/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Express the ratios in the simple... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 18: Problem 10

Express the ratios in the simplest form. $$2000 \mu \mathrm{m} \text { to } 6 \mathrm{mm}$$

Short Answer

Expert verified
The simplified ratio is 1:3.

Step by step solution

01

Convert Units

Convert both quantities to the same unit before comparing them. We have 2000 micro-meters (\(\mu\mathrm{m}\)) and 6 millimeters (\(\mathrm{mm}\)). First, note that 1 millimeter is equal to 1000 micro-meters. Therefore, convert 6 mm to micro-meters: \[6 \times 1000 = 6000 \, \mu\mathrm{m}\]
02

Set Up the Ratio

Write the ratio of the two quantities in the same units. Now, we compare the quantities both in micro-meters: \[\text{Ratio} = \frac{2000 \, \mu\mathrm{m}}{6000 \, \mu\mathrm{m}}\]
03

Simplify the Ratio

Simplify the fraction by finding the greatest common divisor (GCD) of 2000 and 6000. The GCD of 2000 and 6000 is 2000. Divide both the numerator and the denominator by 2000 to simplify:\[\text{Simplified Ratio} = \frac{2000 \div 2000}{6000 \div 2000} = \frac{1}{3}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
Unit conversion is an important skill to master, as it allows you to compare quantities using the same unit of measurement. In this exercise, we started with two different units: micro-meters (\(\mu \mathrm{m}\)) and millimeters (\(\mathrm{mm}\)). To compare these quantities, we needed to convert them into the same unit.

For unit conversion in this context:
  • 1 millimeter is equal to 1000 micro-meters. This is a standard conversion factor that can be found in metric conversion charts.
  • The process involved multiplying the millimeters by the conversion factor to obtain the measurement in micro-meters: \(6 \times 1000 = 6000 \mu \mathrm{m}\).
By standardizing the units, it becomes straightforward to work with and compare them efficiently. Always remember that converting into the same unit helps prevent errors and simplifies the process.
Greatest Common Divisor
Finding the greatest common divisor (GCD) is essential in simplifying ratios or fractions. The GCD of two numbers is the largest positive integer that divides both numbers without a remainder.

To simplify the ratio \(\frac{2000 \, \mu\mathrm{m}}{6000 \, \mu\mathrm{m}}\), we needed to find the GCD of 2000 and 6000.
  • The GCD of 2000 and 6000 is 2000. This means 2000 is the largest number that can exactly divide both.
  • By dividing the numerator and the denominator by their GCD, the ratio is simplified, reducing it to its simplest form.
This step ensures that our ratio representation is as simple as possible, making it easier to understand and work with. Calculating the GCD can involve listing divisors or using the Euclidean algorithm for larger numbers.
Fraction Simplification
Fraction simplification is a process where the numerator and the denominator are reduced by their greatest common divisor (GCD). This ensures that the fraction or ratio is in its simplest form. In our example, we simplified the ratio of \(\frac{2000 \, \mu\mathrm{m}}{6000 \, \mu\mathrm{m}}\) to \(\frac{1}{3}\).

Here's how this works:
  • Divide both the top (numerator) and the bottom (denominator) by their GCD, which is 2000 in this case.
  • The division \(\frac{2000 \div 2000}{6000 \div 2000}\) results in \(\frac{1}{3}\).
By doing this, we achieve a simplified fraction that represents the same ratio or relationship as the original one, but in a more clear and understandable format. This process is crucial for simplifying problems and seeing relationships in data more clearly.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Set up the general equations from the given statements. The electric resistance \(R\) of a wire varies inversely as the square of its diameter \(d\).

Find the required ratios. The percent grade of a road is the ratio of vertical rise to the horizontal change in distance (expressed in percent). If a highway rises \(75 \mathrm{m}\) for each \(1.2 \mathrm{km}\) along the horizontal, what is the percent grade?

Find the required ratios. The Mach number of a moving object is the ratio of its speed to the speed of sound \((1200 \mathrm{km} / \mathrm{h}) .\) Find the Mach number of a military jet that flew at \(7200 \mathrm{km} / \mathrm{h}\)

Solve the given applied problems involving variation. The tangent of the proper banking angle \(\theta\) of the road for a car making a turn is directly proportional to the square of the car's velocity \(v\) and inversely proportional to the radius \(r\) of the turn. If \(7.75^{\circ}\) is the proper banking angle for a car traveling at \(20.0 \mathrm{m} / \mathrm{s}\) around a turn of radius \(300 \mathrm{m},\) what is the proper banking angle for a car traveling at \(30.0 \mathrm{m} / \mathrm{s}\) around a turn of radius \(250 \mathrm{m} ?\) See Fig. 18.9.

Solve the given applied problems involving variation. The force \(F\) between two parallel wires carrying electric currents is inversely proportional to the distance \(d\) between the wires. If a force of \(0.750 \mathrm{N}\) exists between wires that are \(1.25 \mathrm{cm}\) apart, what is the force between them if they are separated by \(1.75 \mathrm{cm} ?\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.