Question

A casting initially weighing 115 lb has \(22.0 \%\) of its material machined off. What is its final weight?

Short answer

The final weight of the casting is 89.7 lb.

Step-by-step solution

Calculate the Weight Removed

To find out how much weight is removed from the casting, calculate 22.0% of the initial weight. Multiply the initial weight of 115 lb by 0.22, the decimal equivalent of 22.0%.

Determine Final Weight

Subtract the weight removed from the initial weight to find the final weight of the casting. This is the original weight minus the product of the original weight and 0.22.

Key concepts

Weight Loss Calculation

When we talk about weight loss calculation, whether in terms of fitness or material reduction as in this example with a casting, it's essential to understand certain steps. First, determine the initial weight of the object, which in this case is 115 lb. Next, identify the percentage of weight to be removed or ‘lost’, here given as 22.0%.

To calculate the actual amount of weight being removed, we convert the percentage to a decimal by dividing by 100, thus 22.0% becomes 0.22. Then, multiply the initial weight by this decimal. This multiplication gives us the weight of the material that will be machined off the casting. It's a simple yet a key calculation required in many practical scenarios.

Percentage Problems

Dealing with percentage problems is a common requirement in mathematics as well as in real-life scenarios such as calculating discounts, interest rates, or weight loss percentages. The concept always follows the same principle: percentages are essentially a fraction of 100. To navigate these problems easily, remember to first convert the percentage to a decimal by dividing by 100.

For instance, to solve our current problem involving the casting, you need to find 22.0% of 115 lb. After converting 22.0% to a decimal (0.22), you then multiply it by the initial weight to find the amount of material removed: \( 115 \times 0.22 = 25.3 \text{l\text{b}} \). Understanding the connection between percentages and decimals is vital for these calculations and mastering this can make solving these problems much simpler.

Basic Algebra

At the heart of percentage problems like our example with the casting is basic algebra. Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is a powerful tool that allows us to solve for unknowns, perform operations with variables, and understand the relationships between quantities. In the context of our weight loss calculation, basic algebra is used to manipulate the expression for the final weight of the casting. Once you have the weight to be removed, you use subtraction: \( \text{initial weight} - (\text{initial weight} \times \text{decimal equivalent of the percentage})\).

Algebraically, this can be represented as: \( W_{\text{final}} = W_{\text{initial}} - (W_{\text{initial}} \times P) \), where \( W_{\text{final}} \) is the final weight, \( W_{\text{initial}} \) is the initial weight, and \( P \) is the decimal equivalent of the percentage. In the given exercise, after determining the removed weight (25.3 lb), we simply subtract it from the initial weight to determine the final weight: \( 115 \text{ lb} - 25.3 \text{ lb} = 89.7 \text{ lb} \). Algebra helps to turn real-world problems into equations that can be easily solved.