Problem 20 - Technical Mathematics with Calculus

Question

Find: \(12.5 \%\) of 72.0 gal.

Step-by-Step Solution

Answer

12.5% of 72.0 gal is 9.0 gal.

Step 1: Convert Percentage to Decimal

To find a percentage of a number, first convert the percentage to a decimal by dividing by 100. So, 12.5% becomes 0.125.

Step 2: Multiply by the Quantity

Multiply the decimal form of the percentage by the quantity you want to find the percentage of. Calculate 0.125 times 72.0 gal.

Step 3: Perform the Multiplication

Perform the multiplication to find the answer: 0.125 * 72.0 = 9.0. So, 12.5% of 72.0 gal is 9.0 gal.

Key Concepts

Converting Percentages to Decimals

Understanding how to convert percentages to decimals is an essential skill in mathematics and it's surprisingly straightforward. A percentage represents a number out of 100. To convert a percentage to a decimal, you simply divide the percentage value by 100.

For instance, to convert 12.5% to a decimal, divide 12.5 by 100, which gives you 0.125. The process of dividing by 100 can also be easily done by moving the decimal point two places to the left. Thus, 12.5 becomes 0.125 when it is expressed as a decimal. It's crucial to remember this basic technique, as it's the first step to working with percentages in various mathematical problems.

Multiplying Decimals

Once you have converted your percentage to a decimal, you may need to use it to find the percentage of a quantity, which involves multiplying decimals. When multiplying decimals, you align the numbers on the right-hand side, ignoring the decimal points, and multiply as you would with whole numbers. After that, you need to count the total number of decimal places in both of the original numbers. The product of your multiplication will have that many decimal places.

For example, to multiply 0.125 (which has three decimal places) by 72.0 (which has one decimal place), we calculate as if they were whole numbers and then place the decimal in the product such that there are four places after the decimal (the combined number of decimal places in both original numbers). This is why the solution to 0.125 times 72.0 results in 9.0.

Mathematical Word Problems

Mathematical word problems allow you to apply your knowledge of concepts such as percentages and decimals in real-life situations. These problems often require several steps to solve, including understanding the problem, converting terms to mathematical expressions, and performing the necessary calculations.

Let's break down a word problem using our previous example. You might see a question such as 'If you have 72.0 gallons of paint, what is 12.5% of that amount?' First, identify the key information: you need to find 12.5% of 72.0 gallons. Next, convert 12.5% to decimal form to make the problem manageable. Then, multiply the decimal by the quantity (in this case, gallons of paint) to find the answer. Word problems often require careful reading and analysis, but with practice, you can develop a systematic approach to solving them efficiently.