Direct proportion questions with answers

Questions

1. Question 1The unitary method: find one unit first

   A car travels 84 km on 6 litres of petrol. Find the distance it travels on 26 litres of petrol.

   Show the worked solution

   1. Step 1: Find the distance for 1 litre first$\text{Distance for 1 litre} = \dfrac{84}{6}$$= 14$
   2. Step 2: Find the distance for the number of litres asked for$\text{Distance for 26 litres} = 14 \times 26$$= 364$The distance is $364$ km.
2. Question 2y = kx: find k, then y

   $y$ is directly proportional to $x$. When $x = 7$, $y = 56$. Find the value of $y$ when $x = 5$.

   Show the worked solution

   1. Step 1: Write the equation connecting y and x, with a constant k$y = kx$
   2. Step 2: Substitute the given pair of values to find k$56 = k(7)$$k = \dfrac{56}{7}$$= 8$
   3. Step 3: Substitute the new value of x to find y$y = 8x$$y = 8(5)$$= 40$
3. Question 3y = kx: find k, then x

   $y$ is directly proportional to $x$. When $x = 12$, $y = 8$. Find the value of $x$ when $y = 2$.

   Show the worked solution

   1. Step 1: Write the equation connecting y and x, with a constant k$y = kx$
   2. Step 2: Substitute the given pair of values to find k$8 = k(12)$$k = \dfrac{8}{12}$$= \dfrac{2}{3}$
   3. Step 3: Substitute the new value of y and solve for x$y = \dfrac{2}{3}x$$2 = \dfrac{2}{3}x$$x = 2 \times \dfrac{3}{2}$$= 3$
4. Question 4Complete a table of values

   $y$ is directly proportional to $x$. Complete the table of values.

   Show the worked solution

   1. Step 1: Find k from the complete column$k = \dfrac{6}{9}$$= \dfrac{2}{3}$
   2. Step 2: Find each missing yWhen $x = 3$:$y = \dfrac{2}{3} \times 3$$= 2$When $x = 12$:$y = \dfrac{2}{3} \times 12$$= 8$When $x = 24$:$y = \dfrac{2}{3} \times 24$$= 16$
5. Question 5Is it direct proportion? (y ÷ x the same?)

   The table shows some values of $x$ and $y$. State whether $y$ is directly proportional to $x$.

   $x$	$2$	$3$	$4$	$5$
   $y$	$16$	$24$	$32$	$40$

   Show the worked solution

   1. Step 1: Choose the test for direct proportionIn direct proportion, $\dfrac{y}{x}$ is the same for every pair.
   2. Step 2: Work it out for each pair$\dfrac{16}{2} = 8$$\dfrac{24}{3} = 8$$\dfrac{32}{4} = 8$$\dfrac{40}{5} = 8$
   3. Step 3: Compare the values and conclude$\dfrac{y}{x}$ is the same for every pair, so $y$ is directly proportional to $x$.
6. Question 6Is it direct proportion? (an equation or a straight-line graph)

   The variables $x$ and $y$ are connected by the equation $y = \dfrac{x}{3}$. Is $y$ directly proportional to $x$?

   Show the worked solution

   1. Step 1: Work out y ÷ x at two values of x (x = 1 and x = 2)When $x = 1$, $y = \dfrac{(1)}{3}$$y = \dfrac{1}{3}$, so $\dfrac{y}{x} = \dfrac{1}{3}$.When $x = 2$, $y = \dfrac{(2)}{3}$$y = \dfrac{2}{3}$, so $\dfrac{y}{x} = \dfrac{1}{3}$.
   2. Step 2: Direct proportion needs y ÷ x the same everywhere (y = kx: a line through the origin)$y = \dfrac{x}{3}$ is $y = kx$ with $k = \dfrac{1}{3}$: $y$ is directly proportional to $x$.
7. Question 7Direct proportion with a change of units

   The cost of a ribbon is directly proportional to its length. 40 cm of the ribbon costs 60 cents. Find the cost, in dollars, of 3 m of the ribbon.

   Show the worked solution

   1. Step 1: Change 3 m to cm, the unit in the pair given$3$ m $= 3 \times 100$$= 300$ cm
   2. Step 2: The cost is (cost for 1 cm) × (the number of cm)$\text{Cost} = \dfrac{60}{40} \times 300$$= 450$ cents
   3. Step 3: Change cents to dollars$\dfrac{450}{100} = 4.5$The cost is $4.50.