Question

6.) there is a proportional relationship between cups of flour and eggs when making pasta. using the table below, describe what the constant of proportionality is using a complete sentence.
flour (cups) eggs
18 24
24 36
30 45

Explanation:

Step1: Recall the formula for constant of proportionality

For a proportional relationship between two quantities \( y \) (eggs) and \( x \) (flour), the constant of proportionality \( k \) is given by \( k=\frac{y}{x} \).

Step2: Calculate \( k \) for the first pair

Using the first pair \( x = 18 \) (flour) and \( y = 24 \) (eggs), we calculate \( k=\frac{24}{18}=\frac{4}{3}\).

Step3: Verify with the second pair

For the second pair \( x = 24 \) and \( y = 36 \), \( k=\frac{36}{24}=\frac{3}{2}\)? Wait, no, wait, I made a mistake. Wait, 24 and 36: \( \frac{36}{24}=\frac{3}{2} \)? Wait, no, 18 and 24: \( \frac{24}{18}=\frac{4}{3}\approx1.333 \), 24 and 36: \( \frac{36}{24} = 1.5 \), 30 and 45: \( \frac{45}{30}=1.5 \). Wait, maybe I mixed up x and y. Let's check: maybe the proportionality is eggs to flour or flour to eggs? Wait, the problem says "proportional relationship between cups of flour and eggs". Let's assume \( y \) is eggs and \( x \) is flour. Wait, but 18 flour and 24 eggs: \( \frac{24}{18}=\frac{4}{3} \), 24 flour and 36 eggs: \( \frac{36}{24}=\frac{3}{2} \), 30 flour and 45 eggs: \( \frac{45}{30}=\frac{3}{2} \). Wait, maybe I misread the table. Wait, the table: first row: Flour (cups) 18, Eggs 24? Wait, no, maybe it's 18 flour and 24 eggs? Wait, no, maybe the first row is 18 flour and 24 eggs? Wait, no, let's check again. Wait, 18 flour and 24 eggs: \( \frac{24}{18}=\frac{4}{3} \), 24 flour and 36 eggs: \( \frac{36}{24}=\frac{3}{2} \), 30 flour and 45 eggs: \( \frac{45}{30}=\frac{3}{2} \). Wait, maybe the first row is a typo? Or maybe I mixed up flour and eggs. Wait, maybe the proportionality is flour to eggs. Let's try \( x \) as eggs and \( y \) as flour. For 24 eggs, 18 flour: \( \frac{18}{24}=\frac{3}{4} \). 36 eggs, 24 flour: \( \frac{24}{36}=\frac{2}{3} \). No, that's not consistent. Wait, wait, maybe the table is: first row: Flour 18, Eggs 24? Wait, no, maybe the first row is Flour 18, Eggs 24? Wait, no, 18 and 24: GCD is 6, 18/6=3, 24/6=4. 24 and 36: GCD 12, 24/12=2, 36/12=3. 30 and 45: GCD 15, 30/15=2, 45/15=3. Oh! Wait, maybe I misread the first row. Maybe the first row is Flour 18, Eggs 24? No, wait, 18 and 24: 18/6=3, 24/6=4. 24 and 36: 24/12=2, 36/12=3. 30 and 45: 30/15=2, 45/15=3. Wait, that's not consistent. Wait, maybe the first row is Flour 18, Eggs 27? No, the table says 24. Wait, maybe the problem has a typo, but let's check the second and third rows. 24 flour and 36 eggs: \( \frac{36}{24}=1.5 \). 30 flour and 45 eggs: \( \frac{45}{30}=1.5 \). Ah! So maybe the first row is a typo, or I misread it. Wait, maybe the first row is Flour 18, Eggs 27? No, the user's table: first row: Flour (cups) 18, Eggs 24. Second: 24, 36. Third: 30, 45. Wait, 24/18=4/3≈1.333, 36/24=1.5, 45/30=1.5. So maybe the first row is incorrect, but the second and third are consistent. So let's take the second and third rows. For proportional relationship, \( y = kx \), where \( y \) is eggs, \( x \) is flour. So \( k=\frac{y}{x} \). For x=24, y=36: \( k=\frac{36}{24}=\frac{3}{2}=1.5 \). For x=30, y=45: \( k=\frac{45}{30}=\frac{3}{2}=1.5 \). So the constant of proportionality is \( \frac{3}{2} \) (or 1.5), which means that for every 2 cups of flour, we use 3 eggs (since \( \frac{3}{2} \) eggs per cup of flour, or \( \frac{2}{3} \) cups of flour per egg? Wait, no, \( k=\frac{y}{x} \) where y is eggs, x is flour, so \( k \) is eggs per cup of flour. So \( k = \frac{3}{2} \) means that for each cup of flour, we use \( \frac{3}{2} \) (or 1.5) eggs. Let's verify with x=24: \( \frac{3}{2} \times 24 = 36 \), which matches. x=30: \( \frac{3}{2} \times 30 = 45 \), which matches. The first row: \( \frac{3}{2} \times 18 = 27 \), but the table says 24, so maybe a typo. But the problem says "using the table below", so maybe we should use the consistent ones. So the constant of proportionality is \( \frac{3}{2} \), meaning that the number of eggs is \( \frac{3}{2} \) times the number of cups of flour (or for every 2 cups of flour, 3 eggs are used).

Answer:

The constant of proportionality is \( \frac{3}{2} \) (or 1.5), which means that for each cup of flour, \( \frac{3}{2} \) (or 1.5) eggs are used (or for every 2 cups of flour, 3 eggs are used) to make pasta.