Question

isaacs favorite drink is a matcha banana smoothie. he wants to make it for a party, but hes not sure how many guests there will be. isaac started making the following chart so he could know how much of each ingredient he would need, depending on the number of guests.
fill in the missing numbers in the table below. use the arrow keys or tab to move from one cell to another. to enter a mixed number, press the \+\ key or click the \mixed number\ button below. to view a hint, click in a cell, then press the \h\ key or click the \hint\ button below.

servings	4	2
bananas	4		8
matcha powder (cups)	$\frac{1}{4}$		$\frac{1}{2}$
ice (ounces)	$7\frac{1}{2}$					30
plain yogurt (pints)	$\frac{1}{2}$			$\frac{3}{4}$

Explanation:

Step1: Find the ratio of servings change

First, we see that when servings change from 4 to 2, the ratio is \( \frac{2}{4}=\frac{1}{2} \). When bananas change from 4 (for 4 servings) to 8 (for some servings), the ratio is \( \frac{8}{4} = 2 \). So we can use the ratio of servings to find the missing values.

Step2: Almond milk for 2 servings

For almond milk, 4 servings have \( 3\frac{1}{2}=\frac{7}{2} \) cups. For 2 servings (ratio \( \frac{1}{2} \)), it's \( \frac{7}{2}\times\frac{1}{2}=\frac{7}{4}=1\frac{3}{4} \) cups.

Step3: Servings for 8 bananas

Bananas: 4 servings have 4 bananas, so per serving is \( \frac{4}{4}=1 \) banana per serving. For 8 bananas, servings = \( \frac{8}{1}=8 \) servings.

Step4: Almond milk for 8 servings

For 8 servings (ratio \( \frac{8}{4}=2 \) from 4 servings), almond milk is \( \frac{7}{2}\times2 = 7 \) cups.

Step5: Almond milk for \( 8\frac{3}{4} \) cups

Let servings be \( x \). \( \frac{7}{2} \) cups for 4 servings, so \( \frac{7}{2}x = 8\frac{3}{4}=\frac{35}{4} \). Solve for \( x \): \( x=\frac{35}{4}\div\frac{7}{2}=\frac{35}{4}\times\frac{2}{7}=\frac{5}{2} = 2.5 \)? Wait, no, wait. Wait, \( 8\frac{3}{4}=\frac{35}{4} \). \( \frac{7}{2} \) cups per 4 servings, so per serving almond milk is \( \frac{7}{2}\div4=\frac{7}{8} \) cups per serving. Then \( \frac{35}{4}\div\frac{7}{8}=\frac{35}{4}\times\frac{8}{7}=10 \)? Wait, no, I messed up. Wait, 4 servings: \( 3\frac{1}{2}=\frac{7}{2} \) cups. So per serving: \( \frac{7}{2}\div4=\frac{7}{8} \) cups. Then for \( 8\frac{3}{4}=\frac{35}{4} \) cups, servings = \( \frac{35}{4}\div\frac{7}{8}=\frac{35}{4}\times\frac{8}{7}=10 \) servings? Wait, no, earlier with bananas, 4 servings have 4 bananas, so 1 banana per serving. So when bananas are 4 for 4 servings, so 1 banana per serving. So for 8 bananas, 8 servings. So let's check the serving ratio with bananas. 4 servings: bananas 4, so 1 banana per serving. So servings = bananas. Wait, 4 servings: 4 bananas, so 1 banana per serving. So that's correct. So for 8 bananas, 8 servings. Then almond milk for 8 servings: 4 servings is \( \frac{7}{2} \), so 8 servings is \( \frac{7}{2}\times2 = 7 \) cups (since 8 is 2 times 4). Then \( 8\frac{3}{4} \) cups of almond milk: let's find servings. \( \frac{7}{2} \) cups for 4 servings, so \( \frac{7}{2} \) cups / 4 servings = \( \frac{7}{8} \) cups per serving. Then \( 8\frac{3}{4}=\frac{35}{4} \) cups. Servings = \( \frac{35}{4}\div\frac{7}{8}=\frac{35}{4}\times\frac{8}{7}=10 \) servings? Wait, no, wait the table has \( 8\frac{3}{4} \) for almond milk, so let's see the serving column. Wait, the serving column: first is 4, 2, then 8 (from bananas), then maybe 10? Wait, let's re - organize.

Wait, let's list the rows:

1. Servings: 4, 2,?, ?,?,?

2. Almond milk: \( 3\frac{1}{2} \),?, ?, ?, \( 8\frac{3}{4} \),?

3. Bananas: 4,?, 8,?, ?,?

4. Matcha powder: \( \frac{1}{4} \),?, \( \frac{1}{2} \),?, ?,?

5. Ice: \( 7\frac{1}{2} \),?, ?, ?, ?, 30

6. Plain yogurt: \( \frac{1}{2} \),?, ?, \( \frac{3}{4} \), ?, ?

Let's use the bananas row first. Bananas: 4 servings → 4 bananas, so 1 banana per serving. So:

- For 2 servings: bananas = 2×1 = 2 bananas.

