Problem 72 - Algebra 1

Question

The largest mammal, a blue whale, has a weight of \(1.3 \times 10^{5}\) kilograms. The smallest mammal, a pygmy shrew, has a weight of \(2.0 \times 10^{-3}\) kilogram. What is the ratio of the weight of a blue whale to the weight of a pygmy shrew?

Step-by-Step Solution

Answer

The ratio of the weight of a blue whale to a pygmy shrew is \(6.5 \times 10^{7}\).

Step 1: Identify the Given Values

The weight of the blue whale is \(1.3 \times 10^{5}\) kg and the weight of the pygmy shrew is \(2.0 \times 10^{-3}\) kg.

Step 2: Set Up the Ratio

The ratio of the weight of a blue whale to a pygmy shrew is expressed as \(\frac{weight of blue whale}{weight of pygmy shrew}\).

Step 3: Substitute the given values into the Ratio

Substitute the given weights into the ratio. This gives: \(\frac{1.3 \times 10^{5}}{2.0 \times 10^{-3}}\).

Step 4: Perform the Calculation

Perform the division operation to obtain the ratio. When you divide \(1.3 \times 10^{5}\) by \(2.0 \times 10^{-3}\), you get \(6.5 \times 10^{7}\)

Key Concepts

Ratios

Ratios are a way to compare two quantities by showing how many times one value contains or is contained within the other. They can be expressed in various forms such as fractions, decimals, or using the colon ":" symbol. In this exercise, we are tasked with finding the ratio between the weights of two animals: a blue whale and a pygmy shrew.

Understanding ratios is helpful because it allows us to make comparisons in a straightforward manner, even when dealing with very large or small numbers. In scientific contexts, expressing quantities as ratios also simplifies the overall understanding by excluding the need for total amounts or absolute changes.

To set up a ratio, place the first quantity over the second as a fraction. So, if our goal is to compare the weight of the blue whale to the pygmy shrew, we calculate their ratio by setting up the fraction: \(\frac{weight \ of \ blue \ whale}{weight \ of \ pygmy \ shrew}.\) This makes ratios a key tool in various fields such as science, engineering, and finance.

Exponents

Exponents are a mathematical way to express repeated multiplication of a number by itself. In scientific notation, exponents play a crucial role in making very large or very small numbers more manageable. For example, in our problem, the weight of the blue whale is written as \(1.3 \times 10^{5}\) and the weight of the pygmy shrew as \(2.0 \times 10^{-3}\).
Exponents tell us how many times to multiply the base number (in this case, 10) by itself. A positive exponent means the base number is multiplied, whereas a negative exponent indicates division by the base number. Here’s a simple breakdown:

• \(10^{5}\) tells us to multiply 10 by itself five times: \(10 \times 10 \times 10 \times 10 \times 10 = 100,000\)
• \(10^{-3}\) tells us to divide 1 by 10 three times: \(\frac{1}{10 \times 10 \times 10} = 0.001\)

Exponents simplify expressions and make it easier to handle multiplication and division of large-scale or small-scale numbers, like those often found in scientific calculations.

Division of Scientific Notation

When dividing numbers in scientific notation, it helps to break the problem into steps: first dividing the coefficients, and then separately managing the powers of ten. Our goal in this exercise is to determine the ratio of the blue whale’s weight to the pygmy shrew’s weight using scientific notation.
We take \(\frac{1.3 \times 10^{5}}{2.0 \times 10^{-3}}\) and divide it. The division can be split into two parts:

• First, divide the numbers: \(\frac{1.3}{2.0} = 0.65\)
• Next, apply the rule of exponents which states: \(a^{m} \div a^{n} = a^{m-n}\). So for the exponents: \(10^{5} \div 10^{-3} = 10^{5 - (-3)} = 10^{8}\)

Putting it all together, we get \(0.65 \times 10^{8}\). To express this in proper scientific notation, we adjust the coefficient by moving the decimal one place to the right, yielding \(6.5 \times 10^{7}\). This method makes it simple to divide large or small numbers, helping you focus on the process without getting lost in massive equations or calculations.