Determine if 12:16 and 72:96 are equivalent ratios.
The correct answer and explanation is:
Correct Answer:
Yes, 12:16 and 72:96 are equivalent ratios.
Explanation (300 words):
To determine whether two ratios are equivalent, we compare them by simplifying both ratios to their lowest terms or by converting them into fractions and checking if they are equal.
Step 1: Simplify the first ratio (12:16)
Write the ratio as a fraction: 1216\frac{12}{16}
Now simplify by dividing both numerator and denominator by their greatest common divisor (GCD). The GCD of 12 and 16 is 4. 12÷416÷4=34\frac{12 ÷ 4}{16 ÷ 4} = \frac{3}{4}
So, 12:16 simplifies to 3:4.
Step 2: Simplify the second ratio (72:96)
Write this as a fraction: 7296\frac{72}{96}
The GCD of 72 and 96 is 24. 72÷2496÷24=34\frac{72 ÷ 24}{96 ÷ 24} = \frac{3}{4}
So, 72:96 also simplifies to 3:4.
Step 3: Compare the simplified ratios
- 12:16 = 3:4
- 72:96 = 3:4
Since both ratios simplify to the same value (3:4), they are equivalent.
Why This Matters:
Equivalent ratios represent the same relationship between two quantities. In real-life scenarios, equivalent ratios are used in cooking recipes, maps, and scaling models. For instance, if you double a recipe, you must maintain the same ratio of ingredients to keep the taste consistent. This is similar to multiplying both parts of a ratio by the same number.
In our case: 12×6=72and16×6=9612 × 6 = 72 \quad \text{and} \quad 16 × 6 = 96
This confirms again that 72:96 is a scaled-up version of 12:16.
Conclusion:
Since both ratios reduce to the same simplest form, 12:16 and 72:96 are indeed equivalent.