Problem 24 - Prealgebra

Question

Use the following information. There are approximately 3.28 feet in 1 meter. Write a proportion that could be used to find the distance in feet of the 110 -meter dash.

Step-by-Step Solution

Answer

The distance is approximately 360.8 feet.

Step 1: Understand the Relationship

We are given that 1 meter is equivalent to approximately 3.28 feet. This relationship will help us set up a proportion to convert meters to feet.

Step 2: Set Up the Proportion

Since 1 meter is equal to 3.28 feet, we can express this as a ratio: \[ \frac{1 \text{ meter}}{3.28 \text{ feet}} = \frac{110 \text{ meters}}{x \text{ feet}} \] where \( x \) represents the unknown distance in feet for 110 meters.

Step 3: Cross Multiply to Solve for x

Use cross multiplication to solve for \( x \). This gives us:\[ 1 \times x = 3.28 \times 110 \] which simplifies to:\[ x = 3.28 \times 110 \]

Step 4: Calculate the Conversion

Now perform the multiplication: \[ x = 3.28 \times 110 = 360.8 \] So, 110 meters is approximately 360.8 feet.

Key Concepts

Unit Conversion

Unit conversion is the process of converting a measure from one unit to another without changing the quantity it represents. It is crucial in science, mathematics, and daily life. Understanding how to switch from one unit to another allows for better interpretation and communication of data. For instance, if you have measurements in meters but need them in feet, you’ll perform a unit conversion.

• Identify the conversion factor: In this example, the conversion factor between meters and feet is 3.28. This means that 1 meter equals approximately 3.28 feet.
• Establish a proportion or use multiplication to convert the units: Multiply the distance in meters by the conversion factor to get the equivalent in feet.

By mastering unit conversion, one can solve problems involving different measurement units effectively and accurately, facilitating better understanding across various contexts.

Cross Multiplication

Cross multiplication is a technique used to solve equations involving fractions or proportions. It is particularly useful when dealing with two ratios set equal to each other, as in our meter-to-feet conversion example.To cross multiply:

• Arrange the terms in the proportion, where the ratios are written as fractions.
• Multiply the diagonal terms across the equals sign. This means multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa.
• Set these products equal to each other: If you have a proportion \( \frac{a}{b} = \frac{c}{d} \), then \( a \times d = b \times c \).

In our example, cross multiplying helps find the number of feet equivalent to 110 meters by equating \( 1 \times x \) and \( 3.28 \times 110 \). This step is crucial for solving proportions.

Ratio

A ratio is a way to compare two quantities of the same kind by dividing one by the other. It describes how much of one thing there is compared to another. Ratios can be expressed in several forms such as 3 to 4, 3:4, or \( \frac{3}{4} \).In the context of converting meters to feet:

• The ratio between meters and feet is given by \( 1:3.28 \), meaning each meter has an equivalent length of approximately 3.28 feet.
• Ratios are useful in setting up a proportion for converting measurements. In our case, \( \frac{1 \, \text{meter}}{3.28 \, \text{feet}} = \frac{110 \, \text{meters}}{x \, \text{feet}} \).

Understanding ratios allows one to construct and manipulate proportions necessary for finding unknown values in conversions and comparisons.

Meters to Feet

The conversion from meters to feet is common in many fields, especially those involving international collaboration, as different countries use different measurement systems.To convert meters to feet, use the known equivalence that 1 meter is approximately 3.28 feet:

• Multiply the number of meters by 3.28 to get the equivalent distance in feet.
• For example, to convert 110 meters to feet, calculate \( 110 \times 3.28 = 360.8 \) feet.

Understanding this conversion can help avoid misunderstandings and ensure accurate communication of measurements in academic, professional, and everyday contexts.