Question

By exercising and changing his diet, Marcus drops from 220 pounds to 180 pounds. a. Is 220 an initial amount, a final amount, or an decrease amount? b. Is 180 an initial amount, a final amount, or an decrease amount? c. With the information given, to find the percent of decrease, do you need to subtract prior to setting up the proportion? Why or why not? d. Write a proportion that can be used to find the percent of decrease. Risa's gross two-week salary is \(\$ 1756.25 .\) If her net pay is \(\$ 1334.75,\) what is the percent of decrease?

Short answer

Answer: The percent of decrease in Risa's salary after deductions is approximately 24%.

Step-by-step solution

a. Identifying 220 as the initial, final, or decrease amount

In this case, Marcus starts at 220 pounds, so 220 is the initial amount.

b. Identifying 180 as the initial, final, or decrease amount

After Marcus exercises and changes his diet, he weighs 180 pounds. So, 180 is the final amount.

c. Subtraction before setting up the proportion

To calculate the percent of decrease, we need to subtract the initial amount from the final amount to find the decrease amount. Then we can set up a proportion to find the percent of decrease.

d. Proportion for the percent of decrease

To calculate the percent of decrease, we can use the proportion: \[\frac{\text{decrease}}{\text{initial amount}} = \frac{\text{percent of decrease}}{100}\] In Marcus's case, the decrease amount is \((220 - 180) = 40\) pounds. Now, we can set up the proportion: \[\frac{40}{220}=\frac{x}{100}\]

e. Risa's salary percent of decrease

First, we need to find the decrease amount in Risa's salary: \(\$1756.25 - \$1334.75 = \$421.50\). Now, we can set up the proportion: \[\frac{\text{decrease}}{\text{initial amount}} = \frac{\text{percent of decrease}}{100}\] \[\frac{421.50}{1756.25}=\frac{x}{100}\] To solve for x, cross-multiply and divide: \[x = \frac{100 \times 421.50}{1756.25} \approx 24\%\] The percent of decrease in Risa's salary after deductions is approximately 24%.

Key concepts

Initial and Final Amounts

Understanding the initial and final amounts in mathematical problems is crucial for analyzing percent changes. An initial amount refers to the original value before any changes take place. In our example, Marcus's weight before his diet and exercise regime was 220 pounds, thus making it the initial amount.

On the other hand, the final amount signifies the value after the change occurs. Post his weight loss journey, Marcus now tips the scales at 180 pounds, representing the final amount. Recognizing these two amounts allows us to evaluate the extent of change Marcus experienced.

Proportion in Mathematics

A proportion in mathematics expresses that two ratios or fractions are equal to each other. It's a statement that two quantities are in balance. For instance, if we have a portion of the cake and express it as a fraction of the entire cake, a proportion would tell us how that fractional part compares to another fraction, perhaps of a different cake.

In the context of percent of decrease, we are using proportions to compare the amount of decrease to the initial amount. The proportion helps us find the 'part per hundred', which is what percentage means.

Percent Change Calculation

Calculating the percent change is a way to express the degree of change over time. To calculate the percent of decrease specifically, you subtract the final amount from the initial amount to get the amount of decrease. Then, you divide this number by the initial amount to find the ratio describing the decrease. Finally, you multiply this ratio by 100 to convert it to a percentage.

Using our previous example with Marcus and Risa, we applied this approach to determine the percentage by which Marcus's weight and Risa's salary decreased. The percent change formula is pivotal in various fields, including finance, science, and health to measure growth or reduction.

Weight Loss Percentage

Understanding weight loss percentage is beneficial not just in mathematical contexts, but also in real-life situations, such as personal health and fitness. Weight loss percentage is calculated similarly to other percentage decreases. It is the ratio of pounds lost to the original weight, converted into a percent format. This measure gives a clear indication of the progress someone has made in their weight loss journey.

For example, Marcus, who started at 220 pounds and reduced his weight to 180 pounds, shows a significant percentage decrease in weight. This percentage is more insightful than just stating the pounds lost because it puts the amount of weight lost in relation to the starting weight, offering a proportional perspective to the change.