Hard GCSE Direct and inverse proportion Questions

Challenging, exam-style GCSE Direct and inverse proportion questions with worked solutions. Stretch yourself on the hardest direct proportion, finding k, reverse substitution, inverse proportion problems.

direct proportionfinding kreverse substitutioninverse proportionmultipliersreasoning
GCSE Foundation34 questionsStep-by-step solutions
Question 1
6 markschallenging
55 workers can tile a floor in 1212 hours. Each worker is paid £9 for every hour that they work. Work out the total wage bill, in pounds, if only 44 workers tile the same floor.
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Worked solution

  1. Decide which kind of proportion this is

    hours∝1workers\text{hours} \propto \frac{1}{\text{workers}}

    Fewer workers take more hours, so hours and workers are inversely proportional.

  2. Write the inverse proportion formula

    h=kwh = \frac{k}{w}

    Here hh is the number of hours and ww is the number of workers.

  3. Substitute the given pair

    12=k512 = \frac{k}{5}

    Put w=5w = 5 and h=12h = 12 into the formula.

  4. Find the constant, the total worker-hours

    k=5×12=60k = 5 \times 12 = 60

    The floor needs 6060 worker-hours of work.

  5. Write the formula for this job

    h=60wh = \frac{60}{w}

    Workers multiplied by hours is always 6060.

  6. Check the formula against the given pair

    605=12\frac{60}{5} = 12

    It gives back 1212 hours, so the constant is right.

  7. Substitute four workers

    h=604h = \frac{60}{4}

    Now use w=4w = 4.

  8. Work out the time taken

    h=15h = 15

    Four workers take 1515 hours.

  9. Check the direction of the change

    5→4 workers (fewer),12→15 hours (more)5 \to 4 \text{ workers (fewer)}, \quad 12 \to 15 \text{ hours (more)}

    Fewer workers take longer, so the answer makes sense.

  10. Work out the total hours worked by all the workers

    4×15=604 \times 15 = 60

    Each of the 44 workers works for 1515 hours, giving 6060 worker-hours in total.

  11. Notice the worker-hours are unchanged

    5×12=60=4×155 \times 12 = 60 = 4 \times 15

    The total worker-hours is the constant of proportionality, so it is the same either way.

  12. Write the formula for the wage bill

    Bill=9×worker-hours\text{Bill} = 9 \times \text{worker-hours}

    Each worker-hour costs £99, so the bill is directly proportional to the total worker-hours.

  13. Work out the wage bill

    Bill=9×60=540\text{Bill} = 9 \times 60 = 540

    The total wage bill is £540540.

  14. Compare with five workers

    9×5×12=5409 \times 5 \times 12 = 540

    Five workers for 1212 hours would cost the same, because the worker-hours are the same. Using fewer workers does not save money here.

  15. State the final answer

    540540

    The answer is 540540.

Answer
540540
Question 2
5 markschallenging
Which one of these relationships gives a graph that is a curve getting closer and closer to both axes but never touching them?
Show worked solution

Worked solution

  1. Recall which relationship gives that curve

    y=kxy = \frac{k}{x}

    Only inverse proportion, y=kxy = \frac{k}{x}, gives a curve that approaches both axes without touching them.

  2. Test the first relationship for a constant product

    y=20x:  xy=20y = \frac{20}{x}: \; xy = 20

    Multiplying gives xy=20xy = 20 for every point, so the product is constant.

  3. Work out some points on it

    (2,10),(4,5),(10,2)(2, 10), \quad (4, 5), \quad (10, 2)

    Each of these has x×y=20x \times y = 20.

  4. Check it never reaches the axes

    20x≠0,x≠0\frac{20}{x} \neq 0, \quad x \neq 0

    yy is never 00 and xx cannot be 00, so the curve never touches an axis.

  5. Test the relationship y equals twenty x

    (1,20),(2,40):  xy=20 then 80(1, 20), \quad (2, 40): \; xy = 20 \text{ then } 80

    The product changes, so this is not inverse proportion. It is direct proportion.

  6. Describe the graph of y equals twenty x

    y=20x passes through (0,0)y = 20x \text{ passes through } (0, 0)

    It is a straight line through the origin, so it touches both axes at the origin.

  7. Test the relationship y equals twenty minus x

    (1,19),(2,18):  xy=19 then 36(1, 19), \quad (2, 18): \; xy = 19 \text{ then } 36

    The product changes, so this is not inverse proportion.

