Question

In 2010, there were about 4 x 107 people living in California and about 1 x 106 people living in Rhode Island.
How many times larger was the population of California than the population of Rhode Island?







Pablo divided 6 x 108 by 2 x 10-4, and got a final answer of 3 x 10-2.

Explain what mistake Pablo made, and give the correct solution.

Answer 1

To find out how many times larger the population of California was compared to Rhode Island, we can set up the following division problem using scientific notation:

Population of California: \( 4 \times 10^7 \)
Population of Rhode Island: \( 1 \times 10^6 \)

We can calculate how many times larger California's population is by dividing the two populations:

\[
\text{Ratio} = \frac{4 \times 10^7}{1 \times 10^6}
\]

When performing the division in scientific notation, we divide the coefficients and subtract the exponents:

\[
\text{Ratio} = \frac{4}{1} \times 10^{7 - 6} = 4 \times 10^1 = 4 \times 10 = 40
\]

So, the population of California was **40 times larger** than the population of Rhode Island.

---

Now for Pablo's calculation, we analyze the division of \( 6 \times 10^8 \) by \( 2 \times 10^{-4} \):

Pablo's calculation was:

\[
\frac{6 \times 10^{8}}{2 \times 10^{-4}}
\]

1. **Dividing the coefficients**:
\[
\frac{6}{2} = 3
\]

2. **Subtracting the exponents** (when dividing, you subtract the exponent in the denominator from the exponent in the numerator):
\[
10^{8 - (-4)} = 10^{8 + 4} = 10^{12}
\]

So, the correct calculation should have been:

\[
3 \times 10^{12}
\]

Pablo mistakenly concluded that the result was \( 3 \times 10^{-2} \). The mistake came from not properly handling the subtraction of the exponent. The correct answer is:

\[
3 \times 10^{12}
\]