Chapter 6 of 7

All 7 chapters
  1. 1Percentage increase and decrease, readunderstood
  2. 2Finding the original amount, readunderstood
  3. 3Simple and compound interest, readunderstood
  4. 4Growth and decay, readunderstood
  5. 5Harder ratio problems, readunderstood
  6. 6Proportion as a formula, readunderstood
  7. 7Graphs of proportion, readunderstood

M4.2.6 Chapter 6

Proportion as a formula

If y is proportional to x, how do I write that as a formula?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Write y = kx for direct proportion and find k
  • Solve problems where y is proportional to x² or to 1/x

A tap fills 5 litres every minute. After 1 minute there are 5 litres. After 2 minutes there are 10. After 3 minutes there are 15. Each time, the litres are the minutes times 5.

Draw it first

Every y is x times the same number

  1. Three rows, each a box joined to a box by an arrow labelled × 5: x = 1 min to y = 5 litres, x = 2 min to y = 10 litres, and x = 3 min to y = 15 litres. Then the formula y = 5x.x = 1 min× 5y = 5 litres

    1 After 1 minute, y is 1 × 5 = 5.

  2. Three rows, each a box joined to a box by an arrow labelled × 5: x = 1 min to y = 5 litres, x = 2 min to y = 10 litres, and x = 3 min to y = 15 litres. Then the formula y = 5x.x = 1 min× 5y = 5 litresx = 2 min× 5y = 10 litres

    2 After 2 minutes, y is 2 × 5 = 10.

  3. Three rows, each a box joined to a box by an arrow labelled × 5: x = 1 min to y = 5 litres, x = 2 min to y = 10 litres, and x = 3 min to y = 15 litres. Then the formula y = 5x.x = 1 min× 5y = 5 litresx = 2 min× 5y = 10 litresx = 3 min× 5y = 15 litres

    3 After 3 minutes, y is 3 × 5 = 15.

  4. Three rows, each a box joined to a box by an arrow labelled × 5: x = 1 min to y = 5 litres, x = 2 min to y = 10 litres, and x = 3 min to y = 15 litres. Then the formula y = 5x.x = 1 min× 5y = 5 litresx = 2 min× 5y = 10 litresx = 3 min× 5y = 15 litresy = 5x

    4 The same number 5 multiplies every x. We write the rule as a formula: y = 5x.

We say y is directly proportional to x, or just proportional. It means y = kx, where k is a fixed number that never changes. The number k is the constant of proportionality. Here k = 5. It must not be 0.

To find k, take any pair of values and divide: k = y ÷ x. Then k stays the same for every pair.

A table with the head x, y, y ÷ x. Rows: 1, 5, 5; 2, 10, 5; 3, 15, 5.xyy ÷ x15521053155
The last column is always 5. That is how you know y is proportional to x.

Worked example

y is proportional to x. When x = 7, y = 35. Find the formula, and find y when x = 12.

  1. Write the formula y = kx.
  2. Put in x = 7 and y = 35: 35 = k × 7, so k = 35 ÷ 7 = 5.
  3. The formula is y = 5x.
  4. When x = 12, y = 5 × 12 = 60.

Answer: y = 5x. When x = 12, y = 60.

Some things grow faster than that. Take a right-angled triangle whose two shorter sides are both x cm. Its area is half of x times x. When x doubles, the area does not double. It becomes four times as big.

Two squares of tiles. The first has 2 tiles along each side and is labelled 4 tiles. The second has 4 tiles along each side and is labelled 16 tiles. The side is doubled, and the area is four times as big.224 tiles4416 tiles
Double the side and the area is 2 × 2 = 4 times as big.

When y is proportional to x², the formula is y = kx². For the triangle, y = 0.5x². Doubling x multiplies y by 2² = 4.

