Problem 75 - Elementary Algebra

Question

For exercises \(67-82\), use the five steps and a proportion. In 2010 , about \(2,465,940\) Americans died. Find the number of Americans who died without a will. Round to the nearest hundred. (Source: www.cdc.gov, Jan. 11, 2012) Seven out of ten Americans die without a will. (Source: extension.umd.edu)

Step-by-Step Solution

Answer

About 1,726,200 Americans died without a will in 2010.

Step 1: Identify the Known Values

The total number of Americans who died in 2010 is given as 2,465,940. It is also stated that seven out of ten Americans die without a will.

Step 2: Set Up the Proportion

Create a proportion based on the information that seven out of ten Americans die without a will. Let x be the number of Americans who died without a will. Thus, the proportion is: \[ \frac{7}{10} = \frac{x}{2,465,940} \ \text{Seven out of ten} = \frac{\text{Number of Americans who died without a will}}{\text{Total number of Americans who died}} \]

Step 3: Cross Multiply

To solve for x, cross multiply the proportion: \[ 7 \times 2,465,940 = 10 \times x \ 17,261,580 = 10x \]

Step 4: Solve for x

Divide both sides of the equation by 10 to isolate x: \[ x = \frac{17,261,580}{10} = 1,726,158 \ \text{Number of Americans who died without a will} = 1,726,158 \]

Step 5: Round to the Nearest Hundred

Finally, round the number to the nearest hundred: \[ 1,726,158 \rightarrow 1,726,200 \] So, approximately 1,726,200 Americans died without a will in 2010.

Key Concepts

proportions

Understanding proportions is crucial in many areas of math, especially when comparing two ratios or fractions. Proportions show that two ratios are equal. In our example, we know that 7 out of 10 Americans die without a will. This ratio is set equal to another ratio representing the unknown number of Americans who died without a will out of the total number who died in 2010. This helps us find unknown values by keeping the relationship between quantities consistent.

cross multiplication

Cross multiplication is a method used to solve proportions. It involves multiplying the numerator of one ratio by the denominator of the other ratio. Here's the step-by-step process:
1. Identify the two fractions or ratios in the proportion.
2. Multiply the numerator of the first fraction by the denominator of the second fraction.
3. Multiply the numerator of the second fraction by the denominator of the first fraction.
4. Set the two products equal to each other and solve for the unknown variable.
In our problem, we cross multiply 7 by 2,465,940, and 10 by x.

solving equations

Once we cross-multiply, we end up with an equation to solve. For example, our equation is 7 times 2,465,940 equals 10 times x. Solving this involves a few steps:
1. Perform the multiplication on both sides.
2. Divide both sides by the number paired with x to isolate x.
3. Simplify the result to find the value of x.
This process helps us determine that approximately 1,726,158 Americans died without a will.

rounding numbers

Rounding numbers makes them simpler and easier to work with, especially when exact precision is not necessary. To round to the nearest hundred:
1. Look at the digit in the tens place.
2. If this digit is 5 or more, increase the hundreds place by 1 and change the tens and ones places to 0.
3. If it's less than 5, keep the hundreds place the same and change the tens and ones places to 0.
In our example, 1,726,158 rounds to 1,726,200 because the tens digit (5) is at least 5.