Direct Proportion - Free Maths Worksheet | Proportions worksheet ... - Free Printable

Problem Analysis

The task involves solving problems related to direct proportionality. In direct proportion, two variables \( x \) and \( y \) are related by the equation:
\[
y = kx
\]
where \( k \) is the constant of proportionality.

We will solve each part step by step.

---

Part 1: Fill in the gaps in each table

#### Table 1:
\[
\begin{array}{|c|c|c|c|}
\hline
x & 4 & 8 & ? \\
\hline
y & 1 & 2 & 3 \\
\hline
\end{array}
\]

- From the given values, we see that \( y \) is directly proportional to \( x \). The constant of proportionality \( k \) can be found using any pair of \( (x, y) \):
\[
k = \frac{y}{x} = \frac{1}{4}
\]
- To find the missing \( x \)-value when \( y = 3 \):
\[
3 = \frac{1}{4} \cdot x \implies x = 3 \cdot 4 = 12
\]

#### Table 2:
\[
\begin{array}{|c|c|c|c|}
\hline
x & 2 & 6 & 7 \\
\hline
y & 10 & ? & ? \\
\hline
\end{array}
\]

- Using the pair \( (x, y) = (2, 10) \), the constant of proportionality \( k \) is:
\[
k = \frac{y}{x} = \frac{10}{2} = 5
\]
- To find the missing \( y \)-values:
- For \( x = 6 \):
\[
y = 5 \cdot 6 = 30
\]
- For \( x = 7 \):
\[
y = 5 \cdot 7 = 35
\]

#### Table 3:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 3 & 5 & 10 & 16 \\
\hline
y & ? & 7.5 & ? & ? \\
\hline
\end{array}
\]

- Using the pair \( (x, y) = (5, 7.5) \), the constant of proportionality \( k \) is:
\[
k = \frac{y}{x} = \frac{7.5}{5} = 1.5
\]
- To find the missing \( y \)-values:
- For \( x = 3 \):
\[
y = 1.5 \cdot 3 = 4.5
\]
- For \( x = 10 \):
\[
y = 1.5 \cdot 10 = 15
\]
- For \( x = 16 \):
\[
y = 1.5 \cdot 16 = 24
\]

#### Table 4:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -2 & 0 & ? & 7 & ? \\
\hline
y & -1 & 0 & 5 & ? & 10 \\
\hline
\end{array}
\]

- Using the pair \( (x, y) = (-2, -1) \), the constant of proportionality \( k \) is:
\[
k = \frac{y}{x} = \frac{-1}{-2} = 0.5
\]
- To find the missing \( x \)- and \( y \)-values:
- For \( y = 5 \):
\[
5 = 0.5 \cdot x \implies x = \frac{5}{0.5} = 10
\]
- For \( x = 7 \):
\[
y = 0.5 \cdot 7 = 3.5
\]
- For \( y = 10 \):
\[
10 = 0.5 \cdot x \implies x = \frac{10}{0.5} = 20
\]

#### Table 5:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & ? & 4 & 7 & 10 & ? \\
\hline
y & 5 & 10 & ? & ? & 37.5 \\
\hline
\end{array}
\]

- Using the pair \( (x, y) = (4, 10) \), the constant of proportionality \( k \) is:
\[
k = \frac{y}{x} = \frac{10}{4} = 2.5
\]
- To find the missing \( x \)- and \( y \)-values:
- For \( y = 5 \):
\[
5 = 2.5 \cdot x \implies x = \frac{5}{2.5} = 2
\]
- For \( x = 7 \):
\[
y = 2.5 \cdot 7 = 17.5
\]
- For \( x = 10 \):
\[
y = 2.5 \cdot 10 = 25
\]
- For \( y = 37.5 \):
\[
37.5 = 2.5 \cdot x \implies x = \frac{37.5}{2.5} = 15
\]

---

Part 2: Solve the problems

#### Problem 2:
\( x \) is directly proportional to \( y \). When \( x = 5 \), \( y = 15 \). Find \( y \) when \( x = 20 \).

- The constant of proportionality \( k \) is:
\[
k = \frac{y}{x} = \frac{15}{5} = 3
\]
- When \( x = 20 \):
\[
y = k \cdot x = 3 \cdot 20 = 60
\]

#### Problem 3:
\( m \) is directly proportional to \( n \). When \( m = 4 \), \( n = 6 \). Find \( n \) when \( m = 10 \).