A table with the head x (cm), y (cm²). Rows: 2, 2; 4, 8; 6, 18.x (cm)y (cm²)2248618
The area y cm² of a right-angled triangle with two sides of x cm. Going from x = 2 to x = 4 multiplies y by 4. Going to x = 6 multiplies it by 9.
A right-angled triangle with its two shorter sides labelled 6 cm and 6 cm, and the label area 18 cm² under it. Beside it is a square of tiles, 6 tiles along each side, labelled 6, 6 and 36 tiles. The triangle is half of the square.6 cm6 cmarea 18 cm²6636 tiles
A right-angled triangle with two sides of 6 cm is half of a 6 cm by 6 cm square. The square holds 36 tiles of 1 cm², so the triangle has an area of 18 cm².

Worked example

A right-angled triangle has two equal shorter sides of x cm. Its area y cm² is proportional to x². When x = 6, y = 18. Find y when x = 10.

  1. Write the formula y = kx².
  2. Put in x = 6 and y = 18: 18 = k × 6², so k = 18 ÷ 36 = 0.5.
  3. The formula is y = 0.5x².
  4. When x = 10, y = 0.5 × 10² = 0.5 × 100 = 50.

Answer: y = 0.5x². When x = 10, y = 50.

The other kind goes the opposite way. A 120 km journey takes 3 hours at 40 km per hour. At twice the speed it takes half the time. The time y hours is 120 divided by the speed x.

We say y is inversely proportional to x, or proportional to 1/x. The formula is y = k ÷ x. Multiply x and y and you always get k. Here x is a speed, so it is above 0.

A table with the head km/h, Hours, Distance (km). Rows: 20, 6, 120; 40, 3, 120; 60, 2, 120; 80, 1.5, 120.km/hHoursDistance (km)206120403120602120801.5120
Speed is in km per hour. The last column is always 120 km. When the speed goes up, the time goes down.

Worked example

The time y hours for a journey is inversely proportional to the speed x km per hour. At 40 km per hour it takes 3 hours. How long does it take at 60 km per hour?

  1. Time is proportional to 1 ÷ speed, so y = k ÷ x.
  2. At 40 km per hour the time is 3 hours: 3 = k ÷ 40.
  3. Multiply both sides by 40: k = 3 × 40 = 120.
  4. The formula is y = 120 ÷ x. Check: 120 ÷ 40 = 3.
  5. At 60 km per hour: y = 120 ÷ 60 = 2.

Answer: 2 hours

See it three ways

With things

A tap fills 5 litres each minute. After 1 minute: 5 litres. After 2 minutes: 10. After 3 minutes: 15.

As a picture

A table with the head Minutes, Litres. Rows: 1, 5; 2, 10; 3, 15; 4, 20.MinutesLitres15210315420

In symbols

y = 5x.

Try it

Rectangles of the same area

You need: squared paper, a pencil.

  1. Draw a rectangle 1 square across and 12 squares up. It holds 12 squares.
  2. Draw more rectangles that also hold 12 squares: 2 across, 3 across, 4 across. Write how many squares go up each time.

What you will see: As the number across goes up, the number up goes down. Across times up is always 12. That is inverse proportion.

If you cannot try it, this is what happens.
A table with the head Across, Up, Squares. Rows: 1, 12, 12; 2, 6, 12; 3, 4, 12; 4, 3, 12; 6, 2, 12.AcrossUpSquares112122612341243126212

A wrong idea many people have

If y is proportional to x², then doubling x doubles y.

Why it feels right: Proportional usually means "when one doubles, the other doubles", and that is how direct proportion works.

The test that breaks it: Take y = 0.5x². When x = 2, y = 2. When x = 4, which is double, y = 8. That is four times as big, not double.

What is true: When y is proportional to x², doubling x multiplies y by 2² = 4. Tripling x multiplies y by 9.

A wrong idea many people have

Inverse proportion means y goes down by the same amount that x goes up.

Why it feels right: One goes up and the other goes down, so it feels as if they trade places evenly.

