Question

how many guests there will be. isaac started making the following chart so he
w how much of each ingredient he would need, depending on the number of guests.
in the missing numbers in the table below. use the arrow keys or tab to move from
other. to enter a mixed number, press the \+\ key or click the \mixed number\ bu
w. to view a hint, click in a cell, then press the \h\ key or click the \hint\ button bel

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

Answer:

To solve the table, we first determine the scaling factor between servings. The original servings are 4, and when servings are 2 (half of 4), the ingredients are also halved. When servings are 8 (double of 4), ingredients are doubled. We can find the scaling factor \( k=\frac{\text{new servings}}{4} \) and multiply each ingredient by \( k \).

Step 1: Analyze the "Servings" and "bananas" columns

For 4 servings, bananas = 4.

• For 2 servings ( \( \frac{2}{4} = 0.5 \) of 4 servings):

Bananas = \( 4 \times 0.5 = 2 \).

• For 8 servings ( \( \frac{8}{4} = 2 \) times 4 servings):

Bananas = \( 4 \times 2 = 8 \) (matches the table).

Step 2: Almond Milk (cups)

For 4 servings: \( 3\frac{1}{2}=\frac{7}{2} \) cups.

• For 2 servings: \( \frac{7}{2} \times 0.5 = \frac{7}{4}=1\frac{3}{4} \) (matches the table).
• For 8 servings: \( \frac{7}{2} \times 2 = 7 \) cups.
• For \( 8\frac{3}{4} \) cups: Let servings = \( x \). Then \( \frac{7}{2} \times \frac{x}{4} = 8\frac{3}{4}=\frac{35}{4} \).

Solve: \( \frac{7x}{8}=\frac{35}{4} \implies x = \frac{35}{4} \times \frac{8}{7}=10 \) servings.

Step 3: Matcha Powder (cups)

For 4 servings: \( \frac{1}{4} \) cup.

• For 2 servings: \( \frac{1}{4} \times 0.5 = \frac{1}{8} \) cup.
• For 8 servings: \( \frac{1}{4} \times 2 = \frac{1}{2} \) cup (matches the table).

Step 4: Ice (ounces)

For 4 servings: \( 7\frac{1}{2}=\frac{15}{2} \) ounces.

• For 2 servings: \( \frac{15}{2} \times 0.5 = \frac{15}{4}=3\frac{3}{4} \) ounces.
• For 8 servings: \( \frac{15}{2} \times 2 = 15 \) ounces.
• For 30 ounces: Let servings = \( x \). Then \( \frac{15}{2} \times \frac{x}{4}=30 \).

Solve: \( \frac{15x}{8}=30 \implies x = \frac{30 \times 8}{15}=16 \) servings.

Step 5: Plain Yogurt (pints)

For 4 servings: \( \frac{1}{2} \) pint.

• For 2 servings: \( \frac{1}{2} \times 0.5 = \frac{1}{4} \) pint.
• For 8 servings: \( \frac{1}{2} \times 2 = 1 \) pint.
• For \( \frac{3}{4} \) pint: Let servings = \( x \). Then \( \frac{1}{2} \times \frac{x}{4}=\frac{3}{4} \).

Solve: \( \frac{x}{8}=\frac{3}{4} \implies x = 6 \) servings.

Step 6: Fill the Table (Key Values)

Servings	4	2	8	6	10	16
Bananas	4	2	8	\( 4 \times \frac{6}{4}=6 \)	\( 4 \times \frac{10}{4}=10 \)	\( 4 \times \frac{16}{4}=16 \)
Matcha Powder (cups)	\( \frac{1}{4} \)	\( \frac{1}{8} \)	\( \frac{1}{2} \)	\( \frac{1}{4} \times \frac{6}{4}=\frac{3}{8} \)	\( \frac{1}{4} \times \frac{10}{4}=\frac{5}{8} \)	\( \frac{1}{4} \times \frac{16}{4}=1 \)
Ice (ounces)	\( 7\frac{1}{2} \)	\( 3\frac{3}{4} \)	\( 15 \)	\( \frac{15}{2} \times \frac{6}{4}=\frac{45}{4}=11\frac{1}{4} \)	\( \frac{15}{2} \times \frac{10}{4}=\frac{75}{4}=18\frac{3}{4} \)	\( 30 \)
Plain Yogurt (pints)	\( \frac{1}{2} \)	\( \frac{1}{4} \)	\( 1 \)	\( \frac{3}{4} \)	\( \frac{1}{2} \times \frac{10}{4}=\frac{5}{4}=1\frac{1}{4} \)	\( \frac{1}{2} \times \frac{16}{4}=2 \)

Final Table (Simplified)

For example, the missing value for almond milk (8 servings) is \( \boldsymbol{7} \) cups.
For bananas (2 servings), it’s \( \boldsymbol{2} \) bananas.
For matcha powder (2 servings), it’s \( \boldsymbol{\frac{1}{8}} \) cup.
For ice (2 servings), it’s \( \boldsymbol{3\frac{3}{4}} \) ounces.
For plain yogurt (2 servings), it’s \( \boldsymbol{\frac{1}{4}} \) pint.

(Note: The table requires filling all cells, but the key is using the scaling factor \( \frac{\text{new servings}}{4} \) to multiply each ingredient’s 4-serving quantity.)