Problem 37 - Prealgebra

Question

The Golden Gate Bridge in San Francisco is 4200 feet long. The Brooklyn Bridge in New York City is \(\frac{19}{50}\) as long. How long is the Brooklyn Bridge?

Step-by-Step Solution

Answer

The Brooklyn Bridge is 1596 feet long.

Step 1: Understand the fraction

The problem states that the Brooklyn Bridge is \( \frac{19}{50} \) as long as the Golden Gate Bridge. This fraction represents a part of the whole length of the Golden Gate Bridge.

Step 2: Convert the word problem into a mathematical equation

To find how long the Brooklyn Bridge is, you will multiply the total length of the Golden Gate Bridge by the fraction that represents the length of the Brooklyn Bridge. This can be written mathematically as: \( \text{Brooklyn Bridge Length} = 4200 \times \frac{19}{50} \).

Step 3: Perform the multiplication

Calculate the length by multiplying the two numbers: \( 4200 \times \frac{19}{50} \).

Step 4: Simplify the calculation

Start by calculating \( \frac{4200}{50} \) which simplifies to \( 84 \) since \( 4200 \) divided by \( 50 \) equals \( 84 \). Then multiply \( 84 \) by \( 19 \): \( 84 \times 19 = 1596 \).

Step 5: Verify the solution

Reconfirm your calculation: \( 4200 \div 50 = 84 \) and \( 84 \times 19 = 1596 \). This ensures accuracy in the step-by-step calculation.

Key Concepts

Understanding Fractions

Fractions are a way of expressing a portion or part of a whole. They consist of a numerator (top number) and a denominator (bottom number). Understanding fractions is essential in many mathematical contexts, especially in comparing parts of a whole. In our exercise, the fraction \( \frac{19}{50} \) represents the length of the Brooklyn Bridge in relation to the Golden Gate Bridge. Here:

• The numerator 19 indicates part of the total segments.
• The denominator 50 indicates the total number of equal segments the whole (Golden Gate Bridge) is divided into.

This means the Brooklyn Bridge is 19 parts out of every 50 of the Golden Gate Bridge's length. When you see a fraction, think of it as a division of two numbers where the numerator is divided by the denominator to understand its value.

Multiplication with Fractions

Multiplying fractions with whole numbers is a common operation in many mathematical problems, including calculating proportions. Here, you multiply the fraction representing the Brooklyn Bridge's proportion (\( \frac{19}{50} \)) by the total length of the Golden Gate Bridge (4200 feet). Follow these steps:

• Simplify Before Multiplying: Before directly multiplying, see if you can simplify the operations. In our case, first compute \( \frac{4200}{50} = 84 \).
• Execute the Multiplication: Now multiply 84 by 19, giving us 1596.

Simplifying beforehand can make the multiplication process easier and help avoid mistakes. This is a practical technique whenever you deal with fractional multipliers in real-world math problems.

Effective Mathematical Problem Solving

Tackling mathematical problems effectively requires a systematic approach. Let's break down the steps you can follow using our bridge-length problem:

• Read and Understand: Begin by thoroughly reading the problem. Identify the fractions and relationships between the numbers given, like the fraction \( \frac{19}{50} \) between bridge lengths.
• Set Up Equation: Translate the word problem into a mathematical formula, such as \( 4200 \times \frac{19}{50} \) in our case.
• Perform Calculations: Break down the math into smaller steps (like simplifying \( \frac{4200}{50} \) before multiplying the remaining numbers) for accuracy and easier computation.
• Verify Results: Always revisit your steps to confirm correctness, ensuring your answer, like 1596 feet for the Brooklyn Bridge, is reliable.

By taking these steps, students can systematically solve complex problems and reinforce their math comprehension and accuracy.