1. **Problem statement:** We are given several proportionality tables with missing values. We need to find the missing values assuming direct proportionality between the quantities. 2. **Formula and rule:** For two quantities $x$ and $y$ that are directly proportional, the ratio $\frac{y}{x}$ is constant. That is, $\frac{y_1}{x_1} = \frac{y_2}{x_2}$. 3. **Step-by-step solutions:** **a)** Given pairs: $(1,2)$, $(2,?)$, $(4,?)$, $(8,?)$. Calculate the constant ratio: $k = \frac{2}{1} = 2$. Find missing prices: $\text{For } x=2: y = k \times 2 = 2 \times 2 = 4$ $\text{For } x=4: y = 2 \times 4 = 8$ $\text{For } x=8: y = 2 \times 8 = 16$ **b)** Given pairs: $(3,9)$, $(?,15)$, $(21,?)$, $(24,?)$. Calculate ratio: $k = \frac{9}{3} = 3$. Find missing values: $\text{For } y=15: x = \frac{y}{k} = \frac{15}{3} = 5$ $\text{For } x=21: y = 3 \times 21 = 63$ $\text{For } x=24: y = 3 \times 24 = 72$ **c)** Given pairs: $(2,?)$, $(6,?)$, $(7,14)$, $(?,30)$. Calculate ratio: $k = \frac{14}{7} = 2$ (km/h). Find missing values: $\text{For } x=2: y = 2 \times 2 = 4$ $\text{For } x=6: y = 2 \times 6 = 12$ $\text{For } y=30: x = \frac{30}{2} = 15$ **d)** Given pairs: $(10,?)$, $(20,400)$, $(40,?)$, $(?,1800)$. Calculate ratio: $k = \frac{400}{20} = 20$ (m/min). Find missing values: $\text{For } x=10: y = 20 \times 10 = 200$ $\text{For } x=40: y = 20 \times 40 = 800$ $\text{For } y=1800: x = \frac{1800}{20} = 90$ **e)** Given pairs: $(?,5)$, $(12,15)$, $(28,?)$, $(40,100)$. Calculate ratio: $k = \frac{15}{12} = \frac{5}{4} = 1.25$ (kg/m). Find missing values: $\text{For } y=5: x = \frac{5}{1.25} = 4$ $\text{For } x=28: y = 1.25 \times 28 = 35$ **f)** Given pairs: $(1,?)$, $(7,28)$, $(?,36)$, $(?,64)$, $(?,100)$. Calculate ratio: $k = \frac{28}{7} = 4$ (g/cm). Find missing values: $\text{For } x=1: y = 4 \times 1 = 4$ $\text{For } y=36: x = \frac{36}{4} = 9$ $\text{For } y=64: x = \frac{64}{4} = 16$ $\text{For } y=100: x = \frac{100}{4} = 25$ **g)** Given pairs: $(1,?)$, $(4,6)$, $(10,?)$, $(?,45)$, $(40,?)$. Calculate ratio: $k = \frac{6}{4} = 1.5$ (price per unit). Find missing values: $\text{For } x=1: y = 1.5 \times 1 = 1.5$ $\text{For } x=10: y = 1.5 \times 10 = 15$ $\text{For } y=45: x = \frac{45}{1.5} = 30$ $\text{For } x=40: y = 1.5 \times 40 = 60$ **h)** Given pairs: $(?,3.5)$, $(3,10.5)$, $(7,?)$, $(10,?)$, $(?,49)$. Calculate ratio: $k = \frac{10.5}{3} = 3.5$ (price per unit). Find missing values: $\text{For } y=3.5: x = \frac{3.5}{3.5} = 1$ $\text{For } x=7: y = 3.5 \times 7 = 24.5$ $\text{For } x=10: y = 3.5 \times 10 = 35$ $\text{For } y=49: x = \frac{49}{3.5} = 14$