Year 11 Extended · Unit 3 Sequences, series, formulae and proportion · Lesson 7 of 7

What does a formula say about size, before any number is put in?

A formula can be turned round to make a different letter the subject, and it can be read for what happens to the answer when an input is multiplied. This lesson does both with the formulae of the unit.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: make x the subject of y = 3x − 7.
    Answerx = (y + 7) ÷ 3.
  2. From Year 10: y is directly proportional to x, and y = 12 when x = 4. Find y when x = 10.
    Answer30.
  3. From Year 9: work out 23 ÷ 4.
    Answer2.
Words used in this lesson
subject
the letter alone on one side of a formula. For example: the s in s = vt.
inverse operation
the operation that undoes another. For example: root undoes square.
varies as
changes with, in a fixed way. For example: y varies as x.
scale factor
the number by which a quantity is multiplied. For example: ×3.
constant
the number that stays the same in a relationship. For example: k.

If the lamp is twice as far away, how much light is left?

A student in Cape Town holds a page 50 cm from a small lamp and reads the light on it with a meter. She moves the page to twice as far, so 100 cm from the lamp. The light spreads over a larger area as the distance grows, and the amount on one square centimetre of page falls.

Predict first: will the light on the page be half as much, a quarter as much, or something else? Then use the figure to see what different powers do to a quantity when its input is multiplied.

1
2
×2y is multiplied by
2y in units (it was 1)

y varies directly as x: when x is 2 times as big, y becomes 2 units (it was 1 unit), a change of ×2.

Show me the steps
  1. y varies directly as x, x 2 times as big: y becomes 2 units and is multiplied by 2.
  2. y varies directly as x cubed, x 2 times as big: y becomes 8 units and is multiplied by 8.
  3. y varies inversely as x, x 4 times as big: y becomes 1/4 of a unit and is divided by 4.
  4. y varies as x to the power p: x multiplied by k multiplies y by k to the power p. Several inputs: multiply the factors.

Two skills, one idea: a formula says how its letters depend on each other. To make a different letter the subject, undo what has been done to it, in reverse order; a letter under a power is freed with a root.

To see how a quantity scales, multiply the input by a factor and see what comes out. If y varies as x to the power p, then x × k gives y × kp. With several inputs, multiply the factors: if y varies as x2 ÷ z, doubling x and tripling z multiplies y by 4 ÷ 3.

Two common slips. Taking the power to be 1 whatever the formula: doubling x in y = cx3 multiplies y by 8, not by 6. And combining changes by adding percentages instead of multiplying factors: a rise of 10% and a fall of 25% is 1.1 × 0.75 = 0.825, a fall of 17.5%, not a fall of 15%.

Worked examples

1. Make d the subject of S = n⁄2 × (2a + (n − 1)d), and check with S = 210, n = 7, a = 15.

  1. Multiply by 2 and divide by n: 2S ÷ n = 2a + (n − 1)d. Subtract 2a: 2S ÷ n − 2a = (n − 1)d.
  2. d = (2S ÷ n − 2a) ÷ (n − 1). Check: (60 − 30) ÷ 6 = 5.

2. Make a the subject of S = a(rn − 1) ÷ (r − 1), and find a when S = 121, r = 3, n = 5.

  1. Multiply by (r − 1): S(r − 1) = a(rn − 1). Divide by (rn − 1): a = S(r − 1) ÷ (rn − 1).
  2. a = 121 × 2 ÷ 242 = 1.

3. F = k m1 m2 ÷ d2. The distance is doubled and m1 is tripled. What happens to F?

  1. Replace d by 2d and m1 by 3m1: F becomes k × 3m1 m2 ÷ (2d)2.
  2. That is 3 ÷ 4 of the original, so F is multiplied by 3⁄4.

Your go. The first step is given. The volume of a cone is V = (1⁄3)πr2h. The radius is tripled and the height is halved. By what factor is the volume multiplied? The radius factor is 32 = 9.

  1. The radius multiplies V by 32 = 9.
  2. The height multiplies V by 1⁄2. Combine the two factors.
Check the answer

9 × 1⁄2 = 4.5, so the volume is multiplied by 4.5.

Check before you go on

1. y is inversely proportional to x2. Then x is multiplied by 5. What happens to y?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The formula v2 = u2 + 2as is used with v = 13, u = 5 and a = 4. Find s.

3. An arithmetic sequence has first term 2 and 10 terms, and the sum of the terms is 155. Use d = (2S ÷ n − 2a) ÷ (n − 1) to find the common difference.

Level 2 · Building

4. P varies as the square root of L. When L = 9, P = 12. Find P when L = 25.

5. z varies as x2 ÷ y. The value of x is doubled and the value of y is multiplied by 8. By what factor is z multiplied?

Level 3 · Stretch

6. The volume of a cylinder varies as r2h. The radius increases by 50% and the height decreases by 20%. By what percentage does the volume increase?

7. Make a the subject of S = n ÷ 2 × (2a + (n − 1)d). Then find a when S = 286, n = 11 and d = 4.

Something to think about

The formula for the sum to infinity, S = a ÷ (1 − r), can be rearranged to r = 1 − a ÷ S. What does this say about r when S is only a little more than a, and when S is very large?

One way to think about it

When S is only a little more than a, a ÷ S is close to 1, so r is close to 0: the terms shrink at once. When S is very large, a ÷ S is close to 0, so r is close to 1: the terms shrink very slowly, so a lot is added before they become small.

Exit ticket

8. y = kx3 ÷ z. The value of x is tripled. By what must z be multiplied so that y does not change?