Question

Write a division problem in which two numbers written in scientific notation have a quotient of 5 x 10-5.






Show how you would solve it.

Answer 1

To create a division problem in which the quotient is \( 5 \times 10^{-5} \), we can denote the two numbers in scientific notation as \( a \) and \( b \), such that:

\[
\frac{a}{b} = 5 \times 10^{-5}
\]

Let's define \( a \) and \( b \) as follows:

Let \( a = 1 \times 10^{-4} \) (which is \( 0.0001 \)) and let \( b \) be the number we need to find.

Now, we need to determine \( b \) such that:

\[
\frac{1 \times 10^{-4}}{b} = 5 \times 10^{-5}
\]

To find \( b \), we can rearrange the equation:

\[
b = \frac{1 \times 10^{-4}}{5 \times 10^{-5}}
\]

Next, we perform the division on the right-hand side:

1. Divide the coefficients:
\[
\frac{1}{5} = 0.2
\]

2. Subtract the exponents from the powers of ten:
\[
10^{-4} \div 10^{-5} = 10^{-4 - (-5)} = 10^{-4 + 5} = 10^{1}
\]

Putting it all together:

\[
b = 0.2 \times 10^{1} = 0.2 \times 10 = 2
\]

Thus, we have defined \( b = 2 \).

Now to check that our values satisfy the original equation:

- We have \( a = 1 \times 10^{-4} \) and \( b = 2 \).
- Calculate \( \frac{a}{b} \):

\[
\frac{1 \times 10^{-4}}{2} = 0.5 \times 10^{-4}
\]

We can rewrite \( 0.5 \) in scientific notation:

\[
0.5 \times 10^{-4} = 5 \times 10^{-5} \quad \text{(since moving the decimal point one place to the right increases the exponent by 1)}
\]

Thus, we confirm:

\[
\frac{1 \times 10^{-4}}{2} = 5 \times 10^{-5}
\]

Therefore, we can conclude that a division problem where two numbers are \( 1 \times 10^{-4} \) and \( 2 \) yields the appropriate quotient:

\[
\frac{1 \times 10^{-4}}{2} = 5 \times 10^{-5}
\]