Math Study Guide for the NLN NEX

Measurement

Measurement is the process of quantifying various attributes, such as length, capacity, and weight, using standard units. It plays a crucial role in various fields, providing a standardized way to express and compare quantities. In the study of measurement, the two primary systems are the United States customary system and the metric system.

Units of Measurement

US Customary System

The US customary system is a measurement system used in the United States. It was developed from the Imperial system (formerly known as the English system). It employs units like inches, feet, pounds, and gallons. This system is prevalent in daily life, and is used across many fields, including construction, cooking, and engineering.

Linear

Linear measurement involves the determination of length or distance. Common linear units in the US customary system include:

• inch (in)—the smallest unit; often used for small measurements
• foot (ft)—equal to \(12\) inches; commonly used in construction and real estate
• yard (yd)—equivalent to three feet; often used for larger measurements

This is the basic equivalency within the system:

\[1 \text{ ft} = 12 \text{ in}\] \[1 \text{ yd} = 3 \text{ ft}\]

Capacity

Capacity refers to the volume of substances a container can hold. Common US units for measuring capacity include:

• fluid ounce (fl oz)—used for measuring liquid volume
• pint (pt)—equal to \(16\) fluid ounces
• gallon (gal)—equivalent to \(128\) fluid ounces or four quarts

This is the basic equivalency within the system:

\[1 \text{ pt} = 16 \text{ fl oz}\] \[1 \text{ gal} = 128 \text{ fl oz}\]

Weight

Weight is a measurement of the mass of an object as it relates to the acceleration of gravity. Common US units for weight include:

• ounce (oz)—suitable for smaller objects
• pound (lb)—equal to \(16\) ounces; widely used in daily life
• ton (t)—equivalent to \(2\text{,}000\) pounds; used for large objects

This is the basic equivalency within the system:

\[1 \text{ lb} = 16 \text{ oz}\] \[1 \text{ t} = 2\text{,}000 \text{lb}\]

Metric System

The metric system, a decimal-based system of measurement, has been widely adopted across the globe for its simplicity and ease of conversion. It is the international standard for scientific, industrial, and everyday measurements.

Linear

Linear measurement in the metric system quantifies length or distance and is extensively used in scientific endeavors, manufacturing, and daily activities. Common linear units in the metric system include:

• millimeter (mm)—the smallest commonly used unit; suitable for precise measurements

• centimeter (cm)—equal to \(10\) millimeters; often used in everyday situations

• meter (m)—equivalent to \(100\) centimeters; commonly used in construction and scientific research

This is the basic equivalency within the system:

\[1 \text{ cm} = 10 \text{ mm}\] \[1 \text{ m} = 100 \text{ cm}\]

Capacity

Again, the metric system has its own units for measuring the volume of liquids or substances. These units are widely used in scientific experiments, medicine, and culinary activities. Common capacity units include:

• milliliter (mL)—used for measuring small volumes

• liter (L)—equal to \(1\text{,}000\) milliliters; widely used in everyday life and scientific contexts

• cubic meter (\(\bf{\text{m}^{3}}\))—equivalent to \(1\text{,}000\) liters; often used for large quantities

This is the basic equivalency within the system:

\[1 \text{ L} = 1\text{,}000 \text{ mL}\] \[1 \text{ m}^{3} = 1\text{,}000 \text{ L}\]

Weight

The metric system’s units for measuring weight are crucial in scientific research, commerce, and health-related fields. Common weight units include:

• milligram (mg)—suitable for measuring small objects

• gram (g)—equal to \(1\text{,}000\) milligrams; widely used in daily life

• kilogram (kg)—equivalent to \(1\text{,}000\) grams; commonly used in scientific and industrial contexts

This is the basic equivalency within the system:

\[1 \text{ g} = 1\text{,}000 \text{ mg}\] \[1 \text{ kg} = 1\text{,}000 \text{ g}\]

Using Scales

In both medical settings and science labs, accurate measurement is essential for various tasks, whether determining a patient’s weight or measuring a chemical substance’s mass. Scales are the tools used for this purpose, and understanding how to use them correctly is crucial for obtaining reliable data.

Types of Scales

Balance Scale

Balance scales are among the oldest and most reliable tools for measuring mass. They consist of a horizontal beam with a pivot in the center and a pan hanging from each end. The object to be measured is placed on one pan, and standard weights are added to the other until the beam is balanced. The total weight of the standard weights represents the mass of the object. This type of scale is highly accurate and commonly used in laboratories to weigh small objects and substances.

When using a balance scale, do the following:

1. Ensure the scale is calibrated and the beam is level before use.
2. Place the object to be weighed on one pan.
3. Gradually add standard weights to the opposite pan until the beam is perfectly balanced.
4. The total weight of the standard weights represents the mass of the object.

Digital Scale

Digital scales are widely used in medical settings and labs due to their ease of use and precision. These scales measure weight electronically and display the result on a digital screen. They are available in various sizes and sensitivities, ranging from those used for weighing patients to those used for precise measurements of small quantities in a lab. Digital scales are also commonly used in pharmacies to weigh medications and provide accurate measurements as small as micrograms (one-millionth of a gram) and milligrams (one-thousandth of a gram) for dosing calculations.

When using digital scales, do the following:

1. Turn on the scale and ensure it is set to zero before placing any object on it.
2. Place the object or substance on the scale’s platform.
3. Read the displayed weight on the digital screen.

Note: If weighing a container and its contents, use the tare function to subtract the weight of the container so only the contents are measured.

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Spring Scale

A spring scale measures weight based on the tension of a spring. When an object is hung from the scale, the spring stretches, and the weight is read directly from the scale’s calibrated markings. These scales are often used for measuring force or weight in a vertical setup, such as in physics experiments.

