Math is one of those love-it-or-hate-it subjects. It is simple for some, but for others, it’s a bottomless pit of despair. Math is not only rife with symbols; it also has many processes — one of which is the backward Z.
The backward Z is a mathematical process that allows you to add two fractions together, even when the denominator is not the same. The process involves the multiplication of two or more denominators until you find a common denominator, ultimately using it to change the numerator.
People who are skilled in math claim that it is simple once you understand the process. Here, we’ll break down the operation of the backward Z so that you’ll never have to stress over fractions again.
The Backward Z Process
Fractions involve a whole divided into equal parts, and we work with a section of that whole. When you divide the whole into the same number of pieces, it is easy to add them together.
For example, if you have two pizzas and cut each one into eight pieces, it is easy to combine the leftovers into one box once you have finished. Let’s say that we have three slices left from one pie and five from another.
Mathematically, we would write that as ⅜ and ⅝. Once combined, you would have a whole number again. This kind of addition is relatively simple.
However, if there are five slices left from each box and you combined the leftovers from both, the result would be an improper fraction where the numerator is larger than the denominator (10 ⁄ 8). You would present this fraction as 1¼ of the two pizzas.
We will discuss more about proper and improper fractions later on, but what makes many math students struggle is when given two or more fractions where the denominator is different. Many stumble when asked to combine those fractions together.
Another challenge that many people go through is understanding symbols in mathematics. Outside of algebra, what are letters doing in math? Find out by reading the article: “What Does the Upside-down A Mean in Math?”
Applying the Backward Z Process
To understand this scenario, we will start with some basic fractions:
Using a visual element, let us assume that these fractions are two cakes. You cut one of the cakes into four quarters and the other cake into ninths. There are three pieces left of the first cake and two pieces left of the other. So how exactly would we do this?
To start the backward Z process, we need to find the common denominator. You can find the common denominator with classic multiplication tables. In this case, we would look at the multiples of both denominators until we find a common one.
The multiplication table of 4 states that 4 times 1 is 4, 4 times 2 is 8, etc. Instead of writing out each step, we can use the number patterns: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, etc.
We now repeat the same step with the 9 times table: 9, 18, 27, 36, 45, 54, etc.
At this stage, you will have noticed that both 4 and 9 have a common multiple of 36. This is our new denominator. Using 36, we will now create new numerators through the backward Z process.
The first part of the backward Z includes some division. Look at your old denominator and your new one and divide them. So, 36 divided by 4 is equal to 9. For the second fraction, 36 divided by 9 is equal to 4.
The next part of the backward Z process involves multiplication. Use the number you got from the division in the previous step and multiply it with the numerator of the original fraction.
In this case, it would be 9 multiplied by 3 in the first fraction and 4 multiplied by 2 for the second fraction. This should give us 27 and 8, respectively.
The final step involves writing our new numerators into the new fraction.
Once you have these, you can add the two new fractions together and simplify as needed. In our scenario, 27 + 8 = 35, and since the denominators are the same, nothing needs to change.
This will give us a new fraction.
This is the backward Z method that we use to add two unlike denominators together. This simple method relies on some mental arithmetic but should be easy enough for most students, especially when dealing with smaller numbers.
As seen by the numbers above, the fraction is quite large and unwieldy. While the thought of having 35 slices of pizza sounds like a dream come true, ideally, we should simplify the fraction to something more manageable.
To do this, you’ll need to identify the greatest common factor.
Identifying the Greatest Common Factor
The greatest common factor refers to the largest number that is divisible into both numbers. Unfortunately, our numbers do not share a common factor except 1, so this fraction is in its simplest form.
However, we can easily simplify more cooperative numbers.
For example, a fraction 8 ⁄ 12 is not in its simplest form.
Let’s examine which numbers can be divided by both. The number 1 works, but it will not simplify our fraction. The number 2 is also divisible by both numbers, and it gives us a fraction 4 ⁄ 6. While this fraction is simplified, it is still not the simplest form.
The number 3 is not divisible to both numbers as the answer is not a whole number. This brings us to the lucky number 4. The number 8 divided by 4 is 2, and 12 divided by 4 is 3. Four is divisible in both numbers, which will give us the simplified fraction of ⅔.
This is one of the easiest methods to simplify fractions.
At this point, you may think that this calculation is simple only because the numbers are small. But you can use divisible numbers for any set of numbers.
Here is another example — we will use the fraction 800 ⁄ 1000. Many numbers can be divisible by the numerator and denominator, but we need to find the largest one. In this case, 1, 2, 4, 5, 8, 10, 20, 40, 50, 100, and 200 are all divisible numbers.