  8. Describe the graph of y equals twenty minus x

    y=20−x meets the x-axis at (20,0)y = 20 - x \text{ meets the } x\text{-axis at } (20, 0)

    It is a straight line and it does touch the axes, so it cannot be the answer.

  9. Test the relationship y equals twenty x plus five

    (1,25),(2,45):  xy=25 then 90(1, 25), \quad (2, 45): \; xy = 25 \text{ then } 90

    The product changes, so this is not inverse proportion. It is a straight line that misses the origin, so it is neither kind of proportion.

  10. Test the relationship y equals x over twenty

    (1,0.05),(2,0.1):  xy=0.05 then 0.2(1, 0.05), \quad (2, 0.1): \; xy = 0.05 \text{ then } 0.2

    The product changes, so this is not inverse proportion.

  11. Describe the graph of y equals x over twenty

    y=120x passes through (0,0)y = \frac{1}{20}x \text{ passes through } (0, 0)

    It is direct proportion with a small constant, so it is a straight line through the origin, not a curve.

  12. Note the trap

    x20≠20x\frac{x}{20} \neq \frac{20}{x}

    Dividing xx by a number is still direct proportion. Only dividing a number by xx gives inverse proportion.

  13. Eliminate every straight line

    four of the five options are straight lines\text{four of the five options are straight lines}

    Only one option is a curve at all.

  14. Confirm the shape of the remaining option

    y=20x is a hyperbolay = \frac{20}{x} \text{ is a hyperbola}

    It falls steeply and then flattens out, staying above the horizontal axis for all positive xx.

  15. State the answer

    y=20xy = \frac{20}{x}

    The relationship y=20xy = \frac{20}{x} gives the curve described.

Answer
y=20xy = \frac{20}{x}
Question 3
6 markschallenging
A recipe for 66 people uses 450450 g of rice. The mass of rice is directly proportional to the number of people. Work out the mass of rice, in grams, needed for 1414 people, and the number of people that 12001200 g of rice would serve. Give the number of people as your answer.
Show worked solution

Worked solution

  1. Write the direct proportion formula

    R=kpR = kp

    Here RR is the mass of rice in grams and pp is the number of people.

  2. Substitute the given pair

    450=k×6450 = k \times 6

    Put p=6p = 6 and R=450R = 450 into the formula.

  3. Find the rice needed for one person

    k=4506=75k = \frac{450}{6} = 75

    Each person needs 7575 g of rice.

  4. Write the formula for the rice

    R=75pR = 75p

    This works for any number of people.

  5. Check the formula against the given pair

    75×6=45075 \times 6 = 450

    It gives back 450450 g, so the constant is right.

  6. Substitute fourteen people

    R=75×14R = 75 \times 14

    Now use p=14p = 14.

  7. Work out the mass of rice

    R=1050R = 1050

    Fourteen people need 10501050 g of rice. This is the first answer.

  8. Check with the multiplier

    146=73,450×73=1050\frac{14}{6} = \frac{7}{3}, \quad 450 \times \frac{7}{3} = 1050

    The number of people is multiplied by 73\frac{7}{3}, so the rice is multiplied by 73\frac{7}{3} too.

  9. Set up the second part

    1200=75p1200 = 75p

    This time the mass of rice is known and the number of people is not.

  10. Divide to find the number of people

    p=120075p = \frac{1200}{75}

    Divide the total rice by the rice needed for one person.

  11. Work out the number of people

    p=16p = 16

    So 12001200 g of rice serves 1616 people.

  12. Check the second answer

    75×16=120075 \times 16 = 1200

    Substituting back gives the right mass of rice.

  13. Check the answer is sensible

    1200>450⇒16>61200 > 450 \Rightarrow 16 > 6

    More rice serves more people, so the direction of the answer is right.

  14. Read the question again

    the answer required is p\text{the answer required is } p

    The question asks for the number of people, not the mass of rice.

  15. State the final answer

    16 people16\text{ people}

    The answer is 1616 people.

Answer
1616
Question 4
5 markschallenging
yy is directly proportional to xx. When x=9x = 9, y=6y = 6. Work out the value of xx when y=26y = 26, and the value of yy when x=12x = 12. Work out the sum of these two answers.
Show worked solution

Worked solution

  1. Identify the type of proportion

    y∝xy \propto x

    Directly proportional means y=kxy = kx: multiply xx by a number and yy is multiplied by the same number.

  2. Write the general formula

    y=kxy = kx

    Here kk is the constant of proportionality. It is the same for every pair of values in the relationship.

