Decimals (Interconversion between a decimal and a fraction)

Welcome to the World of Number Changing!

In this chapter, we are going to learn a very cool "magic trick" in Mathematics: how to turn fractions into decimals and back again! Think of it like a superhero who can change their outfit but stays the same person. Whether we write \( \frac{1}{2} \) or \( 0.5 \), we are talking about the exact same amount.

Understanding how to switch between these two forms helps us compare numbers easily, especially when we are shopping, measuring ingredients, or calculating our test scores. Let’s dive in!

1. Changing a Fraction into a Decimal

The easiest way to remember how to turn a fraction into a decimal is to look at the fraction bar. Did you know that the line in a fraction actually means "divided by"?

For example, \( \frac{3}{4} \) is just another way of saying 3 divided by 4.

How to do it (Step-by-Step):

1. Take the numerator (the top number).
2. Divide it by the denominator (the bottom number).
3. Use long division to find the answer.

Example: Convert \( \frac{3}{8} \) to a decimal.
We calculate \( 3 \div 8 \).
\( 8 \) cannot go into \( 3 \), so we add a decimal point and some zeros: \( 3.000 \div 8 \).
\( 30 \div 8 = 3 \) (remainder 6)
\( 60 \div 8 = 7 \) (remainder 4)
\( 40 \div 8 = 5 \)
So, \( \frac{3}{8} = 0.375 \).

What if the division never ends?

Don't worry if this seems tricky at first! Some divisions keep going forever (like \( \frac{1}{3} \)). According to our P6 syllabus, if a decimal is very long, we usually round off the result to the nearest tenth or nearest hundredth.

Quick Review:
To change a fraction to a decimal: Top \( \div \) Bottom.

2. Changing a Decimal into a Fraction

To turn a decimal back into a fraction, we need to listen to the "name" of the decimal based on its place value.

The Memory Trick:

Count the number of decimal places (numbers after the dot):
- 1 decimal place (e.g., 0.6) = Tenths place. Use 10 as the denominator. \( \frac{6}{10} \)
- 2 decimal places (e.g., 0.25) = Hundredths place. Use 100 as the denominator. \( \frac{25}{100} \)
- 3 decimal places (e.g., 0.125) = Thousandths place. Use 1000 as the denominator. \( \frac{125}{1000} \)

How to do it (Step-by-Step):

1. Identify the place value of the last digit.
2. Write the decimal as a fraction with that place value as the denominator.
3. Simplify the fraction to its lowest terms (reduce it until it can't get any smaller).

Example: Convert 0.65 to a fraction.
- There are 2 decimal places, so the denominator is 100.
- The fraction is \( \frac{65}{100} \).
- Now, simplify! Both numbers can be divided by 5.
- \( 65 \div 5 = 13 \) and \( 100 \div 5 = 20 \).
- So, \( 0.65 = \frac{13}{20} \).

Key Takeaway: Always remember to simplify your final fraction answer!

3. Comparing Fractions and Decimals

Sometimes, a question will ask you which is bigger: \( \frac{3}{5} \) or \( 0.7 \)? It is very hard to compare them when they are in different "outfits."

The Pro Strategy:

The easiest way to compare them is to convert the fraction into a decimal first. Once they both look like decimals, you can compare them easily just like comparing money!

Example: Which is larger, \( \frac{5}{8} \) or \( 0.6 \)?
1. Convert \( \frac{5}{8} \) to a decimal: \( 5 \div 8 = 0.625 \).
2. Now compare: Is \( 0.625 \) larger than \( 0.6 \)?
3. Hint: Think of \( 0.6 \) as \( 0.600 \).
4. Since \( 0.625 \) is bigger than \( 0.600 \), then \( \frac{5}{8} \) is larger than \( 0.6 \).

Did you know?
Comparing decimals is like comparing heights. We line up the decimal points to see which number has a bigger value in the tenths or hundredths place!

4. Common Mistakes to Avoid

1. Forgetting the zeros: When dividing, make sure you align your decimal point correctly. \( 1 \div 2 \) is \( 0.5 \), not \( 5.0 \)!
2. Stopping too early: When simplifying fractions, keep checking if you can divide the top and bottom by 2, 3, or 5 one more time.
3. Miscounting places: Double-check if a decimal has one, two, or three places. For example, \( 0.04 \) is \( \frac{4}{100} \), but \( 0.4 \) is \( \frac{4}{10} \). They are very different!

Summary Checklist

- Fraction to Decimal: Divide the top by the bottom.
- Decimal to Fraction: Use the place value (10, 100, 1000) and simplify.
- Comparing: Change everything to decimals to see which is bigger.
- Rounding: If the decimal is too long, use the \( \approx \) symbol to round to the nearest tenth or hundredth.

You’ve got this! Practice these steps, and soon you'll be switching between fractions and decimals faster than a calculator!