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KS4 Mathematics

KS4 Mathematics. A3 Formulae. A3 Formulae. Contents. A. A3.2 Problems that lead to equations to solve. A. A3.1 Substituting into formulae. A3.3 Changing the subject of a formula. A. A3.4 Manipulating more difficult formulae. A. A3.5 Generating formulae. A. Formulae.

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KS4 Mathematics

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  1. KS4 Mathematics A3 Formulae

  2. A3 Formulae Contents • A A3.2 Problems that lead to equations to solve • A A3.1 Substituting into formulae A3.3 Changing the subject of a formula • A A3.4 Manipulating more difficult formulae • A A3.5 Generating formulae • A

  3. Formulae A formula is a special type of equation that links two or more physical variables. For example in the formula, P =2(l + w) P represents the perimeter of a rectangle and l and w represent its length and width. We can use this formula to work out the perimeter of any rectangle given its length and width. We do this by substituting the values we are given into the formula.

  4. Formulae d S = t Because formulae deal mainly with real-life quantities such as length, mass, temperature or time, the given variables often have units attached. Units shouldn’t be included in the formula itself. The units that have to be used are usually defined in the formula. For example, This formula doesn’t mean much unless we say “S is the average speed in m/s, d is the distance travelled in metres, and t is the time taken in seconds”.

  5. Formulae d S = t 2000 S = 100 Use the formula to find the speed of a car that travels 2 km in 1 minute and 40 seconds. Write the distance and the time using the correct units before substituting them into the formula, 2 kilometres = 2000 metres 1 minute and 40 seconds = 100 seconds Now substitute these numerical values into the formula, We can write the units at the end. = 20 m/s

  6. Substituting into formulae h w l The surface area S of a cuboid is given by the formula S = 2lw + 2lh + 2hw where l is the length, w is the width and h is the height. What is the surface area of a cuboid with a length of 1.5 m, a width of 32 cm and a height of 250 mm?

  7. Substituting into formulae What is the surface area of a cuboid with a length of 1.5 m, a width of 32 cm and a height of 250 mm? Before we can use the formula we must write all of the amounts using the same units. l=150 cm, w = 32 cm and h = 25 cm Next, substitute the values into the formula without the units. S = 2lw + 2lh + 2hw = (2 × 150 × 32) + (2 × 150 × 25) + (2 × 25 × 32) = 9600 + 7500 + 1600 = 18,700 cm2 Don’t forget to write the units in at the end.

  8. Substituting into formulae The distance d, in metres, that an object falls after being dropped is given by the formula, where t is the time in seconds. d = 4.9t2 Suppose a boy drops a rock from a 100 metre high cliff. How far will the rock have fallen after: a) 2 seconds b) 3 seconds c) 5 seconds? When t = 2, When t = 3, When t = 5, 4.9 × 22 4.9 × 32 4.9 × 52 d = d = d = = 4.9 × 4 = 4.9 × 9 = 4.9 × 25 = 19.6 metres = 44.1 metres = 122.5 metres

  9. Substituting into formulae

  10. A3 Formulae Contents A3.1 Substituting into formulae • A • A A3.2 Problems that lead to equations to solve A3.3 Changing the subject of a formula • A A3.4 Manipulating more difficult formulae • A A3.5 Generating formulae • A

  11. Problems that lead to equations to solve Formulae are usually (but not always) arranged so that a single variable is written on the left-hand side of the equals sign. For example, in the formula v = u + at v is called the subject of the formula. If we are given the values of u, a and t, we can find v by substituting these values into the formula. Suppose instead that we are given to values of v, u and a, and asked to find t. When these values are substituted we are left with an equation to solve in t.

  12. Problems that lead to equations to solve For example, suppose v = 20, u = 5 and a = 3. Find t. Substituting these values into v = u + at gives us the equation, 20 = 5 + 3t We can then solve this equation as usual. 5 + 3t = 20 swap both sides: 3t = 15 subtract 5 from both sides: t = 5 seconds divide both sides by 3:

  13. Problems that lead to equations to solve 1 A = (a + b)h 2 a h b The formula used to find the area A of a trapezium with parallel sides a and b and height h is: What is the height of a trapezium with an area of 40 cm2 and parallel sides of length 7 cm and 9 cm?

  14. Problems that lead to equations to solve 1 1 40 = ×16h 40 = (7 + 9)h 2 2 1 A = (a + b)h 2 Substituting A = 40,a = 7 and b = 9 into gives Simplifying, 40 = 8h 8h= 40 swap both sides: h =5 divide by 8: So the height of the trapezium is 5 cm.

  15. Problems that lead to equations to solve

  16. A3 Formulae Contents A3.1 Substituting into formulae • A A3.2 Problems that lead to equations to solve • A A3.3 Changing the subject of a formula • A A3.4 Manipulating more difficult formulae • A A3.5 Generating formulae • A

  17. The subject of a formula Here is a formula you may know from physics: V = IR where V is voltage, I is current and R is resistance. V is called the subject of the formula. The subject of a formula always appears in front of the equals sign without any other numbers or operations. Sometimes it is useful to rearrange a formula so that one of the other variables is the subject of the formula. Suppose, for example, that we want to make I the subject of the formula V = IR.

  18. Changing the subject of the formula V = IR ×R ÷R V I = R V is the subject of this formula The formula: can be written as a function diagram: I V The inverse of this is: I V I is now the subject of this formula So:

  19. Matchstick pattern Look at this pattern made from matchsticks: Pattern Number, n 1 2 3 4 Number of Matches, m 3 5 7 9 The formula for the number of matches, m, in pattern number n is given by the formula: m = 2n + 1 Which pattern number will contain 47 matches?

