Problem 89 - Elementary Algebra

Question

In November, 2011, North Dakota natural gas production was \(15,635,813\) million cubic feet. Because of a shortage of gas processing plants and other infrastructure, more than one-third of the gas is burned off or "flared" instead of being processed and sold. Find the minimum amount of natural gas that was burned off in November 2011. Round to the nearest million cubic feet. (Sources: www.businessweek.com, Jan. 13, 2012; www.dmr.nd.gov, 2011)

Step-by-Step Solution

Answer

At least 5,212 million cubic feet of natural gas was burned off in November 2011.

Step 1 - Identify the Proportion Burned Off

Identify the fraction of natural gas that is burned off. More than one-third of the gas is flared. One-third can be expressed as the fractional value \(\frac{1}{3} \).

Step 2 - Convert to Decimal

Convert the fraction \(\frac{1}{3}\) into a decimal for easier calculation. \(\frac{1}{3} = 0.3333... \) (approximately).

Step 3 - Calculate Minimum Amount Burned Off

Multiply the total production by \(\frac{1}{3}\) to find the minimum amount burned off. Using the total production of 15,635,813 million cubic feet: \[ 15,635,813 \times 0.3333 \approx 5,211,937.32 \]

Step 4 - Round to the Nearest Million

Round the result to the nearest million cubic feet. \[ 5,211,937.32 \approx 5,212,000 \]

Key Concepts

Fraction to Decimal Conversion

Understanding how to convert a fraction to a decimal is a fundamental mathematical skill. When you have a fraction like \(\frac{1}{3}\), you're essentially being asked to divide the numerator (1) by the denominator (3).
We know that: \(\frac{1}{3} = 0.3333...\text{ (and it continues indefinitely)}\).
This is called a repeating decimal because the number 3 repeats endlessly. For practical purposes, we often round repeating decimals to a certain number of decimal places, such as four decimal places, resulting in approximately 0.3333. Using such approximations can make calculations easier while remaining largely accurate.

Multiplication

Multiplication is a basic arithmetic operation that combines equal groups to find the total number of items. In the given problem, we needed to find out how much gas was burned off. The problem states that more than one-third of the gas is flared. To find the exact amount, you multiply the total production by this fraction.
Here's how you can do it step by step:

• First, convert fraction to decimal (as discussed above): \(\frac{1}{3} = 0.3333\).
• Now, multiply the decimal by the total amount of gas produced: \[ 15,635,813 \times 0.3333 \].
• Perform the multiplication: \[ 15,635,813 \times 0.3333 \approx 5,211,937.32 \].

This means around 5,211,937.32 million cubic feet of gas was flared.

Rounding Numbers

Rounding numbers is an essential skill for simplifying complex calculations and making numeric data easier to understand. After calculating the amount of gas burned off, we obtained a value of approximately 5,211,937.32 million cubic feet.
To make this number more digestible, we round to the nearest whole unit, which, in this case, is the nearest million cubic feet.

• Identify the number in the million's place (5,211,937.32).
• Look at the digit immediately to the right (in the hundred thousands place): 9.
• If this digit is 5 or greater, we round up the million's place by one. If it is less than 5, we keep the million's place as it is.

Since 9 is more than 5, we round 5,211,937.32 to 5,212,000. So, the minimum amount of natural gas burned off in November 2011, rounded to the nearest million, is approximately 5,212 million cubic feet.