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Chapter 1: Problem 9

When you convert units, how do you decide which part of the conversion factor is in the numerator and which is in the denominator? [Section 1.6]

Short Answer

Expert verified
To decide which part of the conversion factor goes in the numerator and which part in the denominator, always place the initial units opposite to the desired final units, so that they cancel out. For example, when converting miles to kilometers, set up the conversion factor to cancel the "mile" unit, leaving the "km" unit: \[ 5\,\text{miles} \times \frac{1.609\,\text{km}}{1\,\text{mile}} \] This ensures the correct arrangement of conversion factors and leads to the desired final units.

Step by step solution

01

Identify the units to be converted

To start with, determine the initial units that need to be converted and the desired final units. This information is vital in deciding which conversion factors to use and how to arrange them.
02

Find appropriate conversion factors

Look up the necessary conversion factors that you'll need for converting the initial unit to the final unit. Conversion factors relate two different units and give their equivalent quantities. For example, \( 1\,\text{mile} = 1.609\,\text{km}\).
03

Determine the proper arrangement of conversion factors

To decide which part of the conversion factor goes in the numerator and which part in the denominator, remember that you want to cancel out the initial units and obtain the final units. Therefore, the initial units should always be in the opposite part of the fraction (numerator or denominator) as the desired final units. For instance, if you want to convert miles to kilometers, place the conversion factor in a way that it cancels the "mile" unit and leaves you with the "km" unit.
04

Setting up the conversion factor ratio

Set up the conversion factor ratio with initial units and conversion factors. Make sure that the initial units are opposite to the units in the conversion factor, so they cancel each other out. Example: Convert \ (5\,\text{miles}\) to kilometers. Given, \( 1\,\text{mile} = 1.609\,\text{km}\) Set up the ratio as follows: \[ 5\,\text{miles} \times \frac{1.609\,\text{km}}{1\,\text{mile}} \]
05

Perform the calculation and cancel the units

Do the calculation and ensure the initial units are canceled out. In the example above, the "mile" unit will cancel out, leaving you with the "km" unit: \[ 5\,\cancel{\text{miles}} \times \frac{1.609\,\text{km}}{1\,\cancel{\text{mile}}} = 8.045\,\text{km} \] The result of converting \(5\) miles to kilometers is \(8.045\) kilometers.

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Most popular questions from this chapter

Use appropriate metric prefixes to write the following measurements without use of exponents: (a) \(2.3 \times 10^{-10} \mathrm{~L}\) (b) \(4.7 \times 10^{-6} \mathrm{~g}\), (c) \(1.85 \times 10^{-12} \mathrm{~m}\) (d) \(16.7 \times 10^{6} \mathrm{~s}\); (e) \(15.7 \times 10^{3} \mathrm{~g}\) (f) \(1.34 \times 10^{-3} \mathrm{~m},(\mathrm{~g}) 1.84 \times 10^{2} \mathrm{~cm}\)

Gold is alloyed (mixed) with other metals to increase its hardness in making jewelry. (a) Consider a piece of gold jewelry that weighs \(9.85 \mathrm{~g}\) and has a volume of \(0.675 \mathrm{~cm}^{3} .\) The jewelry contains only gold and silver, which have densities of \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\) and \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\), respectively. If the total volume of the jewelry is the sum of the volumes of the gold and silver that it contains, calculate the percentage of gold (by mass) in the jewelry. (b) The relative amount of gold in an alloy is commonly expressed in units of carats. Pure gold is 24 carat, and the percentage of gold in an alloy is given as a percentage of this value. For example, an alloy that is \(50 \%\) gold is 12 carat. State the purity of the gold jewelry in carats.

(a) Three spheres of equal size are composed of aluminum \(\left(\right.\) density \(\left.=2.70 \mathrm{~g} / \mathrm{cm}^{3}\right),\) silver \(\left(\right.\) density \(\left.=10.49 \mathrm{~g} / \mathrm{cm}^{3}\right),\) and nickel (density \(\left.=8.90 \mathrm{~g} / \mathrm{cm}^{3}\right)\). List the spheres from lightest to (b) Three cubes of equal mass are composed of gold \(\left(\right.\) density \(\left.=19.32 \mathrm{~g} / \mathrm{cm}^{3}\right)\), platinum (density \(\left.=21.45 \mathrm{~g} / \mathrm{cm}^{3}\right)\) and lead (density \(\left.=11.35 \mathrm{~g} / \mathrm{cm}^{3}\right)\). List the cubes from smallest to largest. [Section 1.4]

Water has a density of \(0.997 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C} ;\) ice has a density of \(0.917 \mathrm{~g} / \mathrm{cm}^{3}\) at \(-10^{\circ} \mathrm{C}\). (a) If a soft-drink bottle whose volume is \(1.50 \mathrm{~L}\) is completely filled with water and then frozen to \(-10^{\circ} \mathrm{C},\) what volume does the ice occupy? (b) Can the ice be contained within the bottle?

In the process of attempting to characterize a substance, a chemist makes the following observations: The substance is a silvery white, lustrous metal. It melts at \(649^{\circ} \mathrm{C}\) and boils at \(1105^{\circ} \mathrm{C}\). Its density at \(20{ }^{\circ} \mathrm{C}\) is \(1.738 \mathrm{~g} / \mathrm{cm}^{3}\). The substance burns in air, producing an intense white light. It reacts with chlorine to give a brittle white solid. The substance can be pounded into thin sheets or drawn into wires. It is a good conductor of electricity. Which of these characteristics are physical properties, and which are chemical properties?
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