Inverse proportion questions with answers

Questions

1. Question 1Workers and days: find one worker first

   9 workers take 17 days to paint a block of flats. Find the number of days 3 workers would take, working at the same rate.

   Show the worked solution

   1. Step 1: Find how long ONE worker would take$\text{1 worker} \to 9 \times 17$$= 153$ days
   2. Step 2: Find the number of days for the new number of workers$\text{Number of days} = \dfrac{153}{3}$$= 51$ days
2. Question 2Speed and time

   A car travelling at an average speed of 45 km/h takes 6 hours for a journey. Find the average speed needed to make the same journey in 10 hours.

   Show the worked solution

   1. Step 1: Find the distance of the journey (the same at both speeds)$\text{Distance} = 45 \times 6$$= 270$ km
   2. Step 2: Find the speed for the new time$\text{Speed} = 270 \div 10$$= 27$ km/h
3. Question 3y = k/x: find k, then y

   $y$ is inversely proportional to $x$. When $x = 9$, $y = 12$. Find the value of $y$ when $x = 18$.

   Show the worked solution

   1. Step 1: Write the equation connecting y and x, with a constant k$y = \dfrac{k}{x}$
   2. Step 2: Substitute the given pair of values to find k$12 = \dfrac{k}{9}$$k = 12 \times 9$$= 108$
   3. Step 3: Substitute the new value of x to find y$y = \dfrac{108}{x}$$y = \dfrac{108}{18}$$= 6$
4. Question 4y = k/x: find k, then x

   $y$ is inversely proportional to $x$. When $x = 5$, $y = 3$. Find the value of $x$ when $y = 5$.

   Show the worked solution

   1. Step 1: Write the equation connecting y and x, with a constant k$y = \dfrac{k}{x}$
   2. Step 2: Substitute the given pair of values to find k$3 = \dfrac{k}{5}$$k = 3 \times 5$$= 15$
   3. Step 3: Substitute the new value of y and solve for x$y = \dfrac{15}{x}$$5 = \dfrac{15}{x}$$x = \dfrac{15}{5}$$= 3$
5. Question 5Complete a table of values

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

   Show the worked solution

   1. Step 1: Find k from the complete column$k = 3 \times 6$$= 18$
   2. Step 2: Find each missing yWhen $x = 1$:$y = \dfrac{18}{1}$$= 18$When $x = 6$:$y = \dfrac{18}{6}$$= 3$When $x = 9$:$y = \dfrac{18}{9}$$= 2$
6. Question 6Workers, hours a day and days

   7 workers, working 10 hours a day, take 9 days to build a wall. Find the number of days 6 workers, working 5 hours a day, would take to build the same wall.

   Show the worked solution

   1. Step 1: The whole job in worker-hours: workers × hours a day × days$\text{Total} = 7 \times 10 \times 9$$= 630$ worker-hours
   2. Step 2: The same job: divide by workers × hours a day$\text{Number of days} = \dfrac{630}{6 \times 5}$$= 21$
7. Question 7A change part-way through the job

   10 workers can build a fence in 15 days. They work for 6 days, then 5 more workers join them. Assuming that all the workers work at the same rate, find the number of days the rest of the fence takes to build.

   Show the worked solution

   1. Step 1: The assumption behind the methodWe assume that every worker works at the same rate.
   2. Step 2: The part left, in worker-days: the 10 workers for the days still to go$\text{Left} = 10 \times (15 - 6)$$= 90$ worker-days
   3. Step 3: Share it among the 15 workers there are now$\text{Time} = \dfrac{90}{10 + 5}$$= 6$ days
8. Question 8Is it inverse proportion? (x × y the same?)

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

   $x$	$1$	$4$	$9$	$12$
   $y$	$72$	$18$	$8$	$6$

   Show the worked solution

   1. Step 1: Work out x × y for each pair$1 \times 72 = 72$$4 \times 18 = 72$$9 \times 8 = 72$$12 \times 6 = 72$
   2. Step 2: In inverse proportion, x × y is the same for every pair$xy$ is the same for every pair, so $y$ is inversely proportional to $x$.