When using spring scales, do the following:

1. Ensure the scale is set to zero before use.
2. Attach the object to the hook or platform of the scale.
3. Allow the object to hang freely and wait for the spring to settle.
4. Read the measurement from the scale’s markings.

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Beam Scale

Beam scales, commonly found in medical offices, are used to measure body weight. They are also known as physician’s scales. They consist of a platform on which the person stands and a horizontal beam with sliding weights. The scale achieves balance by moving the weights along the beam, indicating the person’s weight.

When using a beam scale, do the following:

1. Have the person stand on the platform with feet evenly spaced apart.

2. Slide the large weight on the lower beam until it clicks into place at the nearest hundred-pound mark.

3. Adjust the smaller weight on the upper beam until the beam is balanced.

4. Add the values from both weights to determine the total weight.

Measuring Accurately

Accurate measurements are vital in both medical and scientific contexts. In a medical setting, incorrect weight measurements could lead to improper medication dosing or a misdiagnosis. In a lab, precise measurements are necessary to ensure the success of experiments and the validity of results. To maintain accuracy:

• Calibrate the scale regularly to account for any drift in measurements over time.

• Use appropriate scales for the task at hand, ensuring the scale’s capacity and sensitivity match the object being weighed.

• Avoid external factors such as air currents, vibrations, or temperature fluctuations that could affect the scale’s reading.

Using Ratios and Proportions in Measurement

The concepts of ratio and proportion are valuable tools for comparing quantities and solving problems of various types. Ratios and proportions are prevalent in real-world scenarios, making them equally essential for practical applications.

Ratio

A ratio is a way of expressing the relationship between two quantities. It is often written as a fraction, such as \(\frac{a}{b}\), where \(a\) and \(b\) are the two quantities being compared. This can also be written using a colon between the two numbers, \(a:b\). You read either one as “the ratio of \(a\) to \(b\).”

Ratios can be used to compare sizes, amounts, or values. For instance, if you have seven red apples and two green apples, the ratio of red apples to green apples can be written in the following three ways:

\[\frac{7}{2}, 7:2, \text{or } 7 \text{ to } 2\]

Proportion

A proportion is an equation that states two ratios are equal. It is written in the form \(\frac{a}{b} = \frac{c}{d}\). For example, if the ratio of boys to girls in a class is \(\frac{7}{5}\), and another class has a ratio of \(\frac{14}{10}\), we can say the two ratios are in proportion or proportional.

Using Proportions to Solve Problems

Proportions offer a convenient method for solving a variety of problems, both on the NLN NEX and in real life. When faced with a situation involving proportional relationships, setting up and solving a proportion can provide a quick and accurate solution.

For example, if a recipe calls for two cups of flour for every three cups of sugar, and you want to know how much flour you need for nine cups of sugar, you can set up a proportion to find the answer. If the unknown quantity is \(x\), you have:

\[\frac{2}{3} = \frac{x}{9}\]

To find \(x\), simply cross-multiply and do simple algebra:

\[3 \times x = 2 \times 9\] \[3x = 18\] \[x = \frac{18}{3} = 6\]

So, you need six cups of flour for nine cups of sugar.

Using Proportional Relationships to Solve Problems

Proportional relationships are quite useful in both everyday situations and specialized fields such as science and medicine. These relationships allow us to solve problems where two quantities are directly related, meaning that as one quantity changes, the other changes consistently. By understanding and applying proportions, you can solve many practical problems.

In medical settings, determining the correct dosage of medication based on a patient’s weight is a common task that relies heavily on proportional reasoning. For instance, if you know that a certain medication requires a dosage of five milligrams per kilogram of body weight, you would just need to know the patient’s weight, and then you could set up a proportion to find the right dosage.

Let’s see how that would work with an example problem.

A medication is prescribed at \(5\) milligrams per kilogram of body weight. How many milligrams of the medication should be administered to a patient weighing \(70\) kilograms?

Solution

Let \(x\) represent the required dosage in milligrams. You can set up the following proportion:

\[\frac{5 \text{ mg}}{1 \text{ kg}} = \frac{x \text{ mg}}{70 \text{ kg}}\]

Now, solve for \(x\) with cross-multiplication:

\[1 \times x = 5 \times 70\] \[x = 350 \text{ mg}\]

This example shows how using proportions ensures the correct dosage is calculated based on the patient’s specific weight, reducing the risk of underdosing or overdosing.

In chemistry labs, preparing solutions with precise concentrations often involves proportional relationships. Suppose you must prepare a solution by diluting a more concentrated stock solution. Knowing how to set up a proportion can help you determine the correct amount of stock solution and required diluent (like water).

Let’s try another example problem.

You have a \(10\%\) saline solution, and you need to prepare \(200\) milliliters of a \(5\%\) saline solution. How much of the \(10\%\) solution and how much water should you use?

Solution

Let \(x\) represent the volume of the \(10\%\) saline solution needed. You will use the concept of dilution, which is a proportional relationship between the concentration and volume:

\[\frac{10\%}{100} \times x = \frac{5\%}{100} \times 200 \text{ ml}\]

Simplify and solve for \(x\):

\[0.1 \times x = 0.05 \times 200\] \[x = \frac{0.05 \times 200}{0.1}\] \[x = 100 \text{ ml}\]

You need \(100\) milliliters of the \(10\%\) saline solution. Therefore, to make the final \(200\)-milliliter solution, you will add \(100\) milliliters of water.

This problem demonstrates how proportions are essential in creating solutions with the desired concentration, ensuring experiment consistency and accuracy.