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. Van purchases a DVD player on sale. The initial price was \(\$ 175.90 .\) The price after the discount was \(\$ 149.52\) What was the percent of decrease (discount rate)?

Short answer

Answer: To find the amount of decrease in Marcus' weight, subtract the final amount (180 pounds) from the initial amount (220 pounds). The amount of decrease is 220 pounds - 180 pounds = 40 pounds.

Step-by-step solution

a. Determine whether 220 pounds is the initial amount, final amount, or decrease amount.

220 pounds is Marcus' initial weight before exercising and changing his diet.

b. Determine whether 180 pounds is the initial amount, final amount, or decrease amount.

180 pounds is Marcus' final weight after exercising and changing his diet.

c. Determine whether subtraction is necessary before setting up the proportion for the percent of decrease.

To calculate the percent of decrease, we need to know the amount of decrease. Since Marcus' initial and final weights are given and the decrease represents the difference between these two values, we need to subtract the final amount (180 pounds) from the initial amount (220 pounds) before setting up the proportion.

d. Write a proportion that can be used to find the percent of decrease for Marcus' weight loss.

To find the percent of decrease (D%) for Marcus' weight loss, use the following proportion: Decrease / Original amount = Percent of decrease / 100 %

e. Find the percent of decrease for the price of the DVD player.

First, we need to find the decrease in the price of the DVD player. Decrease = Initial price - Final price = \$175.90 - \$149.52 = \$26.38. Now, set up a proportion to find the discount rate (i.e., percent of decrease) for the price of the DVD player: Decrease / Original amount = Percent of decrease / 100 % \$26.38 / \$175.90 = D% / 100 % To find the discount rate (D%), cross-multiply and divide: D% = \$26.38 * 100 % / \$175.90 ≈ 14.99 % The discount rate (percent of decrease) for the DVD player is approximately 14.99%.

Key concepts

Proportion in mathematics

In mathematics, a proportion is an equation that states that two ratios are equivalent. It's a way of expressing one quantity as a fraction of another and is fundamental in understanding how different values relate to each other. For example, if Marcus's weight decreases from 220 pounds to 180 pounds, we express the amount of change in relation to the original amount to calculate the percentage decrease in his weight.

This concept is crucial when we deal with real-world problems, like adjusting recipes, comparing densities, or calculating growth rates. It provides us with a reliable framework to work out the relative changes between numbers, which is especially useful when the comparison is not immediately obvious. In Marcus's case, the decrease is given by the difference between his initial and final weight (220 - 180), and the proportion will compare this decrease to his original weight.

Weight loss calculation

Weight loss calculation is an application of proportion. It involves finding the difference between the initial and final weight to determine the total loss. With Marcus, his initial weight is 220 pounds and his final weight is 180 pounds, so he has lost 40 pounds. The percent decrease is then found by setting up a proportion with his weight loss and original weight.

To calculate this, we use the formula:
Decrease / Original amount = Percent of decrease / 100 %
For Marcus, this looks like: 40 pounds (decrease) / 220 pounds (original weight) = D% / 100%

By simplifying the proportion, you will find the percentage that represents Marcus's weight loss compared to his original weight. This allows us to quantify changes in weight, giving us a clear and universal way to discuss weight loss progress.

Discount rate calculation

Calculating a discount rate, or the percent decrease in price, is another practical application of proportions. To find the discount given when purchasing an item on sale, like Van's DVD player, you subtract the sale price from the original price, then use this number to establish a proportion compared to the original price.

For instance, with the DVD player:
Decrease = Original price - Sale price = \(175.90 - \)149.52 = \(26.38.

Then, you use this decrease to set up a proportion:
\)26.38 / $175.90 = D% / 100%

By cross-multiplying and dividing, you find the discount rate. This tells you what percent of the original price has been subtracted as a discount. Such calculations are essential in understanding how much money you save when buying discounted items and in comparing deals across different products or stores.