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Chapter 1: Problem 31

Carry out these conversions: (a) \(22.6 \mathrm{~m}\) to decimeters, (b) \(25.4 \mathrm{mg}\) to kilograms.

Short Answer

Expert verified
(a) 226 dm, (b) 0.0000254 kg

Step by step solution

01

Understanding the Metric System

The metric system is based on decomposing values using powers of ten. Conversions are made by multiplying or dividing by coefficients like 10, 100, or 1000.
02

Convert Meters to Decimeters

To convert from meters to decimeters, multiply by 10, because 1 meter is equal to 10 decimeters.For example, \(22.6 \text{ meters} = 22.6 \times 10 = 226 \text{ decimeters}\).
03

Convert Milligrams to Kilograms

To convert from milligrams to kilograms, divide by 1,000,000, because 1 milligram is equal to one millionth (\(10^{-6}\)) of a kilogram.For example, \(25.4 \text{ milligrams} = \frac{25.4}{1,000,000} = 0.0000254 \text{ kilograms}\).

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Key Concepts

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

Meters to Decimeters Conversion
The process of converting meters to decimeters is straightforward due to the nature of the metric system, which scales by powers of ten. Each unit within the system is ten times larger or smaller than its adjacent units. In the case of meters and decimeters, this means that one meter equals ten decimeters.

When you want to convert a measurement in meters to decimeters, you simply multiply the number in meters by 10. For instance, if you have 22.6 meters and need to find out how many decimeters that is, the calculation will be:
  • Start with your measurement: 22.6 meters.
  • Multiply by 10 (since 1 meter = 10 decimeters):
  • 22.6 meters x 10 = 226 decimeters.
This approach ensures a quick conversion, aided by the decimal nature of the metric system.
Milligrams to Kilograms Conversion
Converting milligrams to kilograms involves understanding the scale of units within the metric system. Milligrams are much smaller than kilograms, specifically, one milligram is one-millionth of a kilogram. Therefore, converting from milligrams to kilograms requires dividing by 1,000,000.

To illustrate, let's convert 25.4 milligrams to kilograms. Here's how you go about it:
  • Begin with your amount: 25.4 milligrams.
  • Understand the conversion factor: 1 milligram is 10^{-6} kilograms.
  • Divide the milligrams by 1,000,000:
  • 25.4 milligrams 梅 1,000,000 = 0.0000254 kilograms.
By dividing the milligrams by 1,000,000, you scale down to kilograms accurately, bridging the significant size difference between the two units.
Metric System Powers of Ten
The metric system is built around powers of ten, making conversions within the system incredibly efficient and easy to understand. Each step up or down from one metric unit to another involves multiplying or dividing by ten or its multiples. This systematic approach simplifies many mathematical conversions.

For instance:
  • Going from meters to kilometers requires dividing by 1,000, because a kilometer is 1,000 meters.
  • Conversely, increasing granularity from grams to milligrams involves multiplying by 1,000, as there are 1,000 milligrams in a gram.
This power of ten system allows users to switch between different scales with straightforward calculations, minimizing the possibility of error and ensuring precision wherever these units are applied. Understanding this foundation helps in mastering various conversions, not just within the metric system, but also in scientific calculations globally.

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Most popular questions from this chapter

Convert these to nonscientific notation: (a) \(3.256 \times\) \(10^{-5},\) (b) \(6.03 \times 10^{6}\)

Carry out these conversions: (a) A 6.0-ft person weighs 168 lb. Express this person's height in meters and weight in kilograms. \((1 \mathrm{lb}=453.6 \mathrm{~g} ; 1 \mathrm{~m}=\) \(3.28 \mathrm{ft} .\) ) (b) The current speed limit in some states in the United States is 55 miles per hour. What is the speed limit in kilometers per hour? (c) The speed of light is \(3.0 \times 10^{10} \mathrm{~cm} / \mathrm{s} .\) How many miles does light travel in 1 hour? (d) Lead is a toxic substance. The "normal" lead content in human blood is about 0.40 part per million (that is, \(0.40 \mathrm{~g}\) of lead per million grams of blood). A value of 0.80 part per million (ppm) is considered to be dangerous. How many grams of lead are contained in \(6.0 \times 10^{3} \mathrm{~g}\) of blood (the amount in an average adult) if the lead content is \(0.62 \mathrm{ppm} ?\)

Give the SI units for expressing these: (a) length, (b) area, (c) volume, (d) mass, (e) time, (f) force, (g) energy, (h) temperature.

A cylindrical glass tube \(12.7 \mathrm{~cm}\) in length is filled with mercury. The mass of mercury needed to fill the tube is found to be \(105.5 \mathrm{~g}\). Calculate the inner diameter of the tube. (The density of mercury \(=\) \(13.6 \mathrm{~g} / \mathrm{mL} .)\)

Do these statements describe chemical or physical properties? (a) Oxygen gas supports combustion. (b) Fertilizers help to increase agricultural production. (c) Water boils below \(100^{\circ} \mathrm{C}\) on top of a mountain. (d) Lead is denser than aluminum. (e) Uranium is a radioactive element.
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