Ordering Fractions, Decimals, and Percentages
Welcome to this chapter! Have you ever been shopping and seen one sign saying "1/4 off" and another saying "20% off" and wondered which was the better deal? Or perhaps you've looked at sports stats where some are written as decimals and others as percentages? In this section, we are going to learn how to compare these different types of numbers and put them in order from smallest to largest (or vice versa). By the end of these notes, you'll be a pro at making sense of these numbers, no matter how they are written!
1. The Language of Comparison (Symbols)
Before we start ordering numbers, we need to know the "maths shorthand" for comparing them. These symbols are called inequalities.
The Symbols You Need to Know:
\( = \) means Equal to (The values are exactly the same).
\( \neq \) means Not equal to.
\( < \) means Less than.
\( > \) means Greater than.
\( \le \) means Less than or equal to.
\( \ge \) means Greater than or equal to.
Memory Aid: The Hungry Alligator
Don't worry if you get the \( < \) and \( > \) symbols mixed up! Just imagine the symbol is the mouth of a hungry alligator. The alligator is very greedy, so its mouth always opens toward the bigger number because it wants the biggest meal!
Example: \( 10 > 2 \) (The alligator eats the 10) or \( 5 < 12 \) (The alligator eats the 12).
Key Takeaway: Symbols help us show the relationship between two numbers without using words. Always point the "wide end" of the symbol toward the larger value.
2. The "Common Language" Strategy
Comparing a fraction like \( \frac{3}{4} \) to a decimal like \( 0.72 \) is a bit like trying to compare the weight of an apple in grams to the weight of an orange in ounces—it’s confusing because they are in different "languages."
To order them easily, we need to convert them all into the same format. Usually, converting everything into decimals is the easiest way to compare them.
How to Convert to Decimals (Quick Recap):
- From Percentages: Divide by 100 (move the decimal point two places to the left).
Example: \( 75\% = 0.75 \) - From Fractions: Treat the fraction bar as a division sign. Divide the top number by the bottom number.
Example: \( \frac{4}{5} = 4 \div 5 = 0.8 \)
Did You Know?
The word "percent" literally means "per hundred." This is why \( 25\% \) is the same as \( \frac{25}{100} \) or \( 0.25 \). Thinking of them as "parts of a hundred" can make them much less intimidating!
Key Takeaway: Don't try to guess! Convert all numbers into decimals before you start putting them in order.
3. How to Compare Decimals Correctly
Struggling students often make mistakes here. For example, many people think \( 0.09 \) is bigger than \( 0.7 \) because \( 9 \) is bigger than \( 7 \). But this is a trap!
The "Fill the Gaps" Trick
To avoid mistakes, follow these steps:
- Line up the decimal points of all your numbers.
- Add "placeholder zeros" to the end of the numbers so they all have the same amount of digits after the decimal point.
- Now, look at them as if they were whole numbers.
Example: Compare \( 0.7 \) and \( 0.09 \)
Step 1 & 2: Change \( 0.7 \) into \( 0.70 \).
Step 3: Compare \( 0.70 \) and \( 0.09 \).
Now it is obvious that \( 0.70 \) (which is 70) is much bigger than \( 0.09 \) (which is 9).
Quick Review Box:
- \( 0.5 \) is the same as \( 0.50 \) and \( 0.500 \).
- Adding zeros to the right of a decimal doesn't change its value, but it makes it easier to read!
Key Takeaway: Always use placeholder zeros so that every number has the same number of decimal places before you compare them.
4. Dealing with Negative Numbers
The syllabus requires you to order negative numbers as well (like \( -0.9 \)).
Think of Temperature:
Imagine a thermometer. The lower the number, the colder it is.
\( -10^{\circ}C \) is much colder (smaller) than \( -2^{\circ}C \).
So, \( -0.9 \) is smaller than \( -0.1 \).
Common Mistake: Thinking \( -5 \) is bigger than \( -1 \) because 5 is bigger than 1. Remember: with negative numbers, the "bigger" the digit looks, the smaller the actual value is!
Key Takeaway: On a number line, numbers get smaller as you move to the left. \( -0.9 \) is further left than \( -0.5 \), so it is smaller.
5. Step-by-Step Guide: Ordering a Mixed List
Let's try a real exam-style question.
Question: Put these in order from smallest to largest: \( \frac{4}{5}, \frac{3}{4}, 0.72, -0.9 \)
Step 1: Convert everything to decimals.
\( \frac{4}{5} = 0.8 \)
\( \frac{3}{4} = 0.75 \)
\( 0.72 = 0.72 \)
\( -0.9 = -0.9 \)
Step 2: Add placeholder zeros.
\( 0.80 \)
\( 0.75 \)
\( 0.72 \)
\( -0.90 \)
Step 3: Compare and order.
The negative number is definitely the smallest: \( -0.90 \).
Looking at the others: \( 0.72 \) is smaller than \( 0.75 \), and \( 0.75 \) is smaller than \( 0.80 \).
So the order is: \( -0.90, 0.72, 0.75, 0.80 \).
Step 4: Write the final answer using the ORIGINAL numbers.
Answer: \( -0.9, 0.72, \frac{3}{4}, \frac{4}{5} \)
Key Takeaway: Examiners want to see the numbers in the format they gave them to you! Use your decimals to do the work, but convert them back for your final answer.
Final Summary Checklist
- Do I know my symbols? (\( < \) is less than, \( > \) is greater than)
- Have I converted everything to decimals first?
- Did I use placeholder zeros to line up the decimals?
- Did I remember that larger negative numbers are actually "smaller"?
- Did I write my final answer using the original fractions and percentages?
Don't worry if this seems tricky at first—with a bit of practice, you'll be spotting the largest values in your sleep!