Question

1. which proportion is equivalent to \\( \frac { 9 } { 15 } = \frac { 6 } { 10 } \\)? \\( \frac { 10 } { 15 } = \frac { 6 } { 9 } \\) \\( \frac { 6 } { 9 } = \frac { 10 } { 15 } \\) \\( \frac { 10 } { 15 } = \frac { 9 } { 6 } \\) \\( \frac { 6 } { 9 } = \frac { 15 } { 10 } \\)

Explanation:

Step1: Recall cross - multiplication property

For a proportion \(\frac{a}{b}=\frac{c}{d}\), the cross - multiplication property states that \(a\times d = b\times c\). We will use this property to check each option. The original proportion is \(\frac{9}{15}=\frac{6}{10}\).

Step2: Check option 1 (\(\frac{6}{9}=\frac{15}{10}\))

Cross - multiply: \(6\times10 = 60\) and \(9\times15=135\). Since \(60 eq135\), this proportion is not equivalent.

Step3: Check option 2 (\(\frac{10}{15}=\frac{9}{6}\))

Cross - multiply: \(10\times6 = 60\) and \(15\times9 = 135\). Since \(60 eq135\), this proportion is not equivalent.

Step4: Check option 3 (\(\frac{9}{6}=\frac{10}{15}\))

Cross - multiply: \(9\times15=135\) and \(6\times10 = 60\). Since \(135 eq60\), this proportion is not equivalent. Wait, maybe we made a mistake. Wait, the original proportion is \(\frac{9}{15}=\frac{6}{10}\), let's re - express the options. Wait, the first option is \(\frac{6}{9}=\frac{15}{10}\), second \(\frac{10}{15}=\frac{9}{6}\), third \(\frac{9}{6}=\frac{10}{15}\)? No, wait the user's image: Let's re - check. Wait the original proportion is \(\frac{9}{15}=\frac{6}{10}\). Let's use the property of proportion: if \(\frac{a}{b}=\frac{c}{d}\), then \(\frac{a}{c}=\frac{b}{d}\) (alternendo property) or \(\frac{b}{a}=\frac{d}{c}\) (invertendo property) or \(\frac{a + b}{b}=\frac{c + d}{d}\) (componendo) etc.

Wait, the original proportion \(\frac{9}{15}=\frac{6}{10}\), let's simplify both fractions. \(\frac{9}{15}=\frac{3}{5}\), \(\frac{6}{10}=\frac{3}{5}\). Now check the first option: \(\frac{6}{9}=\frac{2}{3}\), \(\frac{15}{10}=\frac{3}{2}\), not equal. Second option: \(\frac{10}{15}=\frac{2}{3}\), \(\frac{9}{6}=\frac{3}{2}\), not equal. Wait, maybe the options are:

Wait the first option is \(\frac{6}{9}=\frac{15}{10}\)? No, maybe the first option is \(\frac{6}{9}=\frac{15}{10}\) (no), second \(\frac{10}{15}=\frac{9}{6}\) (no), third \(\frac{9}{6}=\frac{15}{10}\)? Wait, no, let's use cross - multiplication for the original proportion \(\frac{9}{15}=\frac{6}{10}\), \(9\times10 = 90\), \(15\times6=90\). Now check the first option: \(\frac{6}{9}=\frac{15}{10}\), \(6\times10 = 60\), \(9\times15 = 135\) (not equal). Second option: \(\frac{10}{15}=\frac{9}{6}\), \(10\times6=60\), \(15\times9 = 135\) (not equal). Third option: \(\frac{9}{6}=\frac{15}{10}\)? Wait, no, the first option in the image (from left to right, top to bottom) is:

First option: \(\frac{6}{9}=\frac{15}{10}\)

Second: \(\frac{10}{15}=\frac{9}{6}\)

Third: \(\frac{9}{6}=\frac{10}{15}\)? No, wait the user's problem: "Which proportion is equivalent to \(\frac{9}{15}=\frac{6}{10}\)?" and the options are:

1. \(\frac{6}{9}=\frac{15}{10}\)

2. \(\frac{10}{15}=\frac{9}{6}\)

3. \(\frac{9}{6}=\frac{10}{15}\)

4. \(\frac{10}{15}=\frac{6}{9}\) (Wait, maybe the last option is \(\frac{10}{15}=\frac{6}{9}\))

Wait, let's re - calculate. The original proportion: \(\frac{9}{15}=\frac{6}{10}\), cross - multiply \(9\times10 = 90\), \(15\times6 = 90\).

