Fractions

Imagine you have a delicious chocolate bar, and you want to share it equally with your best friend. You'd break it into two pieces, right? Each piece is a fraction of the whole chocolate bar. A fraction is just a way to show a part of a whole thing.

Think of it like a slice of pizza. If a pizza has 8 slices, and you eat 3 of them, you've eaten 3/8 (three-eighths) of the pizza. The top number (the numerator) tells you how many parts you have, and the bottom number (the denominator) tells you how many total parts make up the whole thing. The line in between just means "out of" or "divided by."

So, 3/8 means "3 parts out of 8 total parts." Easy peasy!

Let's say you're baking cookies, and the recipe calls for 3/4 (three-quarters) of a cup of sugar. What does that mean?

1. Look at the denominator (the bottom number): It's 4. This means your measuring cup needs to be thought of as being divided into 4 equal parts.
2. Look at the numerator (the top number): It's 3. This means you need to fill up 3 of those 4 parts.
3. So, you'd fill your measuring cup almost to the top, but not quite – leaving one quarter empty. You've just used fractions to bake delicious cookies! This is much more precise than just guessing.

Let's learn how to add and subtract fractions, which is like combining or taking away pieces of a pie. The most important rule is that the pieces (the denominators) must be the same size!

1. Find a Common Denominator: If your fractions have different bottom numbers (denominators), you need to make them the same. Think of it like making sure all your pizza slices are the same size before you can count them together. You do this by finding the Least Common Multiple (LCM) of the denominators, which is the smallest number both denominators can divide into evenly. For example, if you have 1/2 and 1/3, the LCM of 2 and 3 is 6.
2. Change the Numerators: Once you've found the common denominator, you need to change the top number (numerator) of each fraction to match. Whatever you multiply the bottom number by to get the new common denominator, you must do the exact same thing to the top number. So, 1/2 becomes 3/6 (because 2 x 3 = 6, so 1 x 3 = 3).
3. Add or Subtract the Numerators: Now that the bottom numbers are the same, you can simply add or subtract the top numbers (numerators). The common denominator stays the same. So, 3/6 + 2/6 = 5/6.
4. Simplify (if possible): Sometimes, your answer can be made simpler. This is like reducing a fraction to its smallest form, where the top and bottom numbers can't be divided by any common number other than 1. For example, 2/4 can be simplified to 1/2.

Multiplying and dividing fractions are actually often easier than adding or subtracting! No need for common denominators here.

Multiplying Fractions (Like finding a fraction of a fraction):

1. Multiply Straight Across: Just multiply the top numbers (numerators) together, and multiply the bottom numbers (denominators) together. For example, (1/2) * (3/4) = (13)/(24) = 3/8.
2. Simplify: Reduce your answer to its simplest form if possible.*

Dividing Fractions (Like seeing how many times one fraction fits into another):

1. "Keep, Change, Flip": This is a super helpful trick! Keep the first fraction the same, change the division sign to a multiplication sign, and flip the second fraction upside down (this is called finding its reciprocal). For example, (1/2) ÷ (1/4) becomes (1/2) * (4/1).
2. Multiply: Now that it's a multiplication problem, just multiply straight across as you learned above. So, (1/2) * (4/1) = 4/2.
3. Simplify: Reduce your answer. 4/2 simplifies to 2.

These are just different ways of saying the same thing! Like having different nicknames for the same person.

• Fraction to Decimal: Divide the top number (numerator) by the bottom number (denominator). Think of the fraction bar as a division sign. So, 1/2 = 1 ÷ 2 = 0.5.
• Decimal to Percentage: Multiply the decimal by 100 and add a percent sign (%). Or, just move the decimal point two places to the right. So, 0.5 = 0.50 = 50%.
• Percentage to Decimal: Divide the percentage by 100 (which means moving the decimal point two places to the left) and remove the percent sign. So, 50% = 0.50.
• Decimal to Fraction: This can be a bit trickier. Look at the place value of the last digit. If it's 0.5, the 5 is in the tenths place, so it's 5/10, which simplifies to 1/2. If it's 0.25, the 5 is in the hundredths place, so it's 25/100, which simplifies to 1/4.
• Percentage to Fraction: First convert the percentage to a decimal, then convert the decimal to a fraction.

Even the smartest students make these! Watch out for them.

• Mistake 1: Adding/Subtracting without a Common Denominator.
  • Why it happens: Students forget that you can only add or subtract pieces that are the same size. You can't add apples and oranges directly.
  • How to avoid it: Always check the bottom numbers (denominators) first. If they're different, find the Least Common Multiple (LCM) and change both fractions before you do anything else.
  • ❌ 1/2 + 1/3 = 2/5 (Wrong!)
  • ✅ 1/2 + 1/3 = 3/6 + 2/6 = 5/6 (Right!)
• Mistake 2: Forgetting to Flip the Second Fraction When Dividing.
  • Why it happens: Students remember to multiply but forget the crucial "flip" step.
  • How to avoid it: Remember the "Keep, Change, Flip" rule every single time you see a division sign with fractions. Write it down if you need to!
  • ❌ 1/2 ÷ 1/4 = 1/8 (Wrong!)
  • ✅ 1/2 ÷ 1/4 = 1/2 * 4/1 = 4/2 = 2 (Right!)
• Mistake 3: Not Simplifying Your Answer.
  • Why it happens: You've done all the hard work, but you forget the final step to make your fraction as neat as possible. The SAT often wants answers in simplest form.
  • How to avoid it: After every calculation, ask yourself: "Can I divide the top and bottom numbers by the same number?" Keep dividing until you can't anymore. Think of it like cleaning up your room after playing.
  • ❌ 6/8 (Could be simpler!)
  • ✅ 3/4 (Simplified!)*