/*! 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 54 A student makes the following ar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 54

A student makes the following argument: "I can prove a dollar equals a penny. Since a dime ( 10 cents) is onetenth of a dollar, I can write: $$10 \phi=\$ 0.1$$ Square both sides of the equation. Since squares of equals are equal, $$100 \phi=\$ 0.1 \text { . }$$ What's wrong with the argument?

Short Answer

Expert verified
The logical mistake is introducing an incorrect factor of 100 after squaring both sides.

Step by step solution

01

- Understand the Given Equation

Identify the initial equation provided by the student: \(10 \times 垄0.01 = \$0.1 \).
02

- Simplify the Left Side of the Equation

Understand that 10 pennies (垄0.01 each) indeed equal 10 cents or \( \$ 0.1 \). The initial assertion is correct. So we have: \( 10 \times 0.01 = 0.1 \).
03

- Square Both Sides

Square both sides of the equation given: \( (10 \times 0.01)^2 = 0.1^2 \). This results in: \( (0.1)^2 = 0.01 \).
04

- Simplify Squared Equation

Simplify both sides of the equation: \( (10 \times 0.01)^2 = (0.1)^2 \). This simplifies to: \( 0.01^2 = 0.01 \).
05

- Identify the Logical Mistake

Realize that the squared terms do not equate the original terms. The mistake in the student's argument lies in the incorrect assumption that squaring the equation: \( 100 \times 0.01^2 = 0.1 \)
06

Conclusion - Error Analysis

Identify the mistake: Squaring the terms has introduced an incorrect factor of 100. Therefore, \( 100 \times 0.01 eq \$0.1 \).

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.

Equation Manipulation
Understanding equation manipulation is crucial in mathematics. It involves transforming an equation into a different form without changing its meaning. This can include operations like squaring both sides, adding or subtracting the same value on both sides, and multiplying or dividing both sides by the same number.

When the student tried to square the equation: \( 10 \times 0.01 = 0.1 \), the correct approach would be to consider both sides of the equation individually and then apply the squaring operation uniformly: \( (0.1)^2 = 0.01 \).

Missteps in equation manipulation often occur when we don't apply transformations correctly or uniformly. To avoid these mistakes, always verify each step and check if both sides of the new equation remain equivalent to the original.
Logical Fallacies in Mathematics
Logical fallacies are errors in reasoning that can invalidate arguments and solutions. In the provided problem, the student made a critical logical error by assuming that squaring terms would maintain their equivalence with the original terms.

This type of fallacy falls under a broader category called 'quantitative fallacies.' Here, the student assumed that \( 100 \times 0.01^2 = 0.1 \) would remain accurate after squaring the terms. But in reality, squaring changes the relationships between quantities.

To prevent logical fallacies:
  • Always check the validity of transformations by breaking down each step.
  • Understand how different operations (like squaring) affect equations.
  • Cross-verify the final results with the original values or units.
Unit Conversion in Physics
Unit conversion is an essential skill in physics, helping us translate quantities from one unit to another consistently and accurately. Understanding standard units and maintaining consistency is key.

In the student's problem, there was a mix-up of units and conversion consistency. While \( 10 \times 0.1 \) correctly identifies 1 dollar equals 10 dimes, the student's method went astray when squaring, due to incorrect handling of units. Squaring impacts both the numerical values and the units themselves.

When converting units:
  • Always confirm the units are correctly converted before and after performing mathematical operations.
  • Take note that squaring or other operations may alter both the units and quantities.
  • Double-check consistency across all steps to ensure correctness.

These principles apply across different areas, from simple currency conversions to complex physical equations.

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

A tourist purchases a car in England and ships it home to the United States. The car sticker advertised that the car's fuel consumption was at the rate of 40 miles per gallon on the open road. The tourist does not realize that the U.K. gallon differs from the U.S. gallon: $$\begin{aligned}1 \text { U.K. gallon } &=4.5459631 \text { liters } \\\1 \text { U.S. gallon } &=3.7853060 \text { liters. }\end{aligned}$$ For a trip of 750 miles (in the United States), how many gallons of fuel does (a) the mistaken tourist believe she needs and (b) the car actually require?

In purchasing food for a political rally, you erroneously order shucked medium-size Pacific oysters (which come 8 to 12 per U.S. pint) instead of shucked medium-size Atlantic oysters (which come 26 to 38 per U.S. pint). The filled oyster container delivered to you has the interior measure of \(1.0 \mathrm{~m} \times 12 \mathrm{~cm} \times 20 \mathrm{~cm}\), and a U.S. pint is equivalent to \(0.4732\) liter. By how many oysters is the order short of your anticipated count?

An astronomical unit (AU) is the average distance of Earth from the Sun, approximately \(1.50 \times 10^{8} \mathrm{~km}\). The speed of light is about \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\). Express the speed of light in terms of astronomical units per minute.

Two types of barrel units were in use in the 1920 s in the United States. The apple barrel had a legally set volume of 7056 cubic inches; the cranberry barrel, 5826 cubic inches. If a merchant sells 20 cranberry barrels of goods to a customer who thinks he is receiving apple barrels, what is the discrepancy in the shipment volume in liters?

Grains of fine California beach sand are approximately spheres with an average radius of \(50 \mu \mathrm{m}\) and are made of silicon dioxide. A solid cube of silicon dioxide with a volume of \(1.00 \mathrm{~m}^{3}\) has a mass of \(2600 \mathrm{~kg}\). What mass of sand grains would have a total surface area (the total area of all the individual spheres) equal to the surface area of a cube \(1 \mathrm{~m}\) on an edge?
See all solutions

Recommended explanations on Physics Textbooks

Thermodynamics

Read Explanation

Famous Physicists

Read Explanation

Kinematics Physics

Read Explanation

Nuclear Physics

Read Explanation

Modern Physics

Read Explanation

Fluids

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.