Which one of the following statements expresses a true proportion?
A) [tex]$2: 1=1: 2$[/tex]
B) [tex]$54: 9=6: 3$[/tex]
C) [tex]$3: 4=9: 12$[/tex]
D) [tex]$7: 9=8: 9$[/tex]
Answer (2)
Check if the ratios in option A, 2 : 1 = 1 : 2 , are equal: 2 × 2 = 4 and 1 × 1 = 1 . Since 4 = 1 , it's false.
Check if the ratios in option B, 54 : 9 = 6 : 3 , are equal: 54 × 3 = 162 and 9 × 6 = 54 . Since 162 = 54 , it's false.
Check if the ratios in option C, 3 : 4 = 9 : 12 , are equal: 3 × 12 = 36 and 4 × 9 = 36 . Since 36 = 36 , it's true.
Check if the ratios in option D, 7 : 9 = 8 : 9 , are equal: 7 = 8 , so it's false. The true proportion is C .
Explanation
Understanding Proportions We need to determine which of the given options expresses a true proportion. A proportion is a statement that two ratios are equal. We will check each option to see if the ratios are equal.
Checking Option A Option A: 2 : 1 = 1 : 2 can be written as 1 2 = 2 1 . To check if this is true, we can cross-multiply: 2 × 2 = 4 and 1 × 1 = 1 . Since 4 = 1 , this proportion is false.
Checking Option B Option B: 54 : 9 = 6 : 3 can be written as 9 54 = 3 6 . Cross-multiplying gives 54 × 3 = 162 and 9 × 6 = 54 . Since 162 = 54 , this proportion is false.
Checking Option C Option C: 3 : 4 = 9 : 12 can be written as 4 3 = 12 9 . Cross-multiplying gives 3 × 12 = 36 and 4 × 9 = 36 . Since 36 = 36 , this proportion is true.
Checking Option D Option D: 7 : 9 = 8 : 9 can be written as 9 7 = 9 8 . Since the denominators are the same, we can compare the numerators. Since 7 = 8 , this proportion is false.
Conclusion Therefore, the only true proportion is option C.
Examples
Proportions are used in everyday life to scale recipes, convert currencies, and understand map scales. For example, if a recipe calls for 2 cups of flour for every 1 cup of sugar, maintaining this proportion is crucial when making a larger or smaller batch. If you want to double the recipe, you would need 4 cups of flour for every 2 cups of sugar, keeping the ratio consistent. This ensures the recipe turns out as intended, demonstrating a practical application of proportional reasoning.
After analyzing each option for equality of ratios, the true proportion is found in Option C: 3 : 4 = 9 : 12 . All other options did not meet the criteria of equivalence. Thus, Option C is the correct answer.
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