Question

question
select the two correct answers.
luis is sorting his marbles. he finds that the ratio of blue marbles to yellow marbles is 15 to 10.
which ratios are equivalent to 15:10?
30:20
2:1
10:5
20:15
3:2

Explanation:

Step1: Simplify the original ratio

First, simplify the ratio \(15:10\) by dividing both terms by their greatest common divisor, which is 5. So, \(\frac{15}{5}:\frac{10}{5}=3:2\). Also, we can multiply both terms by the same non - zero number to get equivalent ratios. If we multiply \(15:10\) by 2, we get \((15\times2):(10\times2) = 30:20\). If we divide \(15:10\) by 5, we get \(3:2\), and if we consider the ratio of yellow to blue (reciprocal in a way, but let's check the options), the ratio of yellow to blue is \(10:15=\frac{10}{5}:\frac{15}{5}=2:3\)? Wait, no, let's check each option:

• Option 30:20: Multiply \(15:10\) by 2, \(15\times2 = 30\), \(10\times2=20\), so \(30:20\) is equivalent to \(15:10\).
• Option 2:1: \(15:10 = 3:2

eq2:1\), so this is not equivalent.

• Option 10:5: Simplify \(10:5 = 2:1

eq3:2\), not equivalent. Wait, wait, maybe I made a mistake. Wait \(15:10=\frac{15}{10}=\frac{3}{2}\). Let's check 3:2, which is the simplified form. Also, 30:20=\(\frac{30}{20}=\frac{3}{2}\), 10:5=\(\frac{10}{5} = 2\), no. Wait, maybe the option "3:2" (I assume the last option is 3:2). And 30:20. Let's re - check:

Original ratio \(15:10\). Let's find the greatest common factor (GCF) of 15 and 10. The factors of 15 are 1, 3, 5, 15. The factors of 10 are 1, 2, 5, 10. GCF is 5. Divide both by 5: \(15\div5 = 3\), \(10\div5=2\), so \(15:10 = 3:2\). Now check each option:

• 30:20: Divide both by 10, \(30\div10 = 3\), \(20\div10 = 2\), so \(30:20=3:2\), equivalent.
• 2:1: \(2:1

eq3:2\)

• 10:5: \(10:5 = 2:1

eq3:2\)

• 20:15: \(20:15=\frac{20}{15}=\frac{4}{3}

eq\frac{3}{2}\)

• 3:2: This is the simplified form of \(15:10\), so it's equivalent.

Wait, maybe the options are 30:20 and 3:2 (the last option). Let's confirm:

For 30:20, \(\frac{30}{20}=\frac{3}{2}\), same as \(\frac{15}{10}=\frac{3}{2}\). For 3:2, it's the reduced form of \(15:10\). So the two correct answers are 30:20 and 3:2 (assuming the last option is 3:2) and 30:20. Wait, maybe I misread the options. Let's re - check the options:

The options are:

• 30:20

• 2:1

• 10:5

• 20:15

• 3:2

So, \(15:10=\frac{15}{10}=\frac{3}{2}\). Let's check each ratio as a fraction:

• 30:20: \(\frac{30}{20}=\frac{3}{2}\), equal to \(\frac{15}{10}\), so equivalent.

• 2:1: \(\frac{2}{1}=2

eq\frac{3}{2}\), not equivalent.

• 10:5: \(\frac{10}{5}=2

eq\frac{3}{2}\), not equivalent.

• 20:15: \(\frac{20}{15}=\frac{4}{3}

eq\frac{3}{2}\), not equivalent.

• 3:2: \(\frac{3}{2}\), equal to \(\frac{15}{10}\), so equivalent.

Also, another way: \(15:10\) can be thought of as blue to yellow. If we reverse it, yellow to blue is \(10:15=\frac{10}{15}=\frac{2}{3}\), but that's not one of the options. Wait, but 30:20 is multiplying both by 2, and 3:2 is dividing both by 5. So the two correct ratios are 30:20 and 3:2.

Wait, also, 15:10 = 3:2, and 30:20 = 3:2 (divide numerator and denominator by 10), and 3:2 is the simplified form. So the two correct answers are 30:20 and 3:2.

Answer:

30:20, 3:2