12:16 and 72:96 Determine if the ratios are equivalent
12:16 and 72:96 Determine if the ratios are equivalent
The Correct Answer and Explanation is:
To determine if the ratios 12:16 and 72:96 are equivalent, we need to simplify each ratio and compare them.
✅ Step 1: Simplify both ratios
1. Simplify 12:16
Find the greatest common divisor (GCD) of 12 and 16:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 16: 1, 2, 4, 8, 16
✅ GCD = 4
Now divide both terms by 4: 124:164=3:4\frac{12}{4} : \frac{16}{4} = 3:4
2. Simplify 72:96
Find the GCD of 72 and 96:
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
✅ GCD = 24
Now divide both terms by 24: 7224:9624=3:4\frac{72}{24} : \frac{96}{24} = 3:4
✅ Step 2: Compare simplified ratios
- 12:16 simplifies to 3:4
- 72:96 simplifies to 3:4
Since both simplify to the same ratio (3:4), the original ratios are equivalent.
📘 Explanation
To determine if two ratios are equivalent, we compare their simplified forms. A ratio is a comparison of two quantities and can be written in various forms such as a:b or a/b. Simplifying a ratio means reducing it to its lowest terms—just like simplifying a fraction—by dividing both parts by their greatest common divisor (GCD).
For the first ratio, 12:16, the GCD is 4. Dividing both numbers by 4, we get 3:4. This means for every 3 parts of the first quantity, there are 4 parts of the second.
For the second ratio, 72:96, the GCD is 24. Dividing both terms by 24 also gives 3:4. This shows that 72:96 represents the same proportional relationship as 12:16.
Another way to check if two ratios are equivalent is by converting them to fractions and simplifying:
- 12/16 = 0.75
- 72/96 = 0.75
Since both fractions are equal, the ratios are equivalent.
Understanding ratios is important in real life because they are used in scaling recipes, comparing prices, or converting measurements. Recognizing equivalent ratios helps in identifying proportional relationships between quantities.
In conclusion, since both 12:16 and 72:96 reduce to the same ratio (3:4), they are equivalent. This confirms that the two original ratios express the same relationship, just on different scales.