- For 8 bananas: servings = 8÷1 = 8 servings.

- For matcha powder: 4 servings → \( \frac{1}{4} \) cup, so per serving: \( \frac{1}{4}\div4=\frac{1}{16} \) cup? No, wait, 4 servings: \( \frac{1}{4} \) cup, so per serving is \( \frac{1}{4}\div4=\frac{1}{16} \)? No, that can't be. Wait, 4 servings: \( \frac{1}{4} \) cup, so for 2 servings (half the servings), matcha powder is \( \frac{1}{4}\times\frac{1}{2}=\frac{1}{8} \) cup. For 8 servings (double the servings from 4), matcha powder is \( \frac{1}{4}\times2=\frac{1}{2} \) cup (which matches the table, so that's correct). So matcha powder: per serving is \( \frac{1}{4}\div4=\frac{1}{16} \)? No, wait, 4 servings: \( \frac{1}{4} \) cup, so 2 servings: \( \frac{1}{4}\times\frac{2}{4}=\frac{1}{8} \) cup (since 2 is half of 4). 8 servings: \( \frac{1}{4}\times\frac{8}{4}=\frac{1}{2} \) cup (which is given, so that's correct).

Ice: 4 servings → \( 7\frac{1}{2}=\frac{15}{2} \) ounces. Per serving: \( \frac{15}{2}\div4=\frac{15}{8} \) ounces per serving. For 30 ounces: servings = 30÷\( \frac{15}{8} \)=30×\( \frac{8}{15} \)=16 servings.

Plain yogurt: 4 servings → \( \frac{1}{2} \) pint. Per serving: \( \frac{1}{2}\div4=\frac{1}{8} \) pint per serving. For \( \frac{3}{4} \) pint: servings = \( \frac{3}{4}\div\frac{1}{8}=\frac{3}{4}\times8 = 6 \) servings.

Now, let's fill the table step by step:

Servings column:

- We know 4, 2, 8 (from bananas), let's find the serving for \( 8\frac{3}{4} \) almond milk. Almond milk: 4 servings → \( \frac{7}{2} \) cups. Let \( x \) be servings for \( 8\frac{3}{4}=\frac{35}{4} \) cups. So \( \frac{7}{2}x=\frac{35}{4} \) → \( x=\frac{35}{4}\times\frac{2}{7}=\frac{5}{2}=2.5 \)? No, that's not right. Wait, no, I think I made a mistake. Wait, \( 3\frac{1}{2} \) cups for 4 servings, so the rate is \( 3\frac{1}{2}\div4=\frac{7}{2}\div4=\frac{7}{8} \) cups per serving. Then \( 8\frac{3}{4}=\frac{35}{4} \) cups. Servings = \( \frac{35}{4}\div\frac{7}{8}=\frac{35}{4}\times\frac{8}{7}=10 \) servings. Ah, there we go. So 10 servings for \( 8\frac{3}{4} \) cups of almond milk.

For ice: 30 ounces. Ice per serving: \( 7\frac{1}{2}\div4=\frac{15}{2}\div4=\frac{15}{8} \) ounces per serving. Servings for 30 ounces: \( 30\div\frac{15}{8}=16 \) servings.

For plain yogurt: \( \frac{3}{4} \) pint. Plain yogurt per serving: \( \frac{1}{2}\div4=\frac{1}{8} \) pint per serving. Servings: \( \frac{3}{4}\div\frac{1}{8}=6 \) servings.

Now let's fill each row:

Servings row:

- 4, 2, 8, 6, 10, 16

Almond milk row:

- 4 servings: \( 3\frac{1}{2} \)
- 2 servings: \( 3\frac{1}{2}\times\frac{2}{4}=3\frac{1}{2}\times\frac{1}{2}=1\frac{3}{4} \)
- 8 servings: \( 3\frac{1}{2}\times\frac{8}{4}=3\frac{1}{2}\times2 = 7 \)
- 6 servings: \( 3\frac{1}{2}\times\frac{6}{4}=3\frac{1}{2}\times\frac{3}{2}=\frac{7}{2}\times\frac{3}{2}=\frac{21}{4}=5\frac{1}{4} \)
- 10 servings: \( 8\frac{3}{4} \) (given)
- 16 servings: \( 3\frac{1}{2}\times\frac{16}{4}=3\frac{1}{2}\times4 = 14 \)

Bananas row:

- 4 servings: 4
- 2 servings: 2 (since 1 per serving)
- 8 servings: 8 (given)
- 6 servings: 6
- 10 servings: 10
- 16 servings: 16

Matcha powder row:

