Fractions are parts of numbers, or portions of a whole. You might remember seeing fraction sticks in school, and creating a fraction out of the number of shaded parts over how many parts there were in total. For example, below is a rectangle divided into 12 equal parts. Six of these pieces are shaded. To write a fraction of this, we put the number of shaded pieces on top for the fraction’s numerator, or top number, and the total number of sections on the bottom for the fractions denominator, or bottom number.
The fraction that represents this is (6 is the numerator and 12 is the denominator). Fractions can sometimes be simplified by dividing the numerator and denominator by the same number. (This is the same as dividing by 1 and does not change the number’s value.) Visually, if we divide both 6 and 12 in by 2, we get .
Notice how both and take up the same amount of the fraction bar. What happens if we divide by ? We would get ―which, visually, is how much all three of the fraction bars for , , and are shaded in. These are called equivalent fractions.
Additionally, since cannot be reduced anymore, it is called the simplified form of .
The first thing you should notice is that both the numerator and the denominator have a common factor of 2 since they are even numbers. Dividing the top and bottom by 2 gives the equivalent fraction . Again, 21 and 28 share a common factor of 7, so dividing the numerator and denominator by 7 gives us the equivalent fraction . Three and four do not have any more common factors, so this is the simplified form of .
Sometimes a fraction will have a whole number in front of it, such as 9. This is called a mixed fraction and can be thought of as the whole number added to the fraction. An example is if you have five quarters. Since it takes four quarters to equal a dollar, you could say you have + or 1 dollars. Another way of writing a mixed fraction is as an improper fraction, where the numerator is larger than the denominator. If we were to write the 1as an improper fraction, we would first write it as + then add the two fractions together, numerator to numerator, but keeping the same denominator. Then 1 as an improper fraction is . Think about this: we have five quarters (the numerator) and it takes four quarters to equal a dollar (the denominator). The denominator, or number of quarters it takes to equal a dollar, is not going to change, but the number of quarters we have might change.
Denali owns a party space that hosts rentals for birthdays, weddings, and other celebrations. One day, while cleaning up after two birthday parties, he finds leftover pizzas, which are served as 8 slices to one pie. One party left behind 2 pizzas. The other party left behind pizzas. How much pizza was left behind in total?
This is a fraction addition problem. To add these fractions together, we have to put them both into improper fraction format. is already in improper fraction format, but 2 will need to be converted. If each pizza has 8 slices and there are 2 whole pizzas left, then that means the whole number 2 = . Then 2 = + = . So the first party left behind pizzas while the second party left behind pizzas. Altogether, there were + = pizzas left over.
But wait! is an improper fraction, and they are called that for a reason. The proper way to write a fraction, when the numerator is larger than the denominator, is as a mixed number. To get there, we divide the top number by the denominator. The number of times it goes in evenly is the whole number, while the remainder is our numerator in the fraction part. For this example, 28 divided by 8 = 3, R 4. Then, as a mixed number, there are 3= 3pizzas leftover.
Subtract and simplify 2-.
Subtraction with fractions is very similar to addition in that the denominator must be the same, and both fractions should be in improper form. First, we’ll convert 2 to an improper fraction. Since our denominator is 15, we multiply 15 by 2 to get 30. So 2 = + = . Next, we want to make it so that has the same denominator as (the least common denominator). Notice that , so we multiply the top and bottom of the fraction by 3, like so . Now we can set up our subtraction problem and subtract the numerators straight across while keeping the same denominator, as follows: 2 – = –= . The final step is to convert from an improper form to a mixed number. We do this by dividing the numerator by the denominator. The whole number is the number of times it goes in evenly while the remainder is the numerator of the fraction. The denominator remains the same. Since 35 divided by 15 = 2 R 5, we simplify = 2= 2.
Up to this point, we’ve done a couple of multiplication steps with the fractions. Explicitly, when fractions need to be multiplied, you work straight across, numerator to numerator and denominator to denominator. When multiplying or dividing to simplify or convert, the numerator and the denominator must be multiplied or divided by the number.
Harriet and Samantha are sisters. In the spring, their father tells them they can each have of the garden to grow whatever they want during the summer, but they have to plan it out first. Harriet decides she wants to grow of her land with corn. How much of the total garden will have her corn growing on it?
