Question

circle all the equivalent ratios in each set.
1:3 2:6 2:3 12:3 2:8 4:1
2:10 4:5 8:10 15:5 9:3 5:1
2:1 9:3 8:4 10:5 6:9 4:6 2:3 3:1
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Explanation:

Step1: Simplify each ratio

To find equivalent ratios, we simplify them to their simplest form. For example, for a ratio \(a:b\), we find the greatest - common divisor (GCD) of \(a\) and \(b\) and divide both by it.

Step2: Analyze the first set

For \(1:3\), it is in simplest form. For \(2:6\), \(\frac{2}{2}:\frac{6}{2}=1:3\), so \(1:3\) and \(2:6\) are equivalent. \(2:3\) is in simplest form and not equivalent to \(1:3\).

Step3: Analyze the second set

For \(12:3\), \(\frac{12}{3}:\frac{3}{3}=4:1\), so \(12:3\) and \(4:1\) are equivalent. \(2:8\), \(\frac{2}{2}:\frac{8}{2}=1:4\) is not equivalent.

Step4: Analyze the third set

For \(2:10\), \(\frac{2}{2}:\frac{10}{2}=1:5\). \(4:5\) is in simplest form. \(8:10\), \(\frac{8}{2}:\frac{10}{2}=4:5\), so \(4:5\) and \(8:10\) are equivalent.

Step5: Analyze the fourth set

For \(15:5\), \(\frac{15}{5}:\frac{5}{5}=3:1\). \(9:3\), \(\frac{9}{3}:\frac{3}{3}=3:1\), so \(15:5\), \(9:3\) and \(3:1\) are equivalent. \(5:1\) is not equivalent to them.

Step6: Analyze the fifth set

For \(2:1\) it is in simplest form. \(9:3\), \(\frac{9}{3}:\frac{3}{3}=3:1\) is not equivalent. \(8:4\), \(\frac{8}{4}:\frac{4}{4}=2:1\), so \(2:1\) and \(8:4\) are equivalent. \(10:5\), \(\frac{10}{5}:\frac{5}{5}=2:1\), so \(2:1\), \(8:4\) and \(10:5\) are equivalent.

Step7: Analyze the sixth set

For \(6:9\), \(\frac{6}{3}:\frac{9}{3}=2:3\). \(4:6\), \(\frac{4}{2}:\frac{6}{2}=2:3\), so \(6:9\), \(4:6\) and \(2:3\) are equivalent.

Answer:

In the first set: Circle \(1:3\) and \(2:6\).
In the second set: Circle \(12:3\) and \(4:1\).
In the third set: Circle \(4:5\) and \(8:10\).
In the fourth set: Circle \(15:5\), \(9:3\) and \(3:1\).
In the fifth set: Circle \(2:1\), \(8:4\) and \(10:5\).
In the sixth set: Circle \(6:9\), \(4:6\) and \(2:3\).