- 4 servings: \( \frac{1}{4} \)
- 2 servings: \( \frac{1}{4}\times\frac{2}{4}=\frac{1}{4}\times\frac{1}{2}=\frac{1}{8} \)
- 8 servings: \( \frac{1}{4}\times\frac{8}{4}=\frac{1}{4}\times2=\frac{1}{2} \) (given)
- 6 servings: \( \frac{1}{4}\times\frac{6}{4}=\frac{1}{4}\times\frac{3}{2}=\frac{3}{8} \)
- 10 servings: \( \frac{1}{4}\times\frac{10}{4}=\frac{1}{4}\times\frac{5}{2}=\frac{5}{8} \)
- 16 servings: \( \frac{1}{4}\times\frac{16}{4}=\frac{1}{4}\times4 = 1 \)

Ice row:

- 4 servings: \( 7\frac{1}{2} \)
- 2 servings: \( 7\frac{1}{2}\times\frac{2}{4}=7\frac{1}{2}\times\frac{1}{2}=3\frac{3}{4} \)
- 8 servings: \( 7\frac{1}{2}\times\frac{8}{4}=7\frac{1}{2}\times2 = 15 \)
- 6 servings: \( 7\frac{1}{2}\times\frac{6}{4}=7\frac{1}{2}\times\frac{3}{2}=\frac{15}{2}\times\frac{3}{2}=\frac{45}{4}=11\frac{1}{4} \)
- 10 servings: \( 7\frac{1}{2}\times\frac{10}{4}=7\frac{1}{2}\times\frac{5}{2}=\frac{15}{2}\times\frac{5}{2}=\frac{75}{4}=18\frac{3}{4} \)
- 16 servings: 30 (given)

Plain yogurt row:

- 4 servings: \( \frac{1}{2} \)
- 2 servings: \( \frac{1}{2}\times\frac{2}{4}=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4} \)
- 8 servings: \( \frac{1}{2}\times\frac{8}{4}=\frac{1}{2}\times2 = 1 \)
- 6 servings: \( \frac{3}{4} \) (given)
- 10 servings: \( \frac{1}{2}\times\frac{10}{4}=\frac{1}{2}\times\frac{5}{2}=\frac{5}{8} \)
- 16 servings: \( \frac{1}{2}\times\frac{16}{4}=\frac{1}{2}\times4 = 2 \)

But let's check the most important missing cells first (the ones in the table as per the image, the first empty in servings is next to 2, then the rest). The first empty in servings is for 8 bananas, which is 8 servings. The first empty in almond milk for 2 servings is \( 1\frac{3}{4} \), for 8 servings is 7, for 10 servings is \( 8\frac{3}{4} \), etc. But maybe the problem expects us to fill the table with the key missing values. Let's focus on the first few:

- Servings for 8 bananas: 8

- Almond milk for 2 servings: \( 1\frac{3}{4} \)

- Almond milk for 8 servings: 7

- Servings for \( 8\frac{3}{4} \) almond milk: 10

- Ice for 16 servings: 30 (given)

- Plain yogurt for 6 servings: \( \frac{3}{4} \) (given)

But to summarize the key fills:

- Servings column: 8 (for 8 bananas)

- Almond milk for 2 servings: \( 1\frac{3}{4} \)

- Almond milk for 8 servings: 7

- Servings for \( 8\frac{3}{4} \) almond milk: 10

- Bananas for 2 servings: 2

- Matcha powder for 2 servings: \( \frac{1}{8} \)

- Ice for 2 servings: \( 3\frac{3}{4} \)

- Plain yogurt for 2 servings: \( \frac{1}{4} \)

But since the problem says "Fill in the missing numbers in the table", we can present the filled table (partial, the key ones):

| Servings | 4 | 2 | 8 | 6 | 10 | 16 |
| --- | --- | --- | --- | --- | --- | --- |
| almond milk (cups) | \( 3\frac{1}{2} \) | \( 1\frac{3}{4} \) | 7 | \( 5\frac{1}{4} \) | \( 8\frac{3}{4} \) | 14 |
| bananas | 4 | 2 | 8 | 6 | 10 | 16 |
| matcha powder (cups) | \( \frac{1}{4} \) | \( \frac{1}{8} \) | \( \frac{1}{2} \) | \( \frac{3}{8} \) | \( \frac{5}{8} \) | 1 |
| ice (ounces) | \( 7\frac{1}{2} \) | \( 3\frac{3}{4} \) | 15 | \( 11\frac{1}{4} \) | \( 18\frac{3}{4} \) | 30 |
| plain yogurt (pints) | \( \frac{1}{2} \) | \( \frac{1}{4} \) | 1 | \( \frac{3}{4} \) | \( \frac{5}{8} \) | 2 |

Answer:

The filled table (key missing values):

• Servings for 8 bananas: \( \boldsymbol{8} \)

• Almond milk for 2