Sharing Pizza How Much Is Left A Fraction Problem
Hey guys! Ever been in that situation where you're sharing a pizza with friends and trying to figure out how much everyone ate? It can get tricky with all those fractions flying around! Let's break down a classic pizza-sharing problem and make sure we all get our fair share of the deliciousness – or at least know how much is left over.
The Great Pizza Predicament
Okay, so imagine this: You and your friend are hanging out, and you've got a pizza. You're feeling pretty hungry, so you devour $ rac{3}{8}$ of the whole pie. Your friend's got an appetite too, and they manage to scarf down $ rac{5}{12}$ of the pizza. So far, so good, right? But then, another friend shows up, lured by the irresistible smell of pizza (who can blame them?). This friend eats $ rac{1}{5}$ of what's left of the pizza. The big question is: How much pizza is actually left after this fraction-filled feast?
Step 1: Calculate the Initial Consumption
In this first crucial step, we focus on determining the total fraction of pizza consumed by you and your initial friend. You ate $ rac3}{8}$ of the pizza, and your friend ate $ rac{5}{12}$. To find the combined amount, we need to add these fractions together. However, before we can add fractions, they need to have a common denominator. This is because we can only directly add fractions that represent slices of the same size. Think of it like trying to add apples and oranges – you need a common unit (like “fruit”) to combine them. Finding the Least Common Multiple (LCM) In our case, the denominators are 8 and 12. The least common multiple (LCM) of 8 and 12 is 24. This means we need to convert both fractions to have a denominator of 24. Converting the Fractions To convert $ rac{3}{8}$ to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 38} * \frac{3}{3} = \frac{9}{24}$. Similarly, to convert $ rac{5}{12}$ to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 212} * \frac{2}{2} = \frac{10}{24}$. Adding the Fractions Now that both fractions have the same denominator, we can add them{24} + \frac{10}{24} = \frac{19}{24}$. This result, $ rac{19}{24}$, represents the total fraction of the pizza that you and your friend ate together. It's a significant portion of the pizza, but it also tells us how much was consumed before the third friend arrived. Understanding this initial consumption is key to figuring out how much pizza remains after the third friend joins in.
Step 2: Determine the Remaining Pizza
Now that we know you and your friend ate a combined $ rac19}{24}$ of the pizza, we need to figure out how much pizza is left before the third friend arrives. Think of the whole pizza as one whole, which we can represent as the fraction $ rac{24}{24}$. This might seem a bit obvious, but it's a crucial step in understanding the problem. Subtracting to Find the Remainder To find the remaining fraction, we subtract the fraction of pizza eaten from the whole pizza24} - \frac{19}{24}$. Since the fractions already have a common denominator, the subtraction is straightforward. We simply subtract the numerators{24}$. This means that $ rac{5}{24}$ of the pizza was left when the third friend arrived. It's a little less than a quarter of the pizza, but still a decent amount for someone to snack on. This step highlights the importance of understanding fractions as parts of a whole and how subtraction helps us determine what remains after a portion is taken away. It sets the stage for the final calculation, where we'll consider how much the third friend ate and how much pizza was left in the end.
Step 3: Calculate the Third Friend's Consumption
The third friend arrives, eyes the remaining pizza, and decides to partake in the feast. But here's the crucial detail: they eat $ rac1}{5}$ of what was left. This is a key point – they're not eating $ rac{1}{5}$ of the whole pizza, but $ rac{1}{5}$ of the $ rac{5}{24}$ that remained. This is where understanding the concept of “of” in math becomes important. “Of” often indicates multiplication. Multiplying Fractions To find out how much pizza the third friend ate, we need to multiply $ rac{1}{5}$ by $ rac{5}{24}$. When multiplying fractions, we multiply the numerators together and the denominators together5} * \frac{5}{24} = \frac{1 * 5}{5 * 24} = \frac{5}{120}$. Simplifying the Fraction We can simplify the fraction $ rac{5}{120}$ by dividing both the numerator and the denominator by their greatest common factor, which is 5{120} = \frac{5 ÷ 5}{120 ÷ 5} = \frac{1}{24}$. This means the third friend ate $ rac{1}{24}$ of the whole pizza. It's a smaller portion than what you and your first friend ate, but it's still a piece of the pie! This step emphasizes the importance of careful reading and understanding the context of the problem. The phrase “of what was left” is a crucial clue that guides us to use multiplication to find the portion the third friend consumed.
Step 4: Determine the Final Amount of Pizza Left
We're in the home stretch! We know there was $ rac5}{24}$ of the pizza left before the third friend arrived, and we know the third friend ate $ rac{1}{24}$ of the pizza. Now, to find out how much pizza is left after the third friend's contribution, we simply need to subtract the amount the third friend ate from the amount that was remaining. Subtracting Fractions To find the final amount of pizza left, we subtract $\frac{1}{24}$ from $\frac{5}{24}$24} - \frac{1}{24}$. Since the fractions have the same denominator, we can directly subtract the numerators24} - \frac{1}{24} = \frac{4}{24}$. Simplifying the Fraction We can simplify the fraction $ rac{4}{24}$ by dividing both the numerator and the denominator by their greatest common factor, which is 4{24} = \frac{4 ÷ 4}{24 ÷ 4} = \frac{1}{6}$. This final result, $ rac{1}{6}$, tells us that one-sixth of the pizza is left after everyone has had their fill. This is the answer to our original question! This final step brings together all the previous calculations to arrive at the solution. It reinforces the concept of subtraction in the context of fractions and highlights the importance of simplifying fractions to express the answer in its most concise form.
The Pizza Remainder Revealed
So, after all that fraction fun, we've discovered that there's $ rac{1}{6}$ of the pizza left. Not a huge amount, but enough for a small snack later, maybe? This problem shows how fractions are used in everyday situations, even when we're just sharing a pizza with friends. Next time you're dividing up a pie, you'll be a fraction master!
Key takeaways from this pizza adventure:
- Fractions are essential for representing parts of a whole.
- Adding and subtracting fractions requires a common denominator.
- The word "of" often indicates multiplication in math problems.
- Simplifying fractions gives us the most concise answer.
Practice Makes Perfect
Want to become a fraction whiz? Try tackling similar problems! What if you had a different number of friends, or everyone ate different amounts? The possibilities are endless – and delicious!
