Work out the value of [tex]$x$[/tex].
[tex]$x-5: 2 x$[/tex] is equivalent to [tex]$1: 3$[/tex]
Answer (2)
Set up the proportion as an equation: 2 x x − 5 = 3 1 .
Cross-multiply to get rid of the fractions: 3 ( x − 5 ) = 2 x .
Expand and simplify the equation: 3 x − 15 = 2 x .
Solve for x : x = 15 . The value of x is 15 .
Explanation
Understanding the Problem We are given the proportion x − 5 : 2 x is equivalent to 1 : 3 . This means that the ratio 2 x x − 5 is equal to the ratio 3 1 . Our goal is to find the value of x that satisfies this proportion.
Setting up the Equation To solve the proportion, we can set up the equation: 2 x x − 5 = 3 1
Cross-Multiplying Now, we can cross-multiply to eliminate the fractions. This means multiplying both sides of the equation by 3 and by 2 x : 3 ( x − 5 ) = 1 ( 2 x ) Expanding the left side gives: 3 x − 15 = 2 x
Isolating x Next, we want to isolate x on one side of the equation. To do this, we can subtract 2 x from both sides: 3 x − 2 x − 15 = 2 x − 2 x x − 15 = 0
Solving for x Finally, we add 15 to both sides of the equation to solve for x : x − 15 + 15 = 0 + 15 x = 15 Therefore, the value of x that satisfies the given proportion is 15 .
Checking the Answer We can check our answer by substituting x = 15 back into the original proportion: 2 ( 15 ) 15 − 5 = 30 10 = 3 1 Since this is the same as the given ratio 1 : 3 , our answer is correct.
Examples
Understanding proportions is incredibly useful in everyday life. For instance, when you're baking, you might need to adjust ingredient quantities based on a recipe that serves a different number of people. If a recipe for 4 people calls for 2 cups of flour, and you want to bake for 6 people, you can set up a proportion to find the new amount of flour needed. This ensures that your baked goods turn out just right, maintaining the correct ratios of ingredients.
The value of x that satisfies the proportion x − 5 : 2 x being equivalent to 1 : 3 is 15 . We derive this value by setting up an equation, cross-multiplying, and isolating x . The solution is confirmed by checking if the original proportions match.
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