Solving: 73/4 Minus 21/6 - Simple Steps!

by Jhon Lennon 41 views

Hey guys! Let's break down how to solve this fraction problem: 73/4 - 21/6. Don't worry, it's easier than it looks! We'll go through it step by step, so you can totally nail it.

Understanding the Problem

So, we have two fractions here: 73/4 and 21/6. Our mission is to subtract the second one from the first. But before we jump into subtraction, we need to make sure these fractions are speaking the same language – in math terms, they need a common denominator. Think of it like trying to compare apples and oranges; you need to find a common unit to compare them properly.

Why Common Denominators Matter

The denominator is the bottom number in a fraction, and it tells us how many equal parts the whole is divided into. When denominators are different, the sizes of the parts are different. You can't directly add or subtract fractions with different denominators because you're not dealing with the same 'size' of pieces. Imagine trying to add a slice from a pizza cut into 4 pieces with a slice from a pizza cut into 6 pieces – they're not the same size, right? That’s why we need a common denominator: to ensure we're working with equal-sized pieces.

Finding the Common Denominator

The easiest way to find a common denominator is to look for the least common multiple (LCM) of the two denominators. In our case, the denominators are 4 and 6. The LCM is the smallest number that both 4 and 6 can divide into evenly. Let's list some multiples of 4 and 6 to find it:

  • Multiples of 4: 4, 8, 12, 16, 20, 24, ...
  • Multiples of 6: 6, 12, 18, 24, 30, ...

Notice that 12 appears in both lists. So, the least common multiple of 4 and 6 is 12. That means we'll convert both fractions to have a denominator of 12. This ensures we're comparing and subtracting equal-sized pieces, making the math accurate and straightforward. Now that we have our common denominator, let's move on to converting the fractions!

Converting the Fractions

Okay, now that we know our common denominator is 12, we need to convert both fractions so they have this denominator. This involves multiplying both the numerator (the top number) and the denominator (the bottom number) of each fraction by a certain number to get the denominator to be 12. It's like scaling the fraction up or down without changing its actual value. Let's tackle each fraction one by one.

Converting 73/4

To convert 73/4 to a fraction with a denominator of 12, we need to figure out what number we need to multiply 4 by to get 12. The answer is 3, because 4 x 3 = 12. So, we multiply both the numerator and the denominator of 73/4 by 3:

(73 x 3) / (4 x 3) = 219/12

So, 73/4 is equivalent to 219/12. We've successfully converted the first fraction to have the desired denominator.

Converting 21/6

Next, we need to convert 21/6 to a fraction with a denominator of 12. We need to find the number that, when multiplied by 6, gives us 12. That number is 2, since 6 x 2 = 12. Therefore, we multiply both the numerator and the denominator of 21/6 by 2:

(21 x 2) / (6 x 2) = 42/12

So, 21/6 is equivalent to 42/12. We've now successfully converted both fractions to have a common denominator of 12. This means we can now perform the subtraction, as both fractions are expressing the same 'size' of pieces. With both fractions now in terms of twelfths, we're ready to subtract!

Subtracting the Fractions

Alright, with both fractions now sporting the same denominator, we can finally subtract them! We have 219/12 and 42/12. To subtract fractions with a common denominator, you simply subtract the numerators and keep the denominator the same. It’s like saying, "I have 219 slices of a pie cut into 12 pieces each, and I'm taking away 42 slices. How many slices do I have left?"

Performing the Subtraction

So, we subtract the numerators:

219 - 42 = 177

The denominator remains the same, which is 12. Therefore, the result of the subtraction is:

177/12

So, 73/4 - 21/6 = 177/12. We've done the subtraction! But hold on, we're not quite finished yet. It's good practice to simplify the fraction if possible, to make it easier to understand and work with. Let's see if we can simplify 177/12.

Simplifying the Result

Now that we've subtracted the fractions, we have the result 177/12. Simplifying a fraction means reducing it to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by that number. Let's find the GCD of 177 and 12.

Finding the Greatest Common Divisor (GCD)

The GCD is the largest number that divides both 177 and 12 without leaving a remainder. One way to find it is to list the factors of both numbers:

  • Factors of 177: 1, 3, 59, 177
  • Factors of 12: 1, 2, 3, 4, 6, 12

The largest number that appears in both lists is 3. So, the GCD of 177 and 12 is 3.

Dividing by the GCD

Now, we divide both the numerator and the denominator by the GCD:

(177 ÷ 3) / (12 ÷ 3) = 59/4

So, 177/12 simplified is 59/4. This is an improper fraction (where the numerator is greater than the denominator), so we can convert it to a mixed number to make it easier to understand. An improper fraction is where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means the fraction represents a value that is one whole or greater. Converting it to a mixed number can make it easier to visualize and understand the quantity the fraction represents.

Converting to a Mixed Number

To convert the improper fraction 59/4 to a mixed number, we divide the numerator (59) by the denominator (4). The quotient (the result of the division) becomes the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part. The denominator stays the same.

Performing the Division

When we divide 59 by 4, we get:

  • Quotient: 14
  • Remainder: 3

So, the mixed number is 14 and 3/4. Therefore, 59/4 is equal to 14 3/4. This tells us that the fraction represents 14 whole units and 3/4 of another unit. Converting improper fractions to mixed numbers helps in understanding the actual value, especially in real-world scenarios. Now, let's summarize our findings!

Final Answer

So, to recap:

  • We started with the problem: 73/4 - 21/6
  • We found a common denominator: 12
  • We converted the fractions: 73/4 = 219/12 and 21/6 = 42/12
  • We subtracted the fractions: 219/12 - 42/12 = 177/12
  • We simplified the result: 177/12 = 59/4
  • We converted to a mixed number: 59/4 = 14 3/4

Therefore, 73/4 - 21/6 = 14 3/4. Awesome job! You've successfully solved the problem! You rock!