Practice With Fraction And Mixed Number Arithmetic

Fraction Style

Fashion styles come and go, but there are certain fashion rules that never change. And you don't have to be a fashion expert to know some of these. For example, socks and sandals? Super comfortable, but not okay. Stripes and plaid? Don't do that. Vertical and horizontal stripes? That's just weird.

Fractions operate in much the same way. Depending on what you're trying to do with your fractions, you may need to color-coordinate. But sometimes you can just pick any two fractions and you're okay. It's all about the operation you're trying to perform. Let's work on practicing arithmetic with fractions and mixed numbers. First, though, let's quickly review what we're talking about.

A fraction is simply a part of a whole number. 1/2, 3/4, 25/26 - these are fractions. 26/26 would just be simplified to 1, which is not a fraction. Fractions consist of two parts, a numerator (which is the top number) and a denominator (which is the bottom number). A mixed number is a whole number and a fraction. Let's say you had 3/2. You could simplify that to 1 ^1/2. That's a mixed number. Okay, let's practice!

Multiplication

Let's start with multiplication. This is the easiest fraction operation. We just multiply the numerators, then multiply the denominators. It's like our closet only has matching colors. So, 1/2 ×1/4 is just 1/8. 2/3 ×3/7? That's 6/21. 5/6 ×7/8? That's 35/48. When you multiply fractions, people are jealous of how good you look no matter what you wear.

Sometimes you need to do a little simplifying at the end. If you multiply 3/4 ×2/3, you get 6/12. That simplifies to 1/2.

With mixed numbers, we can follow the same process, but we first have to convert to an improper fraction. This just means multiplying the whole number times the denominator, then adding it to the numerator. So 1 ^1/2 becomes 3/2. 3 ^2/3 becomes 1^1/3. This is like working with more colors, though maybe they're all Earth tones, so they kind of all still go together.

So what is 4 ^1/2 ×2 ^3/4? 4 ^1/2 becomes 9/2 and 2 ^3/4 becomes 1^1/4. 9/2 ×1^1/4 is 9^9/8. We then convert that back to a mixed number by reversing the process we did before: divide the numerator by the denominator. The answer becomes the whole number and the remainder becomes the numerator. So we have 12 ^3/8.

What about 1 ^1/5 ×1 ^1/6? 1 ^1/5 becomes 6/5 and 1 ^1/6 becomes 7/6. 6/5 ×7/6 is 42/30, or 1 ^12/30. We can simplify that to 1 ^2/5.

Addition And Subtraction

If you want to add or subtract fractions, you need to abide by the fashion police. You can't just add 1/2 to 1/4. That's like going out wearing two kinds of plaid. You're not in a '90s grunge band, are you?

So we need to coordinate. To do that, we need to get the same denominator on both fractions. If we want to add 1/4 and 1/4, we just add the numerators and get 2/4. With 1/2 and 1/4, we need to find the least common denominator, which is 4. We multiply 1/2 ×2/2 to get 2/4. Now we can add them to get 3/4.

What about 3/5 + 2/3? The least common denominator is 15. So we multiply 3/5 ×3/3 to get 9/15. We multiply 2/3 ×5/5 to get 10/15. 9/15 + 10/15 is 19/15. That can be simplified to 1 ^4/15.

What is 5/6 - 1/3? The least common denominator is 6, so we convert 1/3 to 2/6. 5/6 - 2/6 is 3/6, or 1/2.

What about 1/2 - 7/8? 8 is our least common denominator, so 1/2 becomes 4/8. 4/8 - 7/8 is -3/8.

With mixed numbers, it's simplest to convert to improper fractions. So for 3 1/3 - 2 ^1/2, we convert 3 ^1/3 to 10/3 and 2 ^1/2 to 5/2. 6 is our least common denominator. So 10/3 becomes 20/6. 5/2 becomes 15/6. 20/6 - 15/6 is 5/6.

What about 7 ^1/4 + 5 ^4/5? 7 ^1/4 becomes 2^9/4. 5 ^4/5 becomes 2^9/5. Our least common denominator is 20. So 2^9/4 becomes 14^5/20. 29/5 becomes 116/20. Add them together and we have 26^1/20, which is 13 1/20.