Problem 89 - Elementary Algebra

Question

Predict the number of 240 death penalty sentences that were overturned on appeal. The most far-reaching study of the death penalty in the United States has found that two out of three sentences were overturned on appeal, mostly because of serious errors by incompetent defense lawyers or overzealous police officers and prosecutors who withheld evidence. (Source: www .nytimes.com, June 12, 2009)

Step-by-Step Solution

Answer

160 death penalty sentences were overturned on appeal.

Step 1: Understand the problem

First, recognize that we need to predict the number of death penalty sentences overturned on appeal from a total of 240 sentences. The information given states that two out of three sentences, or \( \frac{2}{3} \), were overturned.

Step 2: Convert the fraction to a proportion

Convert the given fraction (two out of three) into a proportion. \( \frac{2}{3} \) can be represented as approximately 0.6667. This proportion indicates the part of the total number of sentences (240) that were overturned on appeal.

Step 3: Calculate the number overturned

Multiply the total number of sentences (240) by the proportion (0.6667). \( 240 \times 0.6667 = 160 \).

Key Concepts

proportion calculations

When dealing with proportion calculations, we translate parts of a whole into a mathematical form that is easy to work with. In this exercise, the given value was 'two out of three sentences' overturned on appeal. This fraction, \(\frac{2}{3}\), represents the proportion of total cases. To predict the number of overturned sentences out of 240 total sentences, we need to use this proportion.

By converting \(\frac{2}{3}\) into a decimal (0.6667), we can easily use it for further calculations. Essentially, what we're doing is finding out what 66.67% of the total 240 sentences would be. Multiplying the total number of sentences (240) by our proportion (0.6667) gives us the predicted amount of overturned sentences. Thus, 240 sentences x 0.6667 yields approximately 160 overturned sentences. This method can be applied to a wide range of problems, making proportion calculations a crucial math skill.

fraction to decimal conversion

Converting fractions to decimals is a key part of many math problems, including this one. Fractions are ratios defined by a numerator (top number) and a denominator (bottom number). Here, the given fraction is \(\frac{2}{3}\). To convert this fraction to a decimal, simply divide the numerator by the denominator.

In this example, dividing 2 by 3 yields approximately 0.6667. This decimal form can then be used in various calculations involving a proportion of a whole.

Using decimals often makes it easier to work with large numbers and apply multipliers in prediction calculations, like finding how many death penalty sentences out of 240 were overturned. It’s also more intuitive for mental math and can serve to cross-verify the predictive results.

problem-solving steps

Breaking down problems into actionable steps helps in understanding and solving them accurately. Here's a quick guide based on our example:

• Step 1: Understand the problem - Recognize what is being asked. Here, we need to predict how many death penalty sentences out of 240 would be overturned based on a given rate.
• Step 2: Convert the fraction to a decimal - The rate of overturned sentences is given as \(\frac{2}{3}\). Converting this to a decimal (0.6667) makes the next steps easier.
• Step 3: Calculate the number overturned - Multiply the total number of cases (240) by the decimal (0.6667). This results in 160 overturned sentences.

This process can be utilized for various similar problems. By understanding the issue, converting necessary fractions, and then calculating the desired value, we systematically solve problems, enhancing our problem-solving skills.