Zoe's Spending: Figuring Out Her Initial Money

Hey guys! Let's dive into a fun math problem about Zoe's spending habits. Zoe splurged a bit, spending some of her money on books and records. She used up 1/3 of her cash on books and another 2/5 on some cool records. After her shopping spree, she checked her wallet and found she had 36 euros left. The big question is: How much money did Zoe have before she started spending? This is a classic fraction problem, and we’re going to break it down step-by-step, making it super easy to understand. So, grab your thinking caps, and let’s figure out Zoe’s initial savings together!

Breaking Down the Problem: Fractions and Finances

To really nail this problem, let's get comfy with fractions. Fractions are just parts of a whole, and they're crucial in understanding how Zoe spent her money. She spent 1/3 on books, meaning her total money was divided into three parts, and she used one of those parts for books. Then, she spent 2/5 on records, which means her total money was also divided into five parts, and she spent two of those parts on records. Think of it like slicing a pizza – each slice is a fraction of the whole pie.

Now, the trick here is that these fractions have different denominators (the bottom number in a fraction). To easily add or compare them, we need a common denominator. This common denominator will help us see the total proportion of money Zoe spent. We'll find this by figuring out the least common multiple (LCM) of 3 and 5. Once we have a common denominator, we can add the fractions together and see what fraction of her money Zoe spent in total. This is going to make the next steps much smoother, trust me!

Finding the Common Ground: Least Common Multiple (LCM)

Okay, let’s talk about finding the Least Common Multiple, or LCM. This is a super important step because it lets us add those fractions together. The LCM of two numbers is the smallest number that both numbers can divide into evenly. For our problem, we need the LCM of 3 and 5. Think of it this way: what's the smallest number that both 3 and 5 can fit into without any leftovers?

To find the LCM, we can list the multiples of each number. Multiples of 3 are: 3, 6, 9, 12, 15, and so on. Multiples of 5 are: 5, 10, 15, 20, and so on. Notice anything? The smallest number that appears in both lists is 15. So, the LCM of 3 and 5 is 15. This means we’re going to convert our fractions so they both have a denominator of 15. This is our common ground, the magic number that lets us compare and add those fractions easily. Stick with me, we’re getting closer to solving Zoe’s spending mystery!

Calculating Zoe's Total Spending: Adding Fractions

Now that we've found our LCM, 15, we're ready to tackle the fractions. Remember, Zoe spent 1/3 of her money on books and 2/5 on records. To add these fractions, we need to convert them so they both have the denominator of 15. This is like speaking the same language – once the denominators match, adding becomes a breeze!

First, let's convert 1/3 to a fraction with a denominator of 15. To do this, we need to multiply both the numerator (the top number) and the denominator (the bottom number) by the same number so that the denominator becomes 15. In this case, we multiply both by 5 (because 3 x 5 = 15). So, 1/3 becomes 5/15. This means that spending 1/3 of her money is the same as spending 5/15 of her money.

Next up, we convert 2/5 to have a denominator of 15. We need to multiply both the numerator and the denominator by 3 (because 5 x 3 = 15). So, 2/5 becomes 6/15. Now we know that spending 2/5 of her money is the same as spending 6/15 of her money.

Now for the fun part: adding the fractions! We add 5/15 (for books) and 6/15 (for records). When you add fractions with the same denominator, you simply add the numerators and keep the denominator the same. So, 5/15 + 6/15 = 11/15. This means Zoe spent a total of 11/15 of her money. We’re on the home stretch now – we know what fraction she spent, and we know how much money she has left. Let’s put it all together!

Figuring Out the Remainder: What Fraction Is Left?

Alright, we’ve figured out that Zoe spent 11/15 of her money on books and records. Great job! But to figure out how much money she started with, we need to know what fraction of her money is left over. Think of her total money as a whole, which we can represent as the fraction 15/15 (since 15 divided by 15 equals 1). If she spent 11/15, the leftover amount is the difference between her total money (15/15) and what she spent (11/15).

To find the remaining fraction, we subtract the fractions: 15/15 - 11/15. Since the denominators are the same, we just subtract the numerators: 15 - 11 = 4. So, the remaining fraction is 4/15. This means Zoe has 4/15 of her original money left. And we know that this 4/15 is equal to 36 euros. We’re almost there! Knowing this piece of the puzzle is key to finding the total amount she started with. Let’s use this information to crack the final code!

The Final Calculation: How Much Did Zoe Start With?

Okay, guys, this is where all our hard work pays off! We know that 4/15 of Zoe's original money is equal to 36 euros. The question now is: if 4/15 of her money is 36 euros, what was the total amount (15/15)?

To find this, we first need to figure out what 1/15 of her money is worth. If 4/15 equals 36 euros, we can find 1/15 by dividing 36 euros by 4. So, 36 ÷ 4 = 9 euros. This means that 1/15 of Zoe's money is 9 euros. We’re getting so close!

Now that we know the value of 1/15, we can easily find the total amount (15/15). To do this, we multiply the value of 1/15 (which is 9 euros) by 15. So, 9 x 15 = 135 euros. And there we have it! Zoe initially had 135 euros. See how breaking it down step by step made it much easier? We tackled those fractions and calculations like pros!

Wrapping Up: Zoe's Money Mystery Solved!

So, there you have it! We successfully solved the mystery of Zoe's spending. By breaking down the problem into smaller, manageable steps, we were able to figure out that Zoe started with 135 euros. We tackled fractions, found the least common multiple, and worked with remainders. You guys rocked it!

Remember, math problems might seem tricky at first, but with a little patience and a step-by-step approach, you can conquer anything. The key is to understand the basics, like fractions, and then apply them to the problem at hand. Great job everyone! Now you’re equipped to solve similar problems and impress your friends with your math skills. Keep practicing, and you’ll become math whizzes in no time!