Solve For R: Easy Math Equation

by Andrew McMorgan 32 views

Hey guys, let's dive into a super straightforward algebra problem that'll get your brains warmed up. We're going to tackle the equation:

r9=1260\frac{r}{9}=\frac{12}{60}

Our mission, should we choose to accept it, is to solve for r. This means we need to find the value of 'r' that makes this equation true. It's like finding a hidden treasure, but the treasure is a number! We've got a few options to choose from: A. $r=5$, B. $r=1.8$, C. $r=20$, and D. $r=6 \frac{2}{3}$. We'll go through this step-by-step, so even if you're just starting out with algebra, you'll totally get it.

Understanding Proportions

First off, what we're looking at here is a proportion. A proportion is simply an equation that states that two ratios are equal. In our case, the ratio on the left is $\frac{r}{9}$ and the ratio on the right is $\frac{12}{60}$. The equals sign in the middle tells us they have the same value. When you're dealing with proportions, there are a few cool tricks you can use to solve for an unknown variable like 'r'. One of the most common methods is cross-multiplication. This is where you multiply the numerator of one fraction by the denominator of the other, and set them equal. It's a super handy technique that works like magic for these kinds of problems. Another approach is to simplify one of the fractions first, if possible, to make the numbers smaller and easier to work with. We'll explore both of these to really nail this down.

Method 1: Cross-Multiplication Magic

Let's start with the most popular method: cross-multiplication. For the equation $\frac{r}{9}=\frac{12}{60}$, we're going to multiply 'r' by 60 and 9 by 12. So, it looks like this:

r×60=9×12r \times 60 = 9 \times 12

Now, let's calculate the right side: $;9 \times 12 = 108$. So our equation becomes:

60r=10860r = 108

To isolate 'r' and find its value, we need to get rid of the 60 that's multiplying it. The opposite of multiplication is division, so we'll divide both sides of the equation by 60.

60r60=10860\frac{60r}{60} = \frac{108}{60}

This simplifies to:

r=10860r = \frac{108}{60}

Now, we need to simplify this fraction. We can see that both 108 and 60 are divisible by common factors. Let's start by dividing both by 12 (since we know 108 = 9 * 12 and 60 = 5 * 12):

108÷1260÷12=95\frac{108 \div 12}{60 \div 12} = \frac{9}{5}

So, $r = \frac{9}{5}$. If we want to express this as a decimal, we divide 9 by 5, which gives us $1.8$. If we want to express it as a mixed number, we see that 5 goes into 9 once with a remainder of 4, so it's $1 \frac{4}{5}$. Let's double-check our options. Option B is $r=1.8$. Bingo! It looks like we've found our answer using cross-multiplication. Pretty neat, huh?

Method 2: Simplify First, Then Solve

Now, let's try another approach to confirm our answer. This time, we'll simplify the fraction $\frac{12}{60}$ before we do any cross-multiplication or other fancy moves. Simplifying fractions is always a good habit, guys, because it makes the numbers way more manageable. So, let's look at $\frac{12}{60}$. Both 12 and 60 are divisible by 12.

12÷1260÷12=15\frac{12 \div 12}{60 \div 12} = \frac{1}{5}

Awesome! So, our original equation $\frac{r}{9}=\frac{12}{60}$ can be rewritten as:

r9=15\frac{r}{9}=\frac{1}{5}

Now, this looks a lot simpler, right? We can solve this in a couple of ways. We could use cross-multiplication again: $;r \times 5 = 9 \times 1$, which gives us $5r = 9$. Dividing both sides by 5, we get $r = \frac{9}{5}$, which is $1.8$. This matches our previous result.

Alternatively, from $\frac{r}{9}=\frac{1}{5}$, we can think about what value of 'r' makes the left side equal to $\frac{1}{5}$. To isolate 'r', we can multiply both sides of the equation by 9:

9×r9=9×159 \times \frac{r}{9} = 9 \times \frac{1}{5}

This simplifies to:

r=95r = \frac{9}{5}

And again, $r = 1.8$. See? Whether you simplify first or use cross-multiplication directly, you end up with the same answer. It's all about consistency and picking the method that makes the most sense to you.

Checking Our Answer

It's always a good practice in math, especially in exams and quizzes, to check your answer to make sure it's correct. This helps prevent silly mistakes and gives you that extra confidence. We found that $r = 1.8$. Let's plug this value back into the original equation: $\frac{r}{9}=\frac{12}{60}$

So, we'll substitute 1.8 for 'r':

1.89=1260\frac{1.8}{9} = \frac{12}{60}

Let's evaluate the left side: $\frac1.8}{9}$. To make this easier, we can multiply both the numerator and the denominator by 10 to get rid of the decimal $\frac{1.8 \times 109 \times 10} = \frac{18}{90}$. Now, we can simplify this fraction. Both 18 and 90 are divisible by 18 $\frac{18 \div 18{90 \div 18} = \frac{1}{5}$.

Now, let's evaluate the right side of the original equation: $\frac{12}{60}$. We already simplified this earlier by dividing both by 12, which gave us $\frac{1}{5}$.

Since both sides of the equation evaluate to $\frac{1}{5}$ when $r=1.8$, our answer is definitely correct! The equation holds true.

Reviewing the Options

Let's quickly look at the given options again:

A. $r=5$: If $r=5$, then $\frac{5}{9} \neq \frac{12}{60}$ (which simplifies to $\frac{1}{5}$).

B. $r=1.8$: We've confirmed that $\frac{1.8}{9} = \frac{1}{5}$ and $\frac{12}{60} = \frac{1}{5}$. This is our correct answer.

C. $r=20$: If $r=20$, then $\frac{20}{9} \neq \frac{12}{60}$.

D. $r=6 \frac2}{3}$ This is $r = 6.66...$ or $\frac{20{3}$ as an improper fraction. If $r=\frac{20}{3}$, then $\frac{20/3}{9} = \frac{20}{27} \neq \frac{12}{60}$.

So, the only option that satisfies the equation is B. $r=1.8$.

Conclusion

Alright guys, we've successfully navigated the equation $\frac{r}{9}=\frac{12}{60}$ and solved for r using two different methods: cross-multiplication and simplifying first. We also took the time to check our answer, which is a crucial step in mastering any math problem. Remember, practice makes perfect, and the more you work through these types of problems, the more comfortable and confident you'll become with algebra. Keep those brains buzzing and happy calculating!