08 Fundamental Math Review

Decimal Places

Decimal places let us quantify measurements with precision. From entering data into a computer to reading doctor prescriptions, we’ll use them all the time. Drug strengths are often written in decimal form consisting of a decimal point and numbers to each side of that decimal point such as 0.25 mcg of calcitriol or 0.1 mg of levothyroxine.

These numbers represent decimal places. Each decimal place to the left of the decimal point indicates an increase in value by a factor of 10 and each decimal place to the right of the decimal point indicates a reduction in value by a factor of 10. For example, 10 lbs of ice weighs 10 times more than 1lb of ice and 5.0 L of water displaces 1,000 times more volume than 0.005 L of water.

Each decimal place has its own unique name:

Fractions

Fractions represent either a portion of something or a greater quantity of it. In our world, that something may be a tablet or perhaps a volume of liquid in a syringe. There are several different ways to write fractions. The first way is called a proper fraction.

proper fraction has a numerator that is less than its denominator. When written out, the numerator is the top half of the fraction and the denominator is the bottom half. The numerator says how many parts of something there is and the denominator says how many individual parts make up 1 whole unit of that something. A neat trick to help you remember which is which is to think of the u in numerator as the u in up and the d in denominator as the d in down.

Let’s dive into some examples:

The proper fraction 1/2 describes 1 part out of the 2 parts that make up 1 unit of something. The proper fraction 3/4 describes 3 parts out of the 4 parts that make up 1 unit of something. As I mentioned earlier, these units can be anything.

let’s say you’re eating 6/12 of a pizza (Yum! Pizza Hut anyone?!); Well, this means that you’re eating 6 slices out of the 12 individual slices that make up 1 unit of your pizza. Get it?! Furthermore, fractions can be simplified. For example, in the case of your delicious pizza the proper fraction 6/12 can be simplified to 1/2 because 6 slices as a unit represent 1 unit out of the 2 units (of 6 slices) that make up 1 pizza. Mamma Mia! Bon Appetit!

In addition to proper fractions, there are improper fractions and mixed fractions. These are just different ways of writing the same fractional value. An improper fraction has a numerator that is greater than or equal to its denominator such as 11/10. A mixed fraction is the combination of a whole number and a proper fraction such as:

$$1 \, \frac{1}{10}$$ $$86 \, \frac{75}{309}$$ $$10 \, \frac{21}{15}$$

The whole number tells us how many whole units (think 10/10 in this example) to add to the proper fraction. Thus:

$$1 \, \frac{1}{10} = \frac{10}{10} + \frac{1}{10} = \frac{11}{10}$$

Also, keep in mind that the numerator of a mixed fraction must be less than its denominator.

Converting a mixed fraction to an improper fraction is simple using the following steps. It should be noted that you should also use these steps before adding, subtracting, multiplying or dividing fractions. A good example is converting

$$1 \, \frac{1}{10}$$ to $$\frac{11}{10}$$

How to convert a mixed fraction to an improper fraction:

  1. Multiply the denominator by the whole number in front.
  2. Add this value to the numerator.
  3. Write the resulting value as the new numerator but don’t change the denominator.
Convert these mixed fractions to improper fractions:
$$5 \, \frac{12}{25}$$ $$7 \, \frac{7}{7}$$ $$1,000 \, \frac{9}{20}$$

The last type of fraction is a decimal fraction. A decimal fraction is simply a fraction written in decimal form. To convert a proper or improper fraction to a decimal fraction, divide the numerator by the denominator. For example, 1/2 is 0.5 in decimal fraction form.

But what if we want to convert a decimal fraction to a proper or mixed fraction? Well, since each decimal place represents a fraction of a multiple of 10 (such as 1/10, 1/100, 1/1,000, 1/10,000), we can rewrite the decimal part in the numerator and its corresponding multiple of 10 in the denominator.

Thus, the decimal fraction 0.0711 as a proper fraction is:

$$\frac{711}{10,000}$$

If our decimal fraction has numbers to the left of the decimal point we can convert it to a mixed fraction by following the steps above and then tacking the numbers to the left of the decimal point as a whole number in front of the fraction.

FWhich makes the decimal fraction 5.0711 in mixed fraction form as:

$$5 \, \frac{711}{10,000}$$

Adding and Subtracting Fractions

When adding and subtracting fractions, the first thing we want to do is obtain their lowest common denominator, also known as the LCD. The LCD is the smallest number that is divisible by both denominators without leaving a remainder. For example, when adding or subtracting 8/3 and 1/7, their LCD is 21 because 21 is the smallest number that is divisible by both denominators, 3 and 7, without leaving a remainder.

Obtaining the LCD lets us rewrite all of the fractions with the same denominator thus allowing easy addition or subtraction of their numerators. After obtaining the LCD, we’ll multiply each individual numerator by the same number each individual denominator was multiplied by to obtain the LCD. Once that’s done that we’ll simply add or subtract the numerators of the resulting fractions. Let’s illustrate this entire process using the example fractions above.

Because 21 is our LCD, 8/3 becomes:

$$\frac{8}{3} \times \frac{7}{7} = \frac{56}{21}$$

and 1/7 becomes:

$$\frac{1}{7} \times \frac{3}{3} = \frac{3}{21}$$

Next, we add these fractions together to obtain:

$$\frac{56}{21} + \frac{3}{21} = \frac{59}{21}$$

If we were subtracting the fractions we would get:

$$\frac{56}{21} – \frac{3}{21} = \frac{53}{21}$$

Multiplying and Dividing Fractions

Multiplying or dividing fractions is even simpler! When multiplying fractions, simply multiply the numerators by one another and the denominators by one another.

For example,

$$\frac{8}{3} \times \frac{1}{7} = \frac{8}{21}$$ and

$$\frac{8}{3} \times \frac{1}{7} \times \frac{2}{3} = \frac{16}{63}$$

When dividing fractions, take the fraction being divided and invert the numerator and the denominator. Then follow the procedure above for multiplying fractions.

For example,

$$\frac{8}{3} \div \frac{1}{7} = \frac{8}{3} \times \frac{7}{1} = \frac{56}{3}$$ and

$$\frac{8}{3} \div \frac{1}{7} \div \frac{2}{3} = \frac{8}{3} \times \frac{7}{1} \times \frac{3}{2} = \frac{168}{6}$$

Lastly, simplify your answers:

$$\frac{168}{6} = \frac{28}{1} = 28$$

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