The test that breaks it: Take y = 24 ÷ x. When x = 4, y = 6. When x = 8, y = 3, which is a fall of 3. When x = 12, y = 2, which is a fall of only 1.

What is true: When x is multiplied by a number, y is divided by the same number. Then x times y stays the same.

Key words

directly proportional
y = kx for a fixed number k. Doubling x doubles y.
constant of proportionality
the fixed number k in y = kx or y = k ÷ x.
inversely proportional
y = k ÷ x for a fixed number k. Doubling x halves y.

Back to your guess

The answer is A: y = kx, where k is a fixed number.

Proportional means y is always the same multiple of x. Double x and y doubles. So y = kx, with k a fixed number. The formula y = k ÷ x is for inverse proportion, where doubling x halves y.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. y is proportional to x. When x = 4, y = 28. Which formula is right?

    Tap your answer.

    1. Not this one. This fits x = 4, y = 28, but adding is not proportion. When x doubles to 8 this gives 32, and y should double to 56. Go to M4.2.6. Try another.

    2. Not this one. You used the value of x as k. The number k is y ÷ x, which is 28 ÷ 4. Go to M4.2.6. Try another.

    3. Not this one. You multiplied 4 by 28. To find k you divide y by x. Go to M4.2.6. Try another.

    4. Yes, that is right. k = y ÷ x = 28 ÷ 4 = 7. So y = 7x.

    See the answer

    y = 7x

    k = y ÷ x = 28 ÷ 4 = 7. So y = 7x.

    • If you chose A: This fits x = 4, y = 28, but adding is not proportion. When x doubles to 8 this gives 32, and y should double to 56. Go to M4.2.6.
    • If you chose B: You used the value of x as k. The number k is y ÷ x, which is 28 ÷ 4. Go to M4.2.6.
    • If you chose C: You multiplied 4 by 28. To find k you divide y by x. Go to M4.2.6.

  2. y is proportional to x. When x = 4, y = 10. Find y when x = 10.

    Tap your answer.

    1. Yes, that is right. k = 10 ÷ 4 = 2.5. Then y = 2.5 × 10 = 25.

    2. Not this one. You added 6 to y because x went up by 6. Proportion multiplies: y is always the same number times x. Go to M4.2.6. Try another.

    3. Not this one. You multiplied 10 by 10. First find k = y ÷ x = 10 ÷ 4, then multiply k by the new x. Go to M4.2.6. Try another.

    See the answer

    25

    k = 10 ÷ 4 = 2.5. Then y = 2.5 × 10 = 25.

    • If you chose B: You added 6 to y because x went up by 6. Proportion multiplies: y is always the same number times x. Go to M4.2.6.
    • If you chose C: You multiplied 10 by 10. First find k = y ÷ x = 10 ÷ 4, then multiply k by the new x. Go to M4.2.6.

  3. y is proportional to x². When x = 2, y = 12. Find y when x = 4.

    Tap your answer.

    1. Not this one. You doubled y because x doubled. That is direct proportion with x. Here y is proportional to x², so doubling x makes y four times as big. Go to M4.2.6. Try another.

    2. Not this one. You multiplied 12 by 4². But y = 12 already includes the 2². Find k first: k = 12 ÷ 2², then use k × 4². Go to M4.2.6. Try another.

    3. Yes, that is right. k = 12 ÷ 2² = 3. Then y = 3 × 4² = 3 × 16 = 48.

    See the answer

    48

    k = 12 ÷ 2² = 3. Then y = 3 × 4² = 3 × 16 = 48.

    • If you chose A: You doubled y because x doubled. That is direct proportion with x. Here y is proportional to x², so doubling x makes y four times as big. Go to M4.2.6.
    • If you chose B: You multiplied 12 by 4². But y = 12 already includes the 2². Find k first: k = 12 ÷ 2², then use k × 4². Go to M4.2.6.

  4. y is inversely proportional to x. When x = 4, y = 6. Find y when x = 8.

    Tap your answer.

