Match the following to the second part of the proportion based on the picture.
[tex]\begin{array}{c|c|c|c}
\frac{15}{18}= & \frac{8}{2 x-17}= & \frac{15}{2 x-17}= \\
\frac{2 x-17}{6} & \frac{18}{6} & \frac{24}{15}
\end{array}[/tex]
Answer (2)
Simplify the fraction 18 15 to 6 5 .
Analyze the first proportion 18 15 = 6 2 x − 17 and solve for x to find x = 11 .
Analyze the second proportion 2 x − 17 8 = 15 24 and solve for x to find x = 11 .
Analyze the third proportion 2 x − 17 15 = 6 18 and solve for x to find x = 11 .
The matches are: 18 15 = 6 2 x − 17 , 2 x − 17 8 = 15 24 , 2 x − 17 15 = 6 18 .
Explanation
Understanding the Problem We are given three proportions with one side already specified, and we need to match each to its corresponding second part from the options: 6 2 x − 17 , 6 18 , and 15 24 .
Simplifying the First Proportion Let's analyze the first proportion: 18 15 = We can simplify the fraction 18 15 by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us 18 ÷ 3 15 ÷ 3 = 6 5 . Now, let's check the options to see if any of them are equal to 6 5 .
Analyzing the Options for the First Proportion The first option is 6 2 x − 17 . We cannot directly determine if this is equal to 6 5 without knowing the value of x . The second option is 6 18 , which simplifies to 3. This is not equal to 6 5 . The third option is 15 24 . Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3, gives us 15 ÷ 3 24 ÷ 3 = 5 8 . This is also not equal to 6 5 . However, let's consider the case where 18 15 = 6 5 = 6 2 x − 17 . If the denominators are equal, then the numerators must be equal as well. So, 2 x − 17 = 5 . Solving for x , we get 2 x = 22 , so x = 11 . This means that when x = 11 , 18 15 = 6 2 x − 17 .
Analyzing the Second Proportion Now let's analyze the second proportion: 2 x − 17 8 = . We need to find a matching fraction from the options. Let's consider the case where 2 x − 17 8 = 6 18 . Since 6 18 = 3 , we have 2 x − 17 8 = 3 . Solving for x , we get 8 = 3 ( 2 x − 17 ) , so 8 = 6 x − 51 . This gives us 6 x = 59 , so x = 6 59 . Now let's consider the case where 2 x − 17 8 = 15 24 . Simplifying 15 24 gives us 5 8 . So, 2 x − 17 8 = 5 8 . This means 2 x − 17 = 5 , so 2 x = 22 , and x = 11 .
Analyzing the Third Proportion Now let's analyze the third proportion: 2 x − 17 15 = . Let's consider the case where 2 x − 17 15 = 6 18 . Since 6 18 = 3 , we have 2 x − 17 15 = 3 . Solving for x , we get 15 = 3 ( 2 x − 17 ) , so 15 = 6 x − 51 . This gives us 6 x = 66 , so x = 11 . Now let's consider the case where 2 x − 17 15 = 15 24 . Simplifying 15 24 gives us 5 8 . So, 2 x − 17 15 = 5 8 . Cross-multiplying, we get 15 ⋅ 5 = 8 ( 2 x − 17 ) , so 75 = 16 x − 136 . This gives us 16 x = 211 , so x = 16 211 . Let's consider the case where 2 x − 17 8 = 15 24 which simplifies to 2 x − 17 8 = 5 8 , so 2 x − 17 = 5 and 2 x = 22 , so x = 11 . Then 2 x − 17 15 = 2 ( 11 ) − 17 15 = 22 − 17 15 = 5 15 = 3 = 6 18 . So, 2 x − 17 15 = 6 18 when x = 11 .
Final Answer From the above analysis, we can conclude that:
18 15 = 6 2 x − 17 when x = 11 . 2 x − 17 8 = 15 24 when x = 11 . 2 x − 17 15 = 6 18 when x = 11 .
Examples
Proportions are used in everyday life to scale recipes, convert measurements, and calculate discounts. For example, if a recipe calls for 2 cups of flour to make 12 cookies, you can use a proportion to determine how much flour you need to make 36 cookies. Setting up the proportion 12 2 = 36 x , you can solve for x to find that you need 6 cups of flour. Understanding proportions helps in making accurate adjustments in various practical situations.
We matched the proportions as follows: 18 15 = 6 2 x − 17 , 2 x − 17 8 = 15 24 , and 2 x − 17 15 = 6 18 using the value x = 11 for all proportions. Therefore, all proportions hold true when x = 11 .
;