To find the answer to this problem, we must multiply by . We multiply straight across the top and bottom numbers to find the answer, then simplify it.
So Harriet’s corn will take up of the garden.
Have you noticed by now that a fraction is just another way to write division? We start with a whole piece and divide it into parts. So when we have to divide fractions by each other, it’s like we are taking a portion of a portion. But isn’t that what we did with the garden above? We took a piece of Harriet’s part of the garden to grow corn on? Yes! To solve a division problem with fractions, we are going to turn them into a multiplication problem. Let’s take a look at the next example.
Divide by .
To solve this problem, we must divide by .
Another way of writing this is
To turn this into a multiplication problem, we multiply the top and bottom by the reciprocal fraction of the denominator. Recall that the reciprocal fraction is the fraction that, when multiplied by the original fraction, equals one.
So the reciprocal of for this problem is
So let’s look at how this is going to help turn our problem into a multiplication problem.
Because of the nature of the reciprocal, the bottom part of the fraction becomes one, and we only have to deal with the top part of the fraction, which is a multiplication problem. From here, we simply multiply across the top and bottom and simplify if needed.
= = 1
When you get the hang of this, a shortcut is simply flipping the divisor fraction into the reciprocal and changing the division sign to a multiplication sign. Let’s do another example.
Divide by .
Our first step is to convert these to improper fractions.
= + =
= + =
and = = 3
Remember the place value chart for whole numbers? There’s a continuation of the chart that delves into numbers smaller than one, in the form of decimals. Here’s a sample of the chart for the number 1.234567:
|Ones||Decimal||Tens||Hundreds||Thousandths||Ten Thousandths||Hundred Thousandths||Millions|
When adding or subtracting decimals, it is the same as adding or subtracting with whole numbers. Just remember to align the ones position or the decimal point. When working down, the decimal point must be carried down into the answer. Multiplying is similar, but you may ignore the decimal point until the very end. Then, the total number of decimal places present between the two numbers you multiplied should also be the number of places present in your answer. For division, you must move the decimal place to the right in the divisor until it is a whole number. However many places you moved it, the decimal place in the dividend must also be moved to the right. Here are three examples showing a subtraction, multiplication, and division problem with decimals.
Multiply: (Note that there are three decimal places in the two factors, or numbers in the problem. Our answer will also have three decimal places.)
Divide 50.125 by .05.
The answer is 1002.5.
Another facet of decimals is that we can convert between decimals and fractions. Remember how we said earlier that fractions are just a form of division? This is evident when converting from a fraction to decimal. To do so, we divide the numerator by the denominator, and work it out with long division.
Convert to a decimal.
What about when we want to convert from a decimal to a fraction? The nice thing about the place value system is that when we want to convert from a decimal to a fraction, the answer is right in the name of the decimal―how we say it.
For example, 0.75 is read aloud as “seventy-five hundredths” or seventy-five over one hundred: .
From here, all we have to do is simplify.
So, = .
Percentages are a way to express a number out of 100. In fact, the root of the word percentage is “per cent” or “per 100”.
Therefore, a percentage such as 42% means 42 out of 100.
As a fraction, it means .
When converting from a percentage to a fraction, simply put the number over one hundred. If we want to convert to a decimal, then once we know the fraction, it’s a matter of using our conversion method for fraction to decimals.
For 42%, = 0.42. Read the two numbers aloud. Note how the pronunciation of the fraction tells us a great deal about how to convert this to a decimal―“forty-two over one hundred” and “forty-two hundredths.”
We simply have to bump over the decimal place from the numerator in the fraction so that the digits reach the hundredths place, the same thing as if we divide by 100.
Sharon is selling tickets for her college’s theater production. There are 540 seats in the theater. If she sells tickets for 66.7% of the seats, how many seats is that (round to the nearest whole number)?
To solve this problem, we must first convert the percentage to a decimal. Sixty-six and seven tenths percent is as a fraction, or 0.667 as a decimal. To find the number of seats, we multiply .667 by 540 and get 360.18. So, seats sold for the theater production. (The symbol means “approximately equal to” and is usually used to indicate the answer has been rounded as opposed to =, the equal sign.) Interpreted another way, 360 seats is 66.7% of the total seats the theater holds.
