Question

1
quantities and are in a proportional relationship. complete the table.

x	y
4	16
3	4 (highlighted)
12 (highlighted)	8

Explanation:

Step1: Find the constant of proportionality

For proportional relationships, \( y = kx \), so \( k=\frac{y}{x} \). Using \( x = 4 \), \( y = 16 \), we get \( k=\frac{16}{4}=4 \). Wait, but let's check the other rows. Wait, maybe I misread. Wait, the table has \( x = 4, y = 16 \); \( x = 3 \), find \( y \); \( x = 12 \), \( y = 8 \)? Wait, no, maybe the proportionality is \( y = kx \) or \( x = ky \)? Wait, let's recalculate. Wait, when \( x = 4 \), \( y = 16 \), so \( k=\frac{y}{x}=\frac{16}{4}=4 \). But then for \( x = 3 \), \( y = 3\times4 = 12 \)? But the table has a blue box on \( y \) when \( x = 3 \), and \( x = 12 \) with \( y = 8 \). Wait, maybe it's inverse proportionality? Wait, inverse proportionality is \( xy = k \). Let's check: \( 4\times16 = 64 \). Then for \( x = 3 \), \( y=\frac{64}{3}\approx21.33 \), which doesn't match. Wait, maybe the table is miswritten? Wait, no, maybe I made a mistake. Wait, the user's table: first row \( x = 4, y = 16 \); second row \( x = 3 \), \( y \) is blue; third row \( x = 12 \), \( y = 8 \). Wait, let's check the third row: if \( x = 12 \), \( y = 8 \), then \( k = xy = 12\times8 = 96 \). Then first row: \( 4\times16 = 64
eq 96 \). So that's not inverse. Wait, maybe direct proportionality but different. Wait, maybe the proportionality is \( x = ky \). Let's try \( x = ky \). For \( x = 4 \), \( y = 16 \), so \( 4 = k\times16 \), \( k=\frac{4}{16}=\frac{1}{4} \). Then for \( x = 3 \), \( 3=\frac{1}{4}y \), so \( y = 12 \). For \( x = 12 \), \( 12=\frac{1}{4}y \), so \( y = 48 \), but the table says \( y = 8 \). No, that's not. Wait, maybe the table has a typo, but assuming it's direct proportionality \( y = kx \), with \( x = 4, y = 16 \), so \( k = 4 \). Then when \( x = 3 \), \( y = 12 \); when \( y = 8 \), \( x = \frac{8}{4}=2 \). But the table has \( x = 12 \) with \( y = 8 \). Wait, maybe the user's table is incorrect, but let's proceed with direct proportionality. Wait, the first row: \( x = 4 \), \( y = 16 \), so \( k = 4 \). So for \( x = 3 \), \( y = 3\times4 = 12 \). For \( y = 8 \), \( x = \frac{8}{4}=2 \). But the table has \( x = 12 \) with \( y = 8 \), which would mean \( k = \frac{8}{12}=\frac{2}{3} \), which doesn't match the first row. Wait, maybe the problem is that the quantities are proportional, so \( \frac{y}{x} = k \). Let's recalculate:

First, find \( k \) from \( x = 4 \), \( y = 16 \): \( k=\frac{y}{x}=\frac{16}{4}=4 \). So the equation is \( y = 4x \).

Step2: Calculate \( y \) when \( x = 3 \)

Using \( y = 4x \), substitute \( x = 3 \): \( y = 4\times3 = 12 \).

Step3: Calculate \( x \) when \( y = 8 \)

Using \( y = 4x \), solve for \( x \): \( x=\frac{y}{4}=\frac{8}{4}=2 \). But the table has \( x = 12 \) with \( y = 8 \), which is a contradiction. Maybe the table has a mistake, but following the first row, the constant of proportionality is 4. So for \( x = 3 \), \( y = 12 \); for \( y = 8 \), \( x = 2 \). But the user's table shows \( x = 12 \) with \( y = 8 \), which would imply \( k=\frac{8}{12}=\frac{2}{3} \), but that doesn't match the first row. Assuming the first row is correct, the second row \( y \) when \( x = 3 \) is 12, and the third row \( x \) when \( y = 8 \) is 2. But maybe the user made a typo in the table. However, based on the first row, the proportionality is \( y = 4x \).

Answer:

For \( x = 3 \), \( y = 12 \); for \( y = 8 \), \( x = 2 \). But based on the table's structure, if we consider the second row ( \( x = 3 \) ), the \( y \) value is \( 12 \), and the third row ( \( x = 12 \) ) should be \( x = 2 \) for \( y = 8 \). But since the problem says "complete the table", and the first row gives \( k = 4 \), the second row \( y = 3\times4 = 12 \), and the third row \( x = 8\div4 = 2 \).