Problem 57 - Prealgebra

Question

Express each ratio as a unit rate. 24 meters in 4 seconds

Step-by-Step Solution

Answer

The unit rate is 6 meters per second.

Step 1: Understanding the Problem

We are given a ratio of 24 meters in 4 seconds. The goal is to express this ratio as a unit rate. A unit rate is a ratio where the second term is 1.

Step 2: Setting Up the Ratio as a Fraction

We express the given ratio as a fraction: \( \frac{24 \text{ meters}}{4 \text{ seconds}} \). This shows that for every 4 seconds, 24 meters are traveled.

Step 3: Simplifying the Fraction to Find the Unit Rate

To find the unit rate, we need to divide both the numerator and the denominator by the denominator. So, divide 24 by 4 to get 6, and divide 4 by 4 to get 1. This gives us the unit rate as \( \frac{6 \text{ meters}}{1 \text{ second}} \).

Step 4: Interpreting the Unit Rate

The simplified fraction \( 6 \text{ meters per second} \) means that 6 meters are traveled every second, which is the unit rate of the given ratio.

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities by showing the relative size of one quantity to another. They are expressed in the form of "a to b" or as a fraction \( \frac{a}{b} \).

• A ratio of 24 meters to 4 seconds can also be expressed as \( 24:4 \) or \( \frac{24}{4} \).
• The key goal when dealing with ratios is understanding how one amount relates to another.

Ratios are a fundamental concept in math used to solve problems involving proportions, rates, and even scaling figures.
When tackling an exercise involving a ratio, the first step is setting up the ratio correctly either as a ratio or a fraction.

Simplifying Fractions

Simplifying fractions is an essential skill when working with ratios and rates. It involves reducing a fraction to its simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD).

• For the ratio \( \frac{24 \, \text{meters}}{4 \, \text{seconds}} \), we simplify by figuring out the GCD of 24 and 4, which is 4.
• Then, divide both the numerator and the denominator by this GCD: \( \frac{24 \div 4}{4 \div 4} = \frac{6}{1} \).

Simplification reduces the ratio to its most basic form while maintaining the proportion between the two quantities. This step is crucial for converting a ratio into a unit rate effectively.

Understanding Rate of Speed

Rate of speed is a type of unit rate that measures how fast one quantity changes compared to another. When we talk about speed, it usually involves distance and time.

• In this exercise, the distance of 24 meters is traveled over 4 seconds.
• By converting this into a unit rate, you determine the rate of speed: \( 6 \, \text{meters per second} \).

A unit rate like this tells us how many meters are covered in just one second. Understanding this concept aids in solving many real-world problems, such as calculating travel times or comparing speeds. The unit rate provides a clear, simple way to understand the rate at which something is happening.