What about when we start with decimal or fraction and want to convert it to a percent? Notationally, to convert a decimal, we simply multiply by 100%, which bumps the decimal place two places to the right and adds the percent sign. For a fraction, we drop the percent sign and write the number as the numerator with 100 as the denominator. Why does this make sense, given what we know about percentages?
Xavier and his friend go out for lunch together. At the end of the meal, Xavier offers to pay. If he leaves a $6.16 tip and the food bill was $34.24, what percentage of the food bill was Xavier’s tip (round to the nearest whole number of the percent)?
To figure out what percentage tip Xavier left, we divide the tip amount, $6.16, by the food bill amount, $34.24. So, $6.16 ÷ $34.24 is approximately equal to 0.179. To convert to a percent, we multiply by 100%, or move the decimal two places to the right and add the percent sign. So 0.179 = 17.9%. Since the problem asks us to round to the nearest whole number, we round up to 18%. Xavier left an 18% tip on the meal bill.
Military time is a way to tell the time by looking at the hours in the day as 24 hours rather than the standard two twelve-hour periods of a.m. (morning) and p.m. (evening). Military time starts at midnight with 0000 and continues up to 2400 (midnight again). This means less confusion if someone tells you to meet at, say 8 o’clock. In the 12-hour clock system, or civilian time, this could be in the morning or the evening. In military time, 0800 hours could only be the morning. Below is a conversion chart between the two systems.
|Civilian Time||Military Time|
|12:00 am||0000 hours|
|1:00 am||0100 hours|
|12:00 pm||1200 hours|
|1:00 pm||1300 hours|
|2:00 am||0200 hours|
|2:00 pm||1400 hours|
|3:00 am||0300 hours|
|3:00 pm||1500 hours|
|4:00 am||0400 hours|
|4:00 pm||1600 hours|
|5:00 am||0500 hours|
|5:00 pm||1700 hours|
|6:00 am||0600 hours|
|6:00 pm||1800 hours|
|7:00 am||0700 hours|
|7:00 pm||1900 hours|
|8:00 am||0800 hours|
|8:00 pm||2000 hours|
|9:00 am||0900 hours|
|9:00 pm||2100 hours|
|10:00 am||1000 hours|
|10:00 pm||2200 hours|
|11:00 am||1100 hours|
|11:00 pm||2300 hours|
The last two digits of the military time are reserved for the minutes, just like in civilian time, and can show from 00 to 59. To find the conversion between military and civilian time, we subtract twelve or count up from twelve. Let’s look at some examples.
Tiana has to report for work at 1945 hours. What time is this in civilian time?
The first two digits, 19, refer to the hour. If we subtract twelve, we’re left with 7. The minutes work the same way in both systems. So Tiana has to work at 7:45 pm.
Sebastian invites his friend to meet for coffee at 2:20. “What time is that in military time?” his friend asks him. How does Sebastian answer?
From 12 noon to 2 p.m. is 2 hours. So adding 2 to 12 gives us 14 as the hour. The minutes stay the same. Sebastian wants to meet at 1420 hours.
Iliana is planning her day. She marks down “Workout―0545 to 0630.” In civilian time, when is Iliana working out?
Since the hours are below 1200, we know this is in the morning, and the hours and minutes are the same as in civilian time. Iliana is working out from 5:45 a.m. to 6:30 a.m.
Roman numerals is the ancient number system used in the Roman Empire. In select situations, they are still used as an alternative to the modern numeral system. Roman numerals are based on the idea of adding, and in special cases, subtracting, to represent each number. The following table provides a list of the most commonly used Roman numerals today and their meaning.
|Roman Numeral||Numerical Value|
Roman numerals are written from left to right, greatest numeral to smallest. As you read them, add together the symbols for the numeric value.
Erica is visiting Rome for vacation. She sees an old building with “MCCLIII” written on it. What number is this?
Looking at our reference chart, we see the first symbol, M, is 1000. CC is 100+100, L is 50, and III is 1+1+1. Adding this together shows that:
1000 + 100 + 100 + 50 + 1 + 1 + 1 = 1253 as the number on the building.
