Genevieve is cutting a 60-inch piece of ribbon into a ratio of [tex]$2: 3$[/tex]. Since 2 inches are frayed at one end of the ribbon, she will need to start 2 inches in. This is indicated as 2 on the number line.
[tex]$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$[/tex]
Where will her cut be located? Round to the nearest tenth.
A. 25.2 in
B. 29.4 in.
C. 35.1 in
D. 40.7 in
Answer (1)
Substitute the given values into the formula: x = ( 2 + 3 2 ) ( 62 − 2 ) + 2 .
Simplify the expression: x = ( 5 2 ) ( 60 ) + 2 .
Calculate the value of x: x = 5 2 × 60 + 2 = 24 + 2 = 26 .
The location of the cut is 26.0 inches.
Explanation
Understanding the Problem We are given the formula x = ( m + n m ) ( x 2 − x 1 ) + x 1 to find the location of the cut on the ribbon. We are also given that the ribbon is cut in a ratio of 2:3, so m = 2 and n = 3 . The ribbon starts at 2 inches, so x 1 = 2 . The ribbon is 60 inches long, so the end point is x 2 = 2 + 60 = 62 .
Substituting the Values Now, we substitute the values into the formula: x = ( 2 + 3 2 ) ( 62 − 2 ) + 2
Simplifying the Expression Simplify the expression: x = ( 5 2 ) ( 60 ) + 2
Calculating the Value of x Calculate the value of x: x = 5 2 × 60 + 2 = 5 120 + 2 = 24 + 2 = 26
Final Answer The location of the cut is 26 inches. Since we need to round to the nearest tenth, we can write it as 26.0 inches.
Examples
Understanding ratios and proportions is crucial in many real-life situations. For example, when baking a cake, you need to maintain the correct ratio of ingredients to ensure the cake turns out as expected. Similarly, in construction, mixing cement requires a specific ratio of cement, sand, and water to achieve the desired strength. In finance, understanding ratios helps in analyzing a company's financial performance and making informed investment decisions. This problem demonstrates how ratios are used to divide a length into specific proportions, which is a fundamental concept in various fields.
