Which proportion correctly shows the equivalence of two fractions?
A) $32 / 116=9 / 29$
B) $18 / 36=144 / 72$
C) $19 / 95=57 / 76$
D) $18 / 36=72 / 144$
Answer (1)
Simplify 116 32 to 29 8 , which is not equal to 29 9 .
Simplify 36 18 to 2 1 and 72 144 to 2 , which are not equal.
Simplify 95 19 to 5 1 and 76 57 to 4 3 , which are not equal.
Simplify 36 18 to 2 1 and 144 72 to 2 1 , which are equal. Therefore, the answer is D .
Explanation
Understanding the Problem We need to determine which of the given proportions correctly shows the equivalence of two fractions. A proportion is correct if the two fractions are equal when simplified. We will simplify each fraction in each option and compare them.
Analyzing Option A Option A: 116 32 = 29 9 .
Simplifying 116 32 , we divide both numerator and denominator by 4: 116 ÷ 4 32 ÷ 4 = 29 8 .
Since 29 8 = 29 9 , option A is incorrect.
Analyzing Option B Option B: 36 18 = 72 144 .
Simplifying 36 18 , we divide both numerator and denominator by 18: 36 ÷ 18 18 ÷ 18 = 2 1 .
Simplifying 72 144 , we divide both numerator and denominator by 72: 72 ÷ 72 144 ÷ 72 = 1 2 = 2 .
Since 2 1 = 2 , option B is incorrect.
Analyzing Option C Option C: 95 19 = 76 57 .
Simplifying 95 19 , we divide both numerator and denominator by 19: 95 ÷ 19 19 ÷ 19 = 5 1 .
Simplifying 76 57 , we divide both numerator and denominator by 19: 76 57 cannot be simplified to 5 1 .
Dividing both numerator and denominator by 19 is not possible since 76 is not divisible by 57. Instead, we can divide 57 by 19 to get 3 and 76 is divisible by 19 to get 4. So, 76 ÷ 19 57 ÷ 19 = 4 3 .
Since 5 1 = 4 3 , option C is incorrect.
Analyzing Option D Option D: 36 18 = 144 72 .
Simplifying 36 18 , we divide both numerator and denominator by 18: 36 ÷ 18 18 ÷ 18 = 2 1 .
Simplifying 144 72 , we divide both numerator and denominator by 72: 144 ÷ 72 72 ÷ 72 = 2 1 .
Since 2 1 = 2 1 , option D is correct.
Conclusion Therefore, the correct proportion is 36 18 = 144 72 .
Examples
Understanding proportions is crucial in everyday life. For instance, when baking, you might need to scale a recipe up or down. If a recipe calls for 2 cups of flour for every 1 cup of sugar, maintaining this proportion is essential for the recipe to turn out correctly. Similarly, in construction, ensuring that the ratio of cement to sand remains consistent is vital for the structural integrity of the concrete. Proportions help us maintain balance and consistency in various practical applications.
