Ratio X To Y Is 1 To 5 Understanding Correct Statements

When dealing with ratios, it's crucial to understand the relationships they represent. Ratios help us compare quantities and express how much of one thing there is compared to another. In this article, we will delve into the ratio ${ x : y = 1 : 5 }$ and dissect the given statements to determine which ones accurately reflect this relationship. We will explore how to interpret ratios, convert them into fractions, and use them to make comparisons. By the end of this discussion, you will have a solid grasp of how to work with ratios and identify correct statements based on them.

Decoding the Ratio x : y = 1 : 5

To begin, let's break down what the ratio ${ x : y = 1 : 5 }$ actually means. This ratio tells us that for every one unit of ${ x }$, there are five units of ${ y }$. In other words, ${ y }$ is five times larger than ${ x }$. This foundational understanding is key to evaluating the statements that follow. We can visualize this relationship by imagining dividing a whole into six parts, where ${ x }$ occupies one part and ${ y }$ occupies the remaining five parts. This visual representation can be incredibly helpful in grasping the proportional relationship between ${ x }$ and ${ y }$.

Converting Ratios to Fractions

The ratio ${ x : y = 1 : 5 }$ can also be expressed as fractions, which provides another way to understand the relationship between ${ x }$ and ${ y }$. We can say that ${ x }$ is ${ \frac{1}{6} }$ of the total (${ x + y }$), and ${ y }$ is ${ \frac{5}{6} }$ of the total. This is because if we consider the total as ${ 1 + 5 = 6 }$ parts, ${ x }$ represents one part out of six, and ${ y }$ represents five parts out of six. Converting ratios to fractions allows us to make direct comparisons and perform calculations more easily. For instance, if we know the total value of ${ x + y }$, we can easily find the individual values of ${ x }$ and ${ y }$ by multiplying the total by their respective fractions.

Expressing x and y in Terms of Each Other

Another useful way to interpret the ratio is to express ${ x }$ and ${ y }$ in terms of each other. From the ratio ${ x : y = 1 : 5 }$, we can directly infer that ${ y = 5x }$. This equation tells us that ${ y }$ is five times ${ x }$. Conversely, we can also say that ${ x = \frac{1}{5}y }$, which means ${ x }$ is one-fifth of ${ y }$. These expressions are invaluable when we need to substitute or compare the values of ${ x }$ and ${ y }$ in different contexts. For example, if we have an equation involving both ${ x }$ and ${ y }$, we can use these relationships to rewrite the equation in terms of a single variable, making it easier to solve.

Analyzing the Statements

Now, let's examine the given statements in light of our understanding of the ratio ${ x : y = 1 : 5 }$. We will analyze each statement individually to determine its correctness. This involves carefully comparing the statement with the relationships we have derived from the ratio. We will use both fractional representations and direct comparisons to validate or refute each statement.

Statement A: y is 5/6 of x

Statement A: ${ y }$ is ${ \frac{5}{6} }$ of ${ x }$

This statement claims that ${ y = \frac{5}{6}x }$. To verify this, we can refer back to our understanding that ${ y = 5x }$. Clearly, ${ \frac{5}{6}x }$ is not equal to ${ 5x }$, so this statement is incorrect. The fraction ${ \frac{5}{6} }$ implies that ${ y }$ is less than ${ x }$, which contradicts the given ratio where ${ y }$ is five times ${ x }$. Therefore, Statement A can be confidently ruled out.

Statement B: x is 1/5 of y

Statement B: ${ x }$ is ${ \frac{1}{5} }$ of ${ y }$

This statement asserts that ${ x = \frac{1}{5}y }$. As we previously established, this is indeed correct. From the ratio ${ x : y = 1 : 5 }$, we directly derived that ${ x }$ is one-fifth of ${ y }$. This can also be seen by rearranging the equation ${ y = 5x }$ to solve for ${ x }$, which gives us ${ x = \frac{1}{5}y }$. Thus, Statement B accurately reflects the relationship between ${ x }$ and ${ y }$.

Statement C: x is 1/6 of y

Statement C: ${ x }$ is ${ \frac{1}{6} }$ of ${ y }$

This statement suggests that ${ x = \frac{1}{6}y }$. This is incorrect. We know that ${ x }$ is ${ \frac{1}{5} }$ of ${ y }$, not ${ \frac{1}{6} }$. The fraction ${ \frac{1}{6} }$ would imply a different ratio between ${ x }$ and ${ y }$ than the one given. To further illustrate, if ${ x }$ were ${ \frac{1}{6} }$ of ${ y }$, the ratio ${ x : y }$ would be ${ 1 : 6 }$, not ${ 1 : 5 }$. Consequently, Statement C is false.

Statement D: y is 5/6 of (x+y)

Statement D: ${ y }$ is ${ \frac{5}{6} }$ of ${ (x + y) }$

This statement claims that ${ y = \frac{5}{6}(x + y) }$. To verify this, we can use our understanding that the total is divided into six parts, with ${ y }$ occupying five of those parts. In other words, ${ y }$ is indeed ${ \frac{5}{6} }$ of the total ${ (x + y) }$. We can also demonstrate this algebraically. Since ${ x = \frac{1}{5}y }$, we have:

${ y = \frac{5}{6}(x + y) = \frac{5}{6}(\frac{1}{5}y + y) = \frac{5}{6}(\frac{6}{5}y) = y }$

This confirms that Statement D is correct.

Conclusion: Identifying the Correct Statements

In summary, after analyzing the given statements in relation to the ratio ${ x : y = 1 : 5 }$, we have identified two correct statements:

• Statement B: ${ x }$ is ${ \frac{1}{5} }$ of ${ y }$
• Statement D: ${ y }$ is ${ \frac{5}{6} }$ of ${ (x + y) }$

Understanding ratios and their implications is a fundamental skill in mathematics. By converting ratios to fractions, expressing variables in terms of each other, and carefully comparing statements, we can confidently determine the accuracy of different claims. This exercise highlights the importance of a thorough and methodical approach when working with ratios and proportions.

• Ratio ${ x : y = 1 : 5 }$
• Proportional relationships
• Converting ratios to fractions
• Expressing variables in terms of each other
• Verifying mathematical statements
• Understanding ratio relationships
• Ratio and proportion problems
• Fractional representation of ratios
• Comparing quantities
• Mathematical analysis of statements

The Ratio x to y is 1 to 5 Understanding Correct Statements

Which two statements are correct given the ratio of x to y is 1 to 5?