Let's solve each part of the question step-by-step.
### Part 1: How many times larger is [tex]\(6 \times 10^9\)[/tex] than [tex]\(2 \times 10^5\)[/tex]?
1. Identify the numbers involved:
- The first number is [tex]\(6 \times 10^9\)[/tex].
- The second number is [tex]\(2 \times 10^5\)[/tex].
2. Set up the ratio to determine how many times larger the first number is compared to the second:
[tex]\[
\text{Ratio} = \frac{6 \times 10^9}{2 \times 10^5}
\][/tex]
3. Simplify the fraction:
- Divide the coefficients: [tex]\(\frac{6}{2} = 3\)[/tex].
- Subtract the exponents of 10: [tex]\(10^9 / 10^5 = 10^{9-5} = 10^4\)[/tex].
4. Combine the results:
[tex]\[
\frac{6 \times 10^9}{2 \times 10^5} = 3 \times 10^4
\][/tex]
5. Calculate the numerical value:
[tex]\[
3 \times 10^4 = 30000
\][/tex]
So, [tex]\(6 \times 10^9\)[/tex] is 30,000 times larger than [tex]\(2 \times 10^5\)[/tex].
### Part 2: How many times smaller is [tex]\(5 \times 10^{-5}\)[/tex] than [tex]\(2.5 \times 10^{-3}\)[/tex]?
1. Identify the numbers involved:
- The third number is [tex]\(5 \times 10^{-5}\)[/tex].
- The fourth number is [tex]\(2.5 \times 10^{-3}\)[/tex].
2. Set up the ratio to determine how many times smaller the first number is compared to the second (i.e., how many times larger the second number is than the first):
[tex]\[
\text{Ratio} = \frac{2.5 \times 10^{-3}}{5 \times 10^{-5}}
\][/tex]
3. Simplify the fraction:
- Divide the coefficients: [tex]\(\frac{2.5}{5} = 0.5\)[/tex].
- Subtract the exponents of 10: [tex]\(10^{-3} / 10^{-5} = 10^{-3-(-5)} = 10^{2}\)[/tex].
4. Combine the results:
[tex]\[
\frac{2.5 \times 10^{-3}}{5 \times 10^{-5}} = 0.5 \times 10^2
\][/tex]
5. Calculate the numerical value:
[tex]\[
0.5 \times 10^2 = 0.5 \times 100 = 50
\][/tex]
So, [tex]\(5 \times 10^{-5}\)[/tex] is 50 times smaller than [tex]\(2.5 \times 10^{-3}\)[/tex].
### Summary
1. [tex]\(6 \times 10^9\)[/tex] is 30,000 times larger than [tex]\(2 \times 10^5\)[/tex].
2. [tex]\(5 \times 10^{-5}\)[/tex] is 50 times smaller than [tex]\(2.5 \times 10^{-3}\)[/tex].