  3. Substitute the pair of values you are given

    6=k×96 = k \times 9

    Put x=9x = 9 and y=6y = 6 into y=kxy = kx.

  4. Solve for the constant of proportionality

    k=69=23k = \frac{6}{9} = \frac{2}{3}

    Dividing yy by xx gives k=23k = \frac{2}{3}. For direct proportion the value of yx\frac{y}{x} is always the same.

  5. Write the formula for this relationship

    y=23xy = \frac{2}{3}x

    Every pair of values in this relationship fits y=23xy = \frac{2}{3}x.

  6. Check the formula against the given pair

    23×9=6\frac{2}{3} \times 9 = 6

    It gives back y=6y = 6 when x=9x = 9, so the value of kk is right.

  7. Substitute the required value of y

    26=23x26 = \frac{2}{3}x

    Put y=26y = 26 into y=23xy = \frac{2}{3}x.

  8. Divide to find x

    x=2623=39x = \frac{26}{\frac{2}{3}} = 39

    So x=39x = 39 when y=26y = 26.

  9. Check the first answer

    23×39=26\frac{2}{3} \times 39 = 26

    Substituting x=39x = 39 back gives y=26y = 26, so it is right.

  10. State the first answer

    x=39x = 39

    The first answer is x=39x = 39.

  11. Substitute the new value of x

    y=23×12y = \frac{2}{3} \times 12

    Now use x=12x = 12 in y=23xy = \frac{2}{3}x.

  12. Work out the value of y

    y=8y = 8

    So y=8y = 8 when x=12x = 12.

  13. State the second answer

    y=8y = 8

    The second answer is y=8y = 8.

  14. Add the two answers

    39+8=4739 + 8 = 47

    The sum of the two answers is 4747.

  15. State the final answer

    4747

    The answer is 4747.

Answer
4747
Question 5
5 markschallenging
It takes 1515 identical machines 88 hours to complete a job. Before the job starts, 55 of the machines break down. Work out how many hours the remaining machines take to complete the whole job.
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Worked solution

  1. Decide which kind of proportion this is

    hours∝1machines\text{hours} \propto \frac{1}{\text{machines}}

    Fewer machines take longer, so hours and machines are inversely proportional.

  2. Write the inverse proportion formula

    t=kmt = \frac{k}{m}

    Here tt is the time in hours and mm is the number of machines.

  3. Substitute the given pair

    8=k158 = \frac{k}{15}

    Put m=15m = 15 and t=8t = 8 into the formula.

  4. Find the constant, the total machine-hours

    k=15×8=120k = 15 \times 8 = 120

    The job needs 120120 machine-hours of work.

  5. Write the formula for this job

    t=120mt = \frac{120}{m}

    Machines multiplied by hours is always 120120.

  6. Check the formula against the given pair

    12015=8\frac{120}{15} = 8

    It gives back 88 hours, so the constant is right.

  7. Work out how many machines are left

    15−5=1015 - 5 = 10

    Five machines break down, so 1010 machines remain.

  8. Substitute the new number of machines

    t=12010t = \frac{120}{10}

    Now use m=10m = 10.

  9. Work out the time

    t=12t = 12

    The 1010 machines take 1212 hours.

  10. Check the product

    10×12=12010 \times 12 = 120

    The machine-hours are unchanged, so the answer is consistent.

  11. Check with the multiplier

    15→10 is ×23,8÷23=1215 \to 10 \text{ is } \times \frac{2}{3}, \quad 8 \div \frac{2}{3} = 12

    The number of machines is multiplied by 23\frac{2}{3}, so the time is divided by 23\frac{2}{3}, which makes it longer.

  12. Check the direction of the change

    15→10 machines (fewer),8→12 hours (more)15 \to 10 \text{ machines (fewer)}, \quad 8 \to 12 \text{ hours (more)}

    Fewer machines needs more hours, so the answer makes sense.

  13. Note the common mistake

    1205=24 is wrong\frac{120}{5} = 24 \text{ is wrong}

    Dividing by the 55 broken machines instead of the 1010 working machines gives 2424 hours, which is wrong.

  14. Note the other common mistake

    8×1015≈5.3 is wrong8 \times \frac{10}{15} \approx 5.3 \text{ is wrong}

    That would be the direct proportion answer, and it says fewer machines finish sooner, which is impossible.

  15. State the final answer

    12 hours12\text{ hours}

    The answer is 1212 hours.

Answer
1212

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