Question

1. kay has a bowl of apples with the apples shown below. if kay picks an apple and then returns it back to the bowl 120 times, how many times, t, can she expect to pick a green apple? choose the two proportions that represent this situation. a. \\(\frac{3}{10} = \frac{t}{120}\\) b. \\(\frac{3}{120} = \frac{10}{t}\\) c. \\(\frac{3}{120} = \frac{t}{10}\\) d. \\(\frac{3}{t} = \frac{10}{120}\\)

Explanation:

First, we count the total number of apples. From the diagram, we have 2 red, 4 yellow, 1 white, and 3 green apples. So total apples \( n = 2 + 4 + 1 + 3 = 10 \). The number of green apples is 3. So the probability of picking a green apple is \( \frac{3}{10} \).

Now, if Kay picks an apple 120 times (with replacement), the expected number of times she picks a green apple is \( t \). The proportion of green apples picked should be equal to the probability of picking a green apple. So we set up the proportion:

Step 1: Determine total and green apples

Total apples: \( 2 + 4 + 1 + 3 = 10 \)
Green apples: \( 3 \)
Probability of green: \( \frac{3}{10} \)

Step 2: Set up the proportion

The proportion of green apples picked (\( \frac{t}{120} \)) should equal the probability of picking a green apple (\( \frac{3}{10} \)). So \( \frac{3}{10} = \frac{t}{120} \), which is option A. Also, we can cross - multiply to get another form: \( \frac{3}{120}=\frac{10}{t} \) is incorrect. Wait, let's re - examine.

Wait, the probability of green is \( \frac{3}{10} \), and the number of trials is 120, the number of successes (green picks) is \( t \). So the proportion is \( \frac{\text{Green apples}}{\text{Total apples}}=\frac{\text{Number of green picks}}{\text{Total picks}} \), so \( \frac{3}{10}=\frac{t}{120} \) (option A) and also, by cross - multiplying \( 3\times120 = 10\times t \), or we can write it as \( \frac{3}{120}=\frac{10}{t} \)? No, wait, let's do the proportion correctly.

The ratio of green apples to total apples is \( \frac{3}{10} \), and the ratio of times green is picked (\( t \)) to total picks (120) is \( \frac{t}{120} \). So \( \frac{3}{10}=\frac{t}{120} \) (option A). Also, if we rearrange \( \frac{3}{120}=\frac{10}{t} \) is wrong. Wait, let's check option D: \( \frac{3}{t}=\frac{10}{120} \), cross - multiplying gives \( 10t = 3\times120 \), which is the same as \( 3\times120 = 10t \), which is the same as \( \frac{3}{10}=\frac{t}{120} \) (since \( \frac{3}{10}=\frac{t}{120}\Rightarrow 3\times120 = 10t\Rightarrow\frac{3}{t}=\frac{10}{120} \)). Wait, maybe I made a mistake earlier.

Wait, total apples: 10, green apples: 3. So probability \( P(\text{green})=\frac{3}{10} \). The expected number of green picks in 120 trials is \( t = P(\text{green})\times120=\frac{3}{10}\times120 \). So the proportion is \( \frac{3}{10}=\frac{t}{120} \) (option A) and also, \( \frac{3}{t}=\frac{10}{120} \) (option D) because cross - multiplying \( 3\times120 = 10\times t \) is the same as \( \frac{3}{10}=\frac{t}{120} \). Wait, but the question says "Choose the two proportions that represent this situation". But looking at the options:

Option A: \( \frac{3}{10}=\frac{t}{120} \)

Option D: \( \frac{3}{t}=\frac{10}{120} \)

Let's verify:

For option A: \( \frac{3}{10}=\frac{t}{120} \), cross - multiply: \( 10t = 3\times120\Rightarrow t = 36 \)

For option D: \( \frac{3}{t}=\frac{10}{120} \), cross - multiply: \( 10t = 3\times120\Rightarrow t = 36 \), same result.

Wait, but maybe I miscounted the apples. Let's count again:

Red: 2, Yellow: 4, White: 1, Green: 3. Total: \( 2 + 4+1 + 3=10 \). Correct. So green is 3 out of 10. So the proportion of green apples is \( \frac{3}{10} \), and the proportion of green picks is \( \frac{t}{120} \). So \( \frac{3}{10}=\frac{t}{120} \) (option A). Also, \( \frac{3}{t}=\frac{10}{120} \) (option D) because \( \frac{3}{10}=\frac{t}{120}\Rightarrow\frac{3}{t}=\frac{10}{120} \) (by cross - multiplying and rearranging).

But let's check the options again:

Option A: \( \frac{3}{10}=\frac{t}{120} \)

Option D: \( \frac{3}{t}=\frac{10}{120} \)

So these two are equivalent.

Answer:

A. \( \boldsymbol{\frac{3}{10}=\frac{t}{120}} \), D. \( \boldsymbol{\frac{3}{t}=\frac{10}{120}} \)