Who Won? Vanessa Or Zack's Game Score!

Oct 19, 2025 by ADMIN 39 views

Hey guys! Let's dive into a fun math problem today that's all about comparing scores. We've got Vanessa and Zack playing a game where the lower score wins. Imagine that! Usually, we're all about getting the highest score, but in this game, it's the opposite. So, we need to figure out who snagged the win by looking at their final scores. Vanessa ended up with a score of $-3 rac{5}{8}$ , and Zack scored $-3 rac{2}{3}$ . Our mission, should we choose to accept it, is to figure out which statement explains who won. This is a great chance to brush up on our fraction comparing skills, especially when dealing with negative numbers. It can get a little tricky, but with a solid strategy, we'll nail it. We'll be looking at different ways to compare these mixed numbers, converting them to decimals, finding common denominators, and really understanding what it means for one negative number to be 'less than' another. So, grab your thinking caps, and let's get started on unraveling this score mystery!

Understanding Negative Mixed Numbers

Alright, let's really get a handle on these scores. We're dealing with negative mixed numbers. That means both Vanessa and Zack have scores that are less than zero. The numbers are $-3 rac{5}{8}$ and $-3 rac{2}{3}$ . When we talk about negative numbers, things get a little counter-intuitive compared to positive numbers. For instance, on a number line, $-5$ is to the left of $-3$ . This means $-5$ is smaller than $-3$ . The further left a number is on the number line, the smaller it is. So, when we're looking for the lower score to win, we're actually looking for the number that is further away from zero in the negative direction.

Vanessa's score is $-3 rac{5}{8}$ . This means she has $-3$ whole units and then an additional $rac{5}{8}$ of a unit further into the negative. Zack's score is $-3 rac{2}{3}$ . This means he has $-3$ whole units and an additional $rac{2}{3}$ of a unit further into the negative. Since both scores start with $-3$ , the winner will be determined by which fraction, $rac{5}{8}$ or $rac{2}{3}$ , is larger. The larger the fractional part added to the negative whole number, the smaller (or lower) the overall score will be. This is the core concept we need to keep in mind as we compare these two numbers.

Converting to Improper Fractions

One of the most reliable ways to compare fractions, especially mixed numbers, is to convert them into improper fractions first. This often makes the comparison process more straightforward. Let's take Vanessa's score: $-3 rac{5}{8}$ . To convert this to an improper fraction, we multiply the whole number (3) by the denominator (8) and then add the numerator (5). So, $3 imes 8 = 24$ , and $24 + 5 = 29$ . This gives us $rac{29}{8}$ . Since the original number was negative, Vanessa's score as an improper fraction is $- rac{29}{8}$ .

Now, let's do the same for Zack's score: $-3 rac{2}{3}$ . We multiply the whole number (3) by the denominator (3), which gives us $3 imes 3 = 9$ . Then, we add the numerator (2), so $9 + 2 = 11$ . This gives us $rac{11}{3}$ . Again, because the original score was negative, Zack's score as an improper fraction is $- rac{11}{3}$ .

So now we are comparing $- rac{29}{8}$ and $- rac{11}{3}$ . The game rules state that the player with the lower score wins. Therefore, we need to determine which of these two improper fractions is smaller. This is where understanding negative numbers on a number line becomes super important. The number that is further to the left on the number line is the smaller one. To compare them effectively, we'll need a common denominator.

Finding a Common Denominator

To compare $- rac{29}{8}$ and $- rac{11}{3}$ using a common denominator, we need to find the least common multiple (LCM) of 8 and 3. The multiples of 8 are 8, 16, 24, 32, ... The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, ... The LCM of 8 and 3 is 24. This is awesome because 24 is a multiple of both 8 and 3, making our conversion easy!

Let's convert $- rac{29}{8}$ to an equivalent fraction with a denominator of 24. We need to multiply 8 by 3 to get 24. So, we must also multiply the numerator, 29, by 3. $29 imes 3 = 87$ . Therefore, $- rac{29}{8}$ is equivalent to $- rac{87}{24}$ .

Now, let's convert $- rac{11}{3}$ to an equivalent fraction with a denominator of 24. We need to multiply 3 by 8 to get 24. So, we must also multiply the numerator, 11, by 8. $11 imes 8 = 88$ . Therefore, $- rac{11}{3}$ is equivalent to $- rac{88}{24}$ .

Now we are comparing $- rac{87}{24}$ and $- rac{88}{24}$ . Since the denominators are the same, we just need to compare the numerators. We are looking for the lower score. Remember, with negative numbers, the one with the larger absolute value is actually the smaller number. Between 87 and 88, 88 is larger. Therefore, $- rac{88}{24}$ is smaller than $- rac{87}{24}$ . This means Zack's score, $- rac{11}{3}$ (which is $- rac{88}{24}$ ), is lower than Vanessa's score, $- rac{29}{8}$ (which is $- rac{87}{24}$ ). So, Zack won!

Converting to Decimals

Another super useful way to compare numbers, especially when you're working with fractions, is to convert them into decimals. This often makes comparisons feel more intuitive, especially if you're more comfortable with decimal values. Let's take Vanessa's score of $-3 rac{5}{8}$ . We already know the whole number part is $-3$ . Now we just need to convert the fractional part, $rac{5}{8}$ , into a decimal. To do this, we divide the numerator (5) by the denominator (8). $5 ilde{A}f%C2%B6> 8$ . Performing this division, we get $5 ilde{A}f%C2%B6> 8 = 0.625$ . So, Vanessa's score as a decimal is $-3.625$ .