    1. Not this one. You doubled y because x doubled. In inverse proportion, doubling x halves y. Go to M4.2.6. Try another.

    2. Yes, that is right. k = x × y = 4 × 6 = 24. Then y = 24 ÷ 8 = 3.

    3. Not this one. You took 4 away from 6 because x went up by 4. Inverse proportion divides: x times y stays the same. Go to M4.2.6. Try another.

    See the answer

    3

    k = x × y = 4 × 6 = 24. Then y = 24 ÷ 8 = 3.

    • If you chose A: You doubled y because x doubled. In inverse proportion, doubling x halves y. Go to M4.2.6.
    • If you chose C: You took 4 away from 6 because x went up by 4. Inverse proportion divides: x times y stays the same. Go to M4.2.6.

  5. A tank fills at 12 litres a minute and is full after 5 minutes. The time is inversely proportional to the speed. How long does it take at 15 litres a minute?

    Tap your answer.

    1. Yes, that is right. The tank holds 12 × 5 = 60 litres. At 15 litres a minute it takes 60 ÷ 15 = 4 minutes.

    2. Not this one. You multiplied the time by 15 ÷ 12, as if it were direct proportion. A faster tap needs less time, not more. Go to M4.2.6. Try another.

    3. Not this one. You took 3 away from 5 because the tap is 3 litres faster. Inverse proportion divides: k = 12 × 5 = 60 litres, so the time is 60 ÷ 15. Go to M4.2.6. Try another.

    See the answer

    4 minutes

    The tank holds 12 × 5 = 60 litres. At 15 litres a minute it takes 60 ÷ 15 = 4 minutes.

    • If you chose B: You multiplied the time by 15 ÷ 12, as if it were direct proportion. A faster tap needs less time, not more. Go to M4.2.6.
    • If you chose C: You took 3 away from 5 because the tap is 3 litres faster. Inverse proportion divides: k = 12 × 5 = 60 litres, so the time is 60 ÷ 15. Go to M4.2.6.

  6. y is proportional to x². When x = 3, y = 36. Which formula is right?

    Tap your answer.

    1. Not this one. You divided y by x, not by x². That gives k for y = kx, which is direct proportion with x. Here y is proportional to x². Go to M4.2.6. Try another.

    2. Not this one. You used x² itself as k. The number k is y ÷ x², which is 36 ÷ 9. Go to M4.2.6. Try another.

    3. Yes, that is right. k = y ÷ x² = 36 ÷ 3² = 36 ÷ 9 = 4. So y = 4x².

    See the answer

    y = 4x²

    k = y ÷ x² = 36 ÷ 3² = 36 ÷ 9 = 4. So y = 4x².

    • If you chose A: You divided y by x, not by x². That gives k for y = kx, which is direct proportion with x. Here y is proportional to x². Go to M4.2.6.
    • If you chose B: You used x² itself as k. The number k is y ÷ x², which is 36 ÷ 9. Go to M4.2.6.

  7. From an earlier chapter

    The number of goats falls by 20% each year from 400. How many goats are there after 2 years?

    See the answer

    256

    The multiplier is 0.8. 400 × 0.8 = 320 after one year, and 320 × 0.8 = 256 after two.

  8. y is proportional to x. When x = 6, y = 18. Find y when x = 15.

    See the answer

    45

    k = 18 ÷ 6 = 3. Then y = 3 × 15 = 45.

  9. y is proportional to x². When x = 3, y = 27. Find y when x = 5.

    See the answer

    75

    k = 27 ÷ 3² = 27 ÷ 9 = 3. Then y = 3 × 5² = 3 × 25 = 75.

Keep it

Copy this into your exercise book.

Proportional to x: y = kx, and y ÷ x is always k. Proportional to x²: y = kx². Inversely proportional to x: y = k ÷ x, and x times y is always k.

Draw: Three small tables, one for each kind, with the column y ÷ x, y ÷ x² or x × y beside each.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (743 KB). It opens on any phone.