Verified Answer :

  • Understand that equivalent ratios are found by multiplying both parts of the original ratio by the same factor \(k\).
  • Scale \(3:5\) by \(k=2\) to get \(3 \times 2 : 5 \times 2 = 6:10\).
  • Scale \(3:5\) by \(k=3\) to get \(3 \times 3 : 5 \times 3 = 9:15\).
  • Use tape diagrams to visualize how splitting original blocks into smaller segments preserves the proportionality, confirming that \(6:10\) and \(9:15\) are equivalent to \(3:5\).
  • Final Answer: \(\boxed{6:10, 9:15}\).

Explanation

  1. Analysis and Definition of Equivalent Ratios

A ratio expresses the proportional relationship between two quantities. For a given ratio \(a:b\), an equivalent ratio is obtained by multiplying both parts by the same non-zero constant \(k\), resulting in \(ka:kb\). In this problem, we are given the base ratio \(3:5\). This means that for every 3 units of the first quantity, there are 5 units of the second quantity. To find equivalent ratios, we can scale this relationship up by any common factor.

  1. Calculating Equivalent Ratios

To generate two ratios equivalent to \(3:5\), we choose two positive constants, \(k_1 = 2\) and \(k_2 = 3\).
Multiplying both terms of the ratio by \(k_1=2\):
\[3 \times 2 : 5 \times 2 = 6:10\]
Multiplying both terms of the ratio by \(k_2=3\):
\[3 \times 3 : 5 \times 3 = 9:15\]
Thus, two ratios equivalent to \(3:5\) are \(6:10\) and \(9:15\).

  1. Proving Equivalence with Tape Diagrams

A tape diagram is a visual tool where we represent the ratio using segments.

  • For the original ratio \(3:5\), we draw 3 blocks for the first quantity and 5 blocks for the second.
  • For the ratio \(6:10\), we can represent this by taking each of the original 3 blocks and splitting them into 2 smaller parts (totaling \(3 \times 2 = 6\) blocks). Similarly, we split each of the original 5 blocks into 2 (totaling \(5 \times 2 = 10\) blocks). Since the total length of the 6 blocks remains proportional to the original 3, and the 10 blocks to the original 5, the ratio \(6:10\) maintains the same relationship as \(3:5\).
  • Similarly, for the ratio \(9:15\), we divide each original block into 3 parts. The first quantity now consists of \(3 \times 3 = 9\) small blocks, and the second consists of \(5 \times 3 = 15\) small blocks. The proportion is preserved, confirming that \(9:15\) is equivalent to \(3:5\).
  1. Conclusion

We have identified two equivalent ratios by scaling the original ratio \(3:5\). By multiplying the terms by 2, we found \(6:10\). By multiplying the terms by 3, we found \(9:15\). Using the concept of dividing blocks in a tape diagram, we demonstrated that these scaled ratios represent the exact same proportions as the original ratio \(3:5\). Thus, the two ratios are \(\boxed{6:10}\) and \(\boxed{9:15}\).

Examples

Imagine you are preparing a fruit punch that follows a strict recipe ratio of 3 cups of orange juice to 5 cups of pineapple juice. If you decide to make a larger batch for a party, you need to keep the taste consistent by scaling the recipe. By doubling the ingredients, you use 6 cups of orange juice and 10 cups of pineapple juice (a 6:10 ratio). By tripling them, you use 9 cups of orange juice and 15 cups of pineapple juice (a 9:15 ratio). In both cases, the concentration of the juices remains the same, ensuring the punch tastes exactly like the original. This is the practical power of equivalent ratios in everyday life.