- The constant of proportionality \( k \) is:
\[
k = \frac{n}{m} = \frac{6}{4} = 1.5
\]
- When \( m = 10 \):
\[
n = k \cdot m = 1.5 \cdot 10 = 15
\]

#### Problem 4:
\( p \) is directly proportional to \( q \). When \( p = 6 \), \( q = 2 \). Find \( q \) when \( p = 4 \).

- The constant of proportionality \( k \) is:
\[
k = \frac{q}{p} = \frac{2}{6} = \frac{1}{3}
\]
- When \( p = 4 \):
\[
q = k \cdot p = \frac{1}{3} \cdot 4 = \frac{4}{3}
\]

#### Problem 5:
\( s \) is directly proportional to \( t \). When \( t = 21 \), \( s = 14 \). Find \( s \) when \( t = 201 \).

- The constant of proportionality \( k \) is:
\[
k = \frac{s}{t} = \frac{14}{21} = \frac{2}{3}
\]
- When \( t = 201 \):
\[
s = k \cdot t = \frac{2}{3} \cdot 201 = 134
\]

#### Problem 6:
\( c \) is directly proportional to \( d \). When \( c = 15 \), \( d = 8 \). Find \( c \) when \( d = 10 \).

- The constant of proportionality \( k \) is:
\[
k = \frac{c}{d} = \frac{15}{8}
\]
- When \( d = 10 \):
\[
c = k \cdot d = \frac{15}{8} \cdot 10 = \frac{150}{8} = 18.75
\]

#### Problem 7:
\( f \) is directly proportional to \( g \). When \( f = 200 \), \( g = 140 \). Find \( g \) when \( f = 350 \).

- The constant of proportionality \( k \) is:
\[
k = \frac{g}{f} = \frac{140}{200} = \frac{7}{10}
\]
- When \( f = 350 \):
\[
g = k \cdot f = \frac{7}{10} \cdot 350 = 245
\]

#### Problem 8:
\( y \) is directly proportional to the square of \( x \). When \( x = 5 \), \( y = 75 \). Find \( y \) when \( x = 4 \).

- The relationship is \( y = kx^2 \). Using \( (x, y) = (5, 75) \):
\[
75 = k \cdot 5^2 \implies 75 = k \cdot 25 \implies k = \frac{75}{25} = 3
\]
- When \( x = 4 \):
\[
y = k \cdot x^2 = 3 \cdot 4^2 = 3 \cdot 16 = 48
\]

#### Problem 9:
\( k \) is directly proportional to the cube of \( j \). When \( j = 2 \), \( k = 40 \). Find \( j \) when \( k = 320 \).

- The relationship is \( k = cj^3 \). Using \( (j, k) = (2, 40) \):
\[
40 = c \cdot 2^3 \implies 40 = c \cdot 8 \implies c = \frac{40}{8} = 5
\]
- When \( k = 320 \):
\[
320 = 5 \cdot j^3 \implies j^3 = \frac{320}{5} = 64 \implies j = \sqrt[3]{64} = 4
\]

#### Problem 10:
\( a \) is directly proportional to the square root of \( b \). When \( b = 9 \), \( a = \frac{3}{2} \). Find \( b \) when \( a = 2 \).

- The relationship is \( a = k\sqrt{b} \). Using \( (b, a) = (9, \frac{3}{2}) \):
\[
\frac{3}{2} = k \cdot \sqrt{9} \implies \frac{3}{2} = k \cdot 3 \implies k = \frac{\frac{3}{2}}{3} = \frac{1}{2}
\]
- When \( a = 2 \):
\[
2 = \frac{1}{2} \cdot \sqrt{b} \implies \sqrt{b} = 2 \cdot 2 = 4 \implies b = 4^2 = 16
\]

---

Final Answers

\[
\boxed{
\begin{array}{l}
\text{Table 1: } x = 12 \\
\text{Table 2: } y = 30, 35 \\
\text{Table 3: } y = 4.5, 15, 24 \\
\text{Table 4: } x = 10, 20; y = 3.5 \\
\text{Table 5: } x = 2, 15; y = 17.5, 25 \\
\text{Problem 2: } y = 60 \\
\text{Problem 3: } n = 15 \\
\text{Problem 4: } q = \frac{4}{3} \\
\text{Problem 5: } s = 134 \\
\text{Problem 6: } c = 18.75 \\
\text{Problem 7: } g = 245 \\
\text{Problem 8: } y = 48 \\
\text{Problem 9: } j = 4 \\
\text{Problem 10: } b = 16 \\
\end{array}
}
\]