  20. Changing the subject of the formula × 2 + 1 ÷ 2 – 1 m – 1 n = 2 m is the subject of this formula The formula: m = 2n + 1 can be written as a function diagram: n m The inverse of this is: n m n is the subject of this formula or

  21. Changing the subject of the formula 47– 1 n = 2 46 n = 2 m– 1 n = 2 To find out which pattern will contain 47 matches, substitute 47 into the rearranged formula. n = 23 So, the 23rd pattern will contain 47 matches.

  22. Changing the subject of the formula F = + 32 subtract 32: 9C 9C 5(F – 32) = 9C multiply by 5: 5(F – 32) 5(F – 32) 5 5 F – 32 = 9 9 divide by 9: = C C = We can also change the subject by performing the same operations on both sides of the equals sign. For example, to make C the subject of

  23. Change the subject of the formula 1

  24. Find the equivalent formulae

  25. A3 Formulae Contents A3.1 Substituting into formulae • A A3.2 Problems that lead to equations to solve • A A3.4 Manipulating more difficult formulae A3.3 Changing the subject of a formula • A • A A3.5 Generating formulae • A

  26. Formulae where the subject appears twice Sometimes the variable that we are making the subject of a formula appears twice. For example, S = 2lw + 2lh + 2hw where S is the surface area of a cuboid, l is its length, w is its width and h is its height. Make w the subject of the formula. To do this we must collect all terms containing w on the same side of the equals sign. We can then isolate w by factorizing.

  27. Formulae where the subject appears twice 2lw + 2hw = S – 2lh subtract 2lh from both sides: w(2l + 2h)= S – 2lh factorize: S – 2lh w = 2l + 2h S = 2lw + 2lh + 2hw Let’s start by swapping the left-hand side and the right-hand side so that the terms with w’s are on the left. 2lw + 2lh + 2hw = S divide by 2l + 2h:

  28. Formulae involving fractions For example, if two resistors with a resistance a and b ohms respectively, are arranged in parallel their total resistance R ohms can be found using the formula, = + 1 1 1 R a b aΩ bΩ When a formula involves fractions we usually remove these by multiplying before changing the subject. Make R the subject of the formula

  29. Formulae involving fractions multiply through by Rab: ab = Rb + Ra simplify: = + 1 1 1 R ab = R(b + a) a b factorize: ab ab Rab Rab Rab = R divide both sides by a + b: = + a + b a + b R a b R =

  30. Formulae involving powers and roots c = √a2 + b2 c2 = a2 + b2 square both sides: c2 – b2 = a2 subtract b2 from both sides: √c2 – b2 = a square root both sides: a = √c2 – b2 The length c of the hypotenuse of a right-angled triangle is given by where a and b are the lengths of the shorter sides. Make a the subject of the formula

  31. Formulae involving powers and roots T = 2π l g The time T needed for a pendulum to make a complete swing is where l is the length of the pendulum and g is acceleration due to gravity. Make l the subject of the formula When the variable that we wish to make the subject appears under a square root sign, we should isolate it on one side of the equation and then square both sides.

  32. Formulae involving powers and roots T = 2π l l l T divide both sides by 2π: = g g g 2π T2 = square both sides: 4π2 T2g T2g = l multiply both sides by g: 4π2 4π2 l =

  33. Change the subject of the formula 2

  34. A3 Formulae Contents A3.1 Substituting into formulae • A A3.2 Problems that lead to equations to solve • A A3.5 Generating formulae A3.3 Changing the subject of a formula • A A3.4 Manipulating more difficult formulae • A • A

  35. Writing formulae m c = p Write a formula to work out, 1) the cost, c, of b boxes of crisps at £3 each c = 3b 2) the distance left, d, of a 500 km journey after travelling k km d = 500–k 3) the cost per person, c, if a meal costing m pounds is shared between p people

  36. Writing formulae a + b + c w = 3 4) the number of seats in a theatre, n, with 25 seats in each row,r n = 25r 5) the age of a boy Andy, a, if he is 5 years older than his sister Betty, b a = b + 5 6) the average weight, w, of Alex who weighs a kg, Bob who weighs b kg and Claire who weighs c kg.

  37. Writing formulae A window cleaner charges a £10 call-out fee plus £7 for every window that he cleans. Write a formula to find the total cost C when n windows are cleaned. C = 7n + 10 Using this formula, how much would it cost to clean all 105 windows of Formula Mansion? We substitute the value into the formula, C = 7 ×105 + 10 = 735 + 10 = 745 It will cost £745.

  38. Writing formulae 7n + 10 = 94 swap both sides: 7n = 84 subtract 10 from both sides: C = 7n + 10 At another house the window cleaner made £94. How many windows did he have to clean? We substitute this value into the formula to give an equation, 94 = 7n + 10 n = 12 He cleaned 12 windows.

  39. Using formulae to write mathematical rules 34 35 44 45 When conducting a mathematical investigation, it is usually necessary to use formulae to write rules and generalizations. For example, Sophie is investigating patterns of shaded squares on a numbered grid. She starts by looking at arrangements of numbers in two by two squares on a 100 square grid. For example, She notices that the sum of the numbers in a two by two square is always equal to four times the number in the top left-hand square plus 22. Sophie writes this as a formula.

  40. Number grid patterns

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