Problem 26 - Technical Mathematics with Calculus

Question

How much metal will be obtained from 375 tons of ore if the metal is \(10.5 \%\) of the ore?

Step-by-Step Solution

Answer

39.375 tons of metal will be obtained.

Step 1: Convert Percentage to Decimal

To calculate the amount of metal from the ore, convert the percentage of metal in the ore to a decimal by dividing by 100. For 10.5%, divide 10.5 by 100.

Step 2: Calculate the Amount of Metal

Multiply the weight of the ore by the decimal to find the amount of metal. Multiply 375 tons by the decimal obtained in Step 1.

Key Concepts

Converting Percentages to Decimals

When it comes to mathematics, understanding how to convert percentages to decimals is crucial. A percentage represents a part per hundred, and converting it to a decimal involves dividing by 100. This simplifies further calculations, as decimals are easier to work with in operations like multiplication and division.

For example, to convert 10.5% into a decimal, you divide 10.5 by 100, which gives you 0.105. This conversion is the first step in many percentage problems, including those involving quantities and weights. It’s important to ensure the accuracy of this step as it sets the stage for subsequent calculations.

Practical Tip:

• Always move the decimal point two places to the left when converting a percentage to a decimal.
• If the percentage is a whole number, don’t forget to include a zero before the decimal point. For example, 7% becomes 0.07.

Calculating Quantities from Percentages

Calculating quantities from percentages is a common requirement in various fields including chemistry, economics, and statistics. Once you have converted the percentage to a decimal, the next step is to use this decimal to find out how much of a certain quantity is represented by that percentage.

Take, for instance, the textbook exercise where we want to calculate the amount of metal in 375 tons of ore if the metal constitutes 10.5% of the ore. After converting the percentage to a decimal (0.105), multiply this by the total quantity, which gives us the formula: \( \text{{quantity of interest}} = \text{{total quantity}} \times \text{{decimal equivalent of the percentage}} \). Therefore, 375 tons multiplied by 0.105 equals 39.375 tons of metal.

Remember:

• Ensure the total quantity and the percentage are about the same concept or unit before multiplying.
• Double-check your calculations, especially the decimal placement, to avoid significant errors.

Weight Conversions in Mathematics

Weight conversions in mathematics are a fundamental concept, especially when dealing with problems involving different units of measurement. In the context of our ore example, the weight is given in tons, which might need to be converted to other units (such as kilograms or pounds) depending on the context or requirements.

Knowing how to convert weights is important when the final answer needs to be reported in a different unit than the one initially given. There are standard conversion factors that can be used, such as 1 ton equals 2000 pounds or approximately 907.185 kilograms.

Key Conversion Tips:

• Always check the conversion factor to ensure it’s accurate and applicable to the units you’re converting between.
• Conversion factors can also be used in reverse; for example, if you know the number of pounds, you can divide by 2000 to convert to tons.