Since 200 is the largest number, it is the greatest common factor, which would leave us with the simplified fraction of ⅘.
This is not the only method that you can use to add unlike denominators together, so let’s take some time to explore other strategies that may work for you.
Other Strategies to Add Unlike Denominators
Another simple strategy for adding unlike denominators is taking the denominator of one fraction and multiplying it with both the numerator and denominator of the other fraction and vice versa (source).
For example, our two fractions are ½ and ⅗. To add these two together, we multiply the denominator of the first fraction with the numerator and denominator of the second. In this case, it would be 2 multiplied by 3, which gives us 6, and 2 multiplied by 5, which provides us with 10.
The new fraction would be 6 ⁄ 10.
On the other side, it would be 5 multiplied by 1, which gives us 5, and 5 multiplied by 2, which gives us 10. That new fraction would be 5 ⁄ 10.
Then you can combine 6 ⁄ 10 and 5 ⁄ 10, which will give you a new fraction of 11 ⁄ 10. Again, we refer to this as an improper fraction due to the numerator being larger than the denominator.
This process is quite similar to the backward Z method, but it takes one step to complete it instead of several.
What Is a Fraction?
Remember when you were a child, and your parents told you that it’s good to share? Even before you ever set foot into a math class, you already had an idea about fractions because you no doubt shared a cookie with your best friend at one point or another.
A fraction is a portion of a whole (source). When you want to discuss only a section of the whole, you can examine it in two ways: decimals or fractions.
Numerators and denominators make up fractions. The numerator is the part of the whole referenced, and the denominator is the number of parts that the whole is divided into (source).
To understand the meaning of any word, including tricky math terms, get The Oxford New Essential Dictionary on Amazon, and you’ll become a whizz at both English and math vocabulary.
Fractions are pretty straightforward, though, and even the most inept at mathematics can understand a half (½), a quarter (¼), and a third (⅓). Fractions generally refer to equal parts, and beginning your understanding of fractions with equal parts of a whole helps you visualize the process.
Examples of Straightforward Fractions
For example, imagine you have an apple. If you split the apple, you will have two halves. If you want to keep the apple for yourself, you have eaten the whole thing. Mathematically, it is ½ + ½ = 1.
If you split the apple between four friends, we would represent this mathematically as ¼ + ¼ + ¼ + ¼ = 1. Fractions are pretty easy to understand in their foundational stage, and many students have spent hours coloring in blocks to represent different fractions.
But when dealing with larger sections of a whole, fractions are different. Imagine that you are eating a pizza after a long day at work. You cut the pizza into eight equal pieces. You eat three of those pieces.
Mathematically, that is ⅛ + ⅛ + ⅛ = ⅜. Therefore, you have eaten three-eights of the pizza, and you still have some leftovers for your work lunch tomorrow.
Confusion with fractions comes in when we do not have an equal part to reference.
Imagine that you have a friend who has no idea how to cut a cake. Instead of cutting a yummy triangular piece, they cut a crescent out of the cake. This piece would not be an equal part, and it would be challenging to write down in its fraction form.
How are Fractions Used?
Many math students will often ask the question, “But when are we ever going to use this in real life?” Besides needing fractions to understand basic mathematical concepts and pass math class, fractions have quite a few real-world purposes (source).
Imagine that you and your friends have gone out to eat for a celebration. We can use fractions to break down the bill and decide who will be paying what.
For cooking and baking, you need fractions to understand how to divide a recipe when needed. Cocktail recipes are quite complicated as well, and only a thorough understanding of fractions will allow you to make a fantastic drink.
Fractions even have a place in medicine. Many medications have to be given based on the weight of a child or adult. Therefore, fractions are vital to know how to provide the correct dosage.
Finally, money is a massive part of our lives. Whether it comes to your payslip and deductions or counting out change, understanding fractions is a vital part of being a functioning adult and knowing when you’re getting a bargain.
You might also wonder whether something is pricey or pricy. The article “Pricey or Pricy: Differences and Usage” will provide you with an answer. This article was written for strategiesforparents.com.
Perhaps you’ve heard the story about how A&W tried to compete with McDonald’s in the 1980s by selling a ⅓ pound burger to compete with McDonald’s Quarter Pounder. Unfortunately, not many people realized that a ⅓ was more than a ¼ and the product flopped.
If only they had read this article!
Fractions are an essential part of our elementary math education, and we need to be aware of how to apply them in different ways. So whether it’s to complete an assignment, split a pizza, or measure time, the backward Z process should make them easier.