2: Check Statement 1: Calculate $\frac{1 \times 10^{-4}}{2 \times 10^{-3}} = \frac{1}{2} \times 10^{-4 - (-3)} = 0.5 \times 10^{-1} = 0.05$. This matches the claim, so Statement 1 is true. Step 3: Check Statement 2: Calculate $\frac{3 \times 10^{5}}{5 \times 10^{4}} = \frac{3}{5} \times 10^{5-4} = 0.6 \times 10^{1} = 6$. The claim says 60, which does not match, so Statement 2 is false. Step 4: Check Statement 3: Calculate $\frac{1 \times 10^{-1}}{4 \times 10^{-5}} = \frac{1}{4} \times 10^{-1 - (-5)} = 0.25 \times 10^{4} = 2500$. This matches the claim, so Statement 3 is true. Step 5: Check Statement 4: Calculate $\frac{2 \times 10^{-4}}{5 \times 10^{-3}} = \frac{2}{5} \times 10^{-4 - (-3)} = 0.4 \times 10^{-1} = 0.04$. The claim says 4, which does not match, so Statement 4 is false.