- Identify the ratios of red to green blocks for both groups: \(\frac{15}{45}\) and \(\frac{30}{90}\).
- Simplify the ratios by dividing the numerator and denominator by their greatest common divisors, resulting in \(\frac{1}{3}\) for both.
- Establish that since both ratios are equal, the correct representation is \(\frac{15}{45} = \frac{30}{90}\).
- Final Answer: \(\boxed{\frac{15}{45} = \frac{30}{90}}\).
Explanation
- Analyzing the Block Ratios
We are given two groups of children with collections of colored blocks. The first group has \(15\) red blocks and \(45\) green blocks. The second group has \(30\) red blocks and \(90\) green blocks. We need to determine the correct relationship between the proportions of red to green blocks in these two groups. To compare the proportions, we first define the ratio of red to green blocks for each group. For the first group, the ratio is \(\frac{15}{45}\). For the second group, the ratio is \(\frac{30}{90}\).
- Simplifying and Comparing Proportions
To compare the proportions, we simplify each fraction by finding the greatest common divisor of the numerator and denominator. For the first group, the greatest common divisor of \(15\) and \(45\) is \(15\). Thus, \(\frac{15}{45} = \frac{15 \div 15}{45 \div 15} = \frac{1}{3}\). For the second group, the greatest common divisor of \(30\) and \(90\) is \(30\). Thus, \(\frac{30}{90} = \frac{30 \div 30}{90 \div 30} = \frac{1}{3}\). Since both ratios simplify to \(\frac{1}{3}\), we can conclude that the proportions are equal: \(\frac{15}{45} = \frac{30}{90}\).
- Selecting the Correct Proportion Statement
Comparing our findings with the given options, we see that the expression \(\frac{15}{45} = \frac{30}{90}\) correctly describes the relationship between the relative amounts of colored blocks in both groups. Therefore, this is the correct choice. The final answer is \(\boxed{\frac{15}{45} = \frac{30}{90}}\).
Examples
In real-life scenarios, understanding ratios is crucial for scaling recipes or mixing materials. For instance, if a cookie recipe requires 2 cups of sugar for every 3 cups of flour, this ratio of 2:3 must be maintained regardless of whether you are making a small batch or a large batch. By keeping the proportion \(\frac{2}{3} = \frac{\text{Sugar}}{\text{Flour}}\) constant, you ensure your baked goods turn out perfectly every time, just as maintaining the ratio of red to green blocks keeps the relative proportion consistent between the two groups.