There are a few special cases where instead of adding, we subtract to find the meaning of the Roman numeral. These special cases occur when we want to avoid stringing four instances of a symbol together (like IIII for 4 or VIIII for 9). Instead, we use one of the symbols for a power of ten (I, X, C, M) and place it before a symbol with a greater numeric value. An example of the first time this would occur is for the value of four. This is written as IV, or “5 minus 1.” The next time this occurs is for the number nine, IX, or “10 minus 1.” Forty is denoted by XL, or “50 minus 10,” while ninety is XC, or “100 minus 10.”
Jerome is in class and wants to toss a note inconspicuously to let his friend know which anniversary of a big football game finale is coming up. He knows it’s going to be the 459th year the game has been held. How would he write this number in Roman numerals?
First, we break up the number by place value. Written this way, 459 becomes 400 + 50 + 9 or (500–100) + 50 + (10–1). In Roman numerals, we write this as CDLIX. So Jerome would write “FOOTBALL CDLIX!!!” in his note to his friend.
Any measurement of an object will be taken in a certain unit. Whether it’s the volume of a unit of blood in milliliters, the height of a person in inches, their weight in pounds, or a medicine in grams, there is a unit attached to it so that we have an idea of exactly how much that measurement means. Very often, however, the unit the measurement is given in is not the unit we need to know. In that case, we must convert to a different unit. The following table provides some common conversion factors or measurements that are equivalent and used to calculate conversions.
1 gallon= 4 quarts
1 quart = 2 pints
1 pint= 2 cups
1 cup= 8 ounces
1 cup= 16 tablespoons
1 gallon= 3.78541 liters
1 pound= 453.592 grams
1 ounce=28.3495 grams
1 kilogram= 2.20462 pounds
1 kilometer= 0.621371 miles
1 mile= 5280 feet
1 mile/hour= 1.60934 kilometers/hour
1 inch= 2.54 centimeters
1 kilometer= 1000 meters
1 meter= 100 centimeters
1 meter= 1000 millimeters
1 liter=1000 milliliters
1 kilogram= 1000 grams
1 gram= 1000 milligrams
1 year= 365 days
1 year= 52 weeks
1 day= 24 hours
1 hour= 60 minutes
1 minute= 60 seconds
Some of these conversion factors might look familiar, while others are new. Usually people find that the more often they use a conversion factor, the easier it is to recall it. But how do we use these statements to convert a measurement from one unit to another?
To convert something not readily known, we can write out a chain of factors in fraction form and multiply or divide as needed until the old units cancel out and we are left with the new units. This relies on the idea that 1 = . Let’s look at an example.
Jordan takes a measurement of a patient’s height as 5 feet and 8 inches. However, when he goes to mark the chart, he realizes the measurement should have been in centimeters. Convert the patient’s height to centimeters.
First, we will convert 5 feet to inches, then add it to the remaining 8 inches.
= = 60 in
Notice how the feet on the top and bottom, as units, cancel each other out, since = 1. Jordan’s patient, then, is 68 inches tall (60 inches + 8 inches). Next, we do another conversion multiplication to go from inches (in) to centimeters (cm).
= =172.72 cm
Again, notice how the inches in the top of the first fraction cancel out with the inches from the conversion factor of the second fraction. At the end of the problem, we are only left with centimeters, which is what we wanted. Jordan should document that the patient is 172.72 centimeters tall.
If there are more than one conversion factors that need to be used to convert a measurement, then we can create a conversion chain of multiplication problems using the conversion factors.
Shawna received a container, which measures 250 ounces, to store chemicals. However, her lab uses the metric system, liters, to measure everything. How many liters should she label the container as, rounded to three decimal places?
This conversion requires the following steps based on the conversion factors pulled from the table above: ounces to cups to pints to quarts to gallons to liters. Here is what the conversion chain looks like.
= approximately 7.393 L
Note how the ounces on top cross out with the ounces on the bottom, the cups on top with the cups on bottom, etc., until all that is left is liters for the units. So Shawna should label the 250-ounce container as holding about 7.393 liters.