Now check option 1: \(\frac{6}{9}=\frac{15}{10}\), cross - multiply \(6\times10 = 60\), \(9\times15 = 135\) (not equal).

Option 2: \(\frac{10}{15}=\frac{9}{6}\), cross - multiply \(10\times6 = 60\), \(15\times9 = 135\) (not equal).

Option 3: \(\frac{9}{6}=\frac{15}{10}\), cross - multiply \(9\times10 = 90\), \(6\times15 = 90\) (equal). Wait, maybe I misread the option. If the third option is \(\frac{9}{6}=\frac{15}{10}\), then it's equivalent. But according to the user's image, the second option (from the top, second circle) is \(\frac{9}{6}=\frac{10}{15}\)? No, maybe the correct option is the first one? Wait no, let's use the property of proportion: if \(\frac{a}{b}=\frac{c}{d}\), then \(\frac{b}{a}=\frac{d}{c}\) (invertendo) or \(\frac{a}{c}=\frac{b}{d}\) (alternendo).

Original: \(\frac{9}{15}=\frac{6}{10}\), by alternendo, \(\frac{9}{6}=\frac{15}{10}\), which is \(\frac{9}{6}=\frac{15}{10}\), cross - multiply \(9\times10 = 90\), \(6\times15 = 90\), so this is equivalent. Wait, maybe the first option is \(\frac{6}{9}=\frac{15}{10}\) (no), the second option (the second circle) is \(\frac{9}{6}=\frac{10}{15}\)? No, I think there is a mis - reading. Wait, the original proportion is \(\frac{9}{15}=\frac{6}{10}\), let's simplify both sides: \(\frac{3}{5}=\frac{3}{5}\). Now check the options:

Option 1: \(\frac{6}{9}=\frac{2}{3}\), \(\frac{15}{10}=\frac{3}{2}\) (not equal)

Option 2: \(\frac{10}{15}=\frac{2}{3}\), \(\frac{9}{6}=\frac{3}{2}\) (not equal)

Option 3: \(\frac{9}{6}=\frac{3}{2}\), \(\frac{10}{15}=\frac{2}{3}\) (not equal)

Option 4: \(\frac{10}{15}=\frac{2}{3}\), \(\frac{6}{9}=\frac{2}{3}\) (equal). Ah! Maybe the last option is \(\frac{10}{15}=\frac{6}{9}\). Let's check: \(\frac{10}{15}=\frac{2}{3}\), \(\frac{6}{9}=\frac{2}{3}\), and cross - multiply \(10\times9 = 90\), \(15\times6 = 90\). So the equivalent proportion is \(\frac{10}{15}=\frac{6}{9}\) (the last option, the fourth circle with \(\frac{10}{15}=\frac{6}{9}\)).

Wait, the user's image shows:

First circle: \(\frac{6}{9}=\frac{15}{10}\)

Second circle: \(\frac{9}{6}=\frac{10}{15}\)

Third circle: \(\frac{10}{15}=\frac{9}{6}\)

Fourth circle: \(\frac{10}{15}=\frac{6}{9}\)

Wait, let's check the fourth option: \(\frac{10}{15}=\frac{6}{9}\). Cross - multiply: \(10\times9 = 90\), \(15\times6 = 90\). So this is equivalent.

Answer:

The equivalent proportion is \(\frac{10}{15}=\frac{6}{9}\) (the last option, the fourth circle - shaped option with \(\frac{10}{15}=\frac{6}{9}\))