Solving Proportions A Step-by-Step Guide For 47/39 = B/2

Hey guys! Ever stumbled upon a proportion problem and felt a bit lost? Don't worry, you're not alone! Proportions might seem tricky at first, but with a little guidance, they can become super straightforward. In this article, we're going to break down the proportion ${ \frac{47}{39} = \frac{b}{2} }$ step by step, so you'll not only solve this specific problem but also understand the general method for tackling proportions.

Understanding Proportions

So, what exactly is a proportion? At its heart, a proportion is simply a statement that two ratios are equal. Think of a ratio as a comparison between two quantities. For instance, if you have 47 apples and 39 oranges, the ratio of apples to oranges is 47:39, which we can also write as the fraction ${ \frac{47}{39} }$. A proportion then says that this ratio is the same as another ratio. In our case, we're saying that ${ \frac{47}{39} }$ is equal to ${ \frac{b}{2} }$, where 'b' is some unknown quantity we need to find.

Proportions are incredibly useful in everyday life. Imagine you're baking a cake and the recipe calls for certain ratios of ingredients. If you want to make a bigger cake, you'll need to maintain those proportions to ensure it tastes just as delicious. Or, think about scaling maps – the distances on the map are proportional to the actual distances on the ground. Understanding proportions helps us solve problems in cooking, construction, travel, and many other areas.

When we look at a proportion like ${ \frac{47}{39} = \frac{b}{2} }$, we're essentially saying that the relationship between 47 and 39 is the same as the relationship between 'b' and 2. This understanding is key to solving the problem. We need to figure out what 'b' must be to keep this relationship true. To do this, we use a technique called cross-multiplication, which we'll dive into in the next section. But for now, remember that proportions are all about maintaining equal ratios, and they're a powerful tool for solving a wide variety of problems.

The Cross-Multiplication Method

Alright, let's get to the nitty-gritty of solving our proportion: ${ \frac{47}{39} = \frac{b}{2} }$. The most common and efficient way to solve this type of problem is using the cross-multiplication method. This method is based on a fundamental property of proportions: if ${ \frac{a}{b} = \frac{c}{d} }$, then ${ ad = bc }$. In simpler terms, we multiply the numerator of the first fraction by the denominator of the second fraction, and set that equal to the product of the denominator of the first fraction and the numerator of the second fraction.

Applying cross-multiplication to our proportion ${ \frac{47}{39} = \frac{b}{2} }$, we get:

${ 47 \times 2 = 39 \times b }$

This simplifies to:

${ 94 = 39b }$

Now, we have a simple equation to solve for 'b'. To isolate 'b', we need to divide both sides of the equation by 39:

${ \frac{94}{39} = \frac{39b}{39} }$

This gives us:

${ b = \frac{94}{39} }$

So, we've found that the value of 'b' that makes the proportion true is ${ \frac{94}{39} }$. Cross-multiplication is a powerful technique because it transforms a proportion into a standard algebraic equation, which we can then solve using familiar methods. It's a reliable and straightforward way to tackle proportion problems, and it works for all types of proportions, whether the numbers are whole numbers, fractions, or decimals.

The beauty of cross-multiplication lies in its simplicity and efficiency. It eliminates the fractions in the proportion, making the equation easier to handle. This method is a cornerstone of solving proportions and is used extensively in various fields of mathematics and science. Understanding and mastering cross-multiplication is crucial for anyone dealing with proportional relationships.

Step-by-Step Solution

Let's walk through the solution to our proportion ${ \frac{47}{39} = \frac{b}{2} }$ step-by-step, just to make sure we've got it nailed down. This will help solidify your understanding and give you a clear process to follow for future proportion problems.

Step 1: Write down the proportion.

Our proportion is ${ \frac{47}{39} = \frac{b}{2} }$. It's always a good idea to start by clearly writing out the proportion you're trying to solve. This helps you visualize the problem and ensures you don't miss any details.

Step 2: Apply cross-multiplication.

As we discussed, cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and setting that equal to the product of the denominator of the first fraction and the numerator of the second fraction. So, we multiply 47 by 2 and 39 by b:

${ 47 \times 2 = 39 \times b }$

Step 3: Simplify the equation.

Now, let's perform the multiplications:

${ 94 = 39b }$

We've now transformed our proportion into a simple linear equation.

Step 4: Isolate the variable.

To solve for 'b', we need to get it by itself on one side of the equation. Since 'b' is being multiplied by 39, we need to divide both sides of the equation by 39:

${ \frac{94}{39} = \frac{39b}{39} }$

Step 5: Simplify to find the value of b.

Simplifying the right side of the equation, we get:

${ \frac{94}{39} = b }$

So, our solution is ${ b = \frac{94}{39} }$. This means that the value of 'b' that makes the proportion ${ \frac{47}{39} = \frac{b}{2} }$ true is ${ \frac{94}{39} }$.

By following these five steps, you can confidently solve any proportion problem using cross-multiplication. Remember, the key is to understand the principle behind proportions – that they represent equal ratios – and to apply the cross-multiplication method systematically.

Choosing the Correct Answer

Now that we've solved the proportion ${ \frac{47}{39} = \frac{b}{2} }$ and found that ${ b = \frac{94}{39} }$, let's look at the answer choices provided and select the correct one. This step is crucial to ensure that we not only understand the solution process but also can identify the correct answer in a multiple-choice setting.

The answer choices were:

A. ${ \frac{94}{39} }$ B. ${ \frac{78}{47} }$ C. ${ \frac{47}{78} }$ D. ${ \frac{39}{94} }$

By comparing our solution, ${ b = \frac{94}{39} }$, with the answer choices, it's clear that option A, ${ \frac{94}{39} }$, is the correct answer. The other options represent different values and would not satisfy the original proportion.

Choosing the correct answer in a multiple-choice question is more than just finding the solution; it's about verifying that your solution matches one of the provided options. It's always a good idea to double-check your work and compare your answer carefully to the choices given. Sometimes, answer choices might be designed to trick you, so a thorough comparison is essential.

In this case, options B, C, and D are all variations of the numbers 47, 39, and their reciprocals. This is a common tactic in multiple-choice questions – to include answers that might result from a common mistake, such as inverting a fraction or performing the wrong operation. By carefully following our step-by-step solution and comparing it to the answer choices, we can avoid these traps and confidently select the correct answer.

Real-World Applications of Proportions

So, we've conquered the proportion ${ \frac{47}{39} = \frac{b}{2} }$, but you might be wondering,