Now, let's do the same for Zack's score of $-3 rac{2}{3}$ . The whole number part is $-3$ . We need to convert $rac{2}{3}$ into a decimal. Dividing 2 by 3 gives us $2 ilde{A}f%C2%B6> 3 = 0.666...$ . This is a repeating decimal, often written as $0.ar{6}$ . So, Zack's score as a decimal is $-3.666...$ (or $-3.ar{6}$ ).

Now we are comparing $-3.625$ and $-3.666...$ . Remember, in this game, the lower score wins. With negative numbers, the number that is further to the left on the number line is the smaller one. Both scores start with $-3$ . So we look at the decimal part. We are comparing $0.625$ and $0.666...$ . Since $0.666...$ is greater than $0.625$ , the number $-3.666...$ is less than $-3.625$ . In other words, $-3.666...$ is further to the left on the number line than $-3.625$ . This means Zack's score is lower than Vanessa's score. Therefore, Zack won the game!

Comparing Decimal Values

Let's explicitly compare the decimal values we found: $-3.625$ for Vanessa and $-3.666...$ for Zack. When comparing negative numbers, we look for the number that has the larger magnitude (the larger distance from zero). This might sound confusing, but think about it on a number line. Zero is in the middle. $-1$ is to the left of zero, $-2$ is to the left of $-1$ , and so on. The further left you go, the smaller the number gets. So, $-4$ is smaller than $-3$ .

In our case, both numbers are between $-3$ and $-4$ . We look at the digits after the decimal point. Vanessa has $.625$ . Zack has $.666...$ . To compare these, we can think of them as having the same number of decimal places by adding zeros. So, Vanessa's score is $-3.6250$ and Zack's score is $-3.6666...$ . Comparing them digit by digit from left to right after the decimal point:

The tenths place: Both have a 6. They are equal.
The hundredths place: Vanessa has a 2, and Zack has a 6. Since 6 is greater than 2, Zack's number is larger in magnitude after the decimal point.

This means that $-3.666...$ (Zack's score) is further away from zero than $-3.625$ (Vanessa's score). Therefore, $-3.666...$ is the lower score. Since the lower score wins, Zack is the winner!

Which Statement Explains Who Won?

Now that we've done the math, let's look at how we would explain who won. The core idea is that the player with the score that is numerically less than the other player's score is the winner. We've established that Zack's score of $-3 rac{2}{3}$ is lower than Vanessa's score of $-3 rac{5}{8}$ .

Here are a few ways a statement could explain this:

Using Decimal Comparison: "Zack won because his score of approximately $-3.67$ is less than Vanessa's score of $-3.63$ . When comparing negative numbers, the number with the greater absolute value (the one further from zero) is the lower number."
Using Common Denominators: "Zack won because his score of $-3 rac{2}{3}$ can be rewritten as $- rac{11}{3}$ , which is equivalent to $- rac{88}{24}$ . Vanessa's score of $-3 rac{5}{8}$ can be rewritten as $- rac{29}{8}$ , which is equivalent to $- rac{87}{24}$ . Since $- rac{88}{24}$ is less than $- rac{87}{24}$ , Zack's score is lower."
Direct Comparison of Fractions: "Zack won because $rac{2}{3}$ is greater than $rac{5}{8}$ . Since both scores start with $-3$ , the larger fractional part results in a smaller overall negative number. Therefore, $-3 rac{2}{3}$ is less than $-3 rac{5}{8}$ ."

Each of these statements accurately reflects our findings. The key is to correctly compare the two negative mixed numbers and then apply the game's rule: the lower score wins. In this case, Zack's score is unequivocally the lower score, making him the champion of this particular game!

Why Comparing Negatives Matters

It's crucial to get the hang of comparing negative numbers, guys. It's a concept that trips up a lot of people because it's the opposite of comparing positive numbers. With positive numbers, bigger means more. With negative numbers, bigger means less! For example, $5$ is greater than $3$ , but $-5$ is less than $-3$ . This is because $-5$ is further to the left on the number line. When you're comparing $-3 rac{5}{8}$ and $-3 rac{2}{3}$ , you're essentially asking which number is further left on the number line. Since $rac{2}{3}$ is a larger fraction than $rac{5}{8}$ (in terms of value), adding it to $-3$ makes the number more negative (i.e., further left) than adding $rac{5}{8}$ to $-3$ . So, $-3 rac{2}{3}$ is indeed less than $-3 rac{5}{8}$ . This is why Zack wins!

Final Check: Visualizing on a Number Line

To really cement this, let's picture a number line. Mark $-3$ and $-4$ . Both scores fall between these two integers. Now, we need to see which one is closer to $-4$ (further left). We are comparing the fractional parts: $rac{5}{8}$ and $rac{2}{3}$ . We found that $rac{2}{3}$ is approximately $0.666...$ and $rac{5}{8}$ is $0.625$ . So, we're comparing $-3.666...$ and $-3.625$ . Since $0.666...$ is greater than $0.625$ , the number $-3.666...$ is further from zero than $-3.625$ . On the negative side, further from zero means smaller. So, Zack's score is smaller. Boom! Zack wins. It’s all about remembering that on the negative side, bigger magnitude equals smaller value. Keep practicing this, and you'll be a pro at number line comparisons in no time!

So, to wrap it all up, Zack took home the win in this particular game because his score was mathematically lower than Vanessa's. It’s a fantastic example of how crucial it is to understand fraction and negative number comparisons. Great job working through this, everyone!