Year 9 End of Year Revision - Percentages, Compound Measures, Graphs

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In this revision material for Year 9, you will cover topics such as percentages, compound measures, and graphs. Practice calculating percentage change, compound interest, measure conversions, graph equations, and more to prepare for your end-of-year exams.

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Dr Frost Year 9 End of Year Revision

Percentages Be careful: Are you trying to find the new value or the old value? In the first case, you multiply, in the second case, you divide. Percentage change is based on the old value. A jumper is bought in for 30 and marked up by 40%. What is it sold for? I put 15,000 into a savings account. It accrues 2.6% interest. What is in my account in one year s time? Answer: 30 x 1.4 = 42 ? Answer: 15390 ? After one year the value of a care fell by 20% to 9600. What was its original value? Lucy made 20% profit on the picture frame she sold at 35. What did she buy it in for? Answer: 12000 ? ? Answer: 29.17

Percentages The interest rate for a savings account is 2.5% p.a. with compound interest. The principal is 1500. How much do I have in 10 years time? My Bentley depreciates in value 10% each year. It is bought new for 150,000. How much is it worth in 5 years time? ? Answer: 150,000 x 0.95 = 88573.50 Answer: 1500 x 1.02510 = 1920.13 ?

Compound Measures A cat travels at 15km/s. It races around a 50km track. How much time did it take him? Answer = 3.33s ? The density of a hamster is 1.3kg/m3. Its volume is 0.03m3. What is the hamster s mass? Answer = 0.039kg ?

Graphs Match the graphs with the equations, and identify what type of equation it is. 1 5 9 ? ? ? ? ? ? ? ? ? ? y = -2x3 + x2 + 6x y = 4x y = 2x - 3 y = x2 + x 2 y = 5 2x2 6 Cubic 11 Exponential 9 Straight Line 1 Quadratic 2 Quadratic 6 10 2 8 Reciprocal y = 2x3 y = 5 x y = x3 7x + 6 y = -3x3 5 Cubic 10 Straight Line 3 Cubic 4 Cubic 3 7 11 ? 8 7 Reciprocal 4

Graphs y = x3 2x2 - 5x + 6 x y -3 -24 0 ? -2 -1 8 0 6 1 0 2 -4 3 0 4 18 ? ? ? ? ? ? ? When sketching, ensure you sketch a curvy line (i.e. don t join up your points with lines), or you ll lose a mark.

Changing the Subject Change the subject of the formula to the indicated letter. ? ? ? ? (b) ? ? ? ? ? ?

Changing the Subject ? ? ? ? ? ? ? ? ? ?

Changing the Subject The following require you to factorise at some point. Make a the subject of the formula: a = n2-Pn n = _3a_ ? ? P-1 a+1 ? ?

Simultaneous Equations You can either use elimination or substitution. 3x + 2y = 10 5x 2y = 14 3x + 2y = 4 4x + 3y = 7 ? ? x = 3, y = 0.5 x = -2, y = 5

Probability Question: Give there s 5 red balls and 2 blue balls. What s the probability that after removing two balls from the bag, we have a red ball and a blue ball? 4 6 ? R 5 7 ? R B 2 6 ? 5 6 R ? 2 7 B ? B 1 6 ? Answer = 10 21?

Probability Matching outcomes What s the probability that when I roll 10 dice, I see the same number on every die? ? ? ?? ? ? ???? ?????? = ???= Total outcomes ? What s the probability that when I roll 10 dice, the total of the dice is 10? ? ? ??? ?? ?? = ??? Difficult:What s the probability that when I roll 3 dice, I see exactly two sixes. ? ??? ?? =?? ? ?? ??= ?

Probability If I have two dice, one numbered 1, 2, 3 and the other numbered 2, 3, 4, what s the probability the sum is at least 5? Second Die + 2 3 4 First Die 1 3 4 5 ? 2 4 5 6 3 5 6 7 6 2 9 3 p(sum 5) = = ?

Sequences Determine the formula for the following sequences. 5, 8, 11, 14, 17, ... 10, 8, 6, 4, 2, 0, -2, ... Un = 3n + 2 Un = 12 2n ? ? 3, 9, 17, 27, 39, ... 1, 3, 6, 10, 15, ... Un = n2 + 3n - 1 Un = 0.5n(n+1) ? ?

Expanding brackets Expand the following. (x+1)(x-2) = x2 x 2 (x-4)(x-8) = x2 12x + 32 (x+1)(y+1) = xy + x + y + 1 (x2+1)(y2-1) = x2y2 x2 + y2 1 (2x+1)(2x-1) = 4x2 1 (x + 1)(x + y + 1) = x2 + xy + 2x + y + 1 x(y-x)-y(x-y) = y2 x2 ? ? ? ? This is known as the: difference of two squares ? ? ? ?

Factorisation Factorise the following x2 + 7x + 12 x2 + 2x 3 x2 10x + 24 = (x 4)(x 6) 2x2 5x 12 12x2 + 5x 3 x2 9 4 y2 x3 x 16x2y2 9z4 x4 + 2x2 143 = (x + 4)(x + 3) = (x 1)(x + 3) ? ? ? ? ? ? ? ? = (2x + 3)(x 4) = (4x + 3)(3x 1) = (x + 3)(x 3) = (2 + y)(2 y) = x(x + 1)(x 1) = (4xy + 3z2)(4xy 3z2) = (x2 + 13)(x2 11) ? ?

y 4 Enlargement 3 2 Image 1 x -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Object -1 -2 -3 Enlarge the shape by a scale factor of 2 about the point (0,-2) -4

y 4 Enlargement 3 2 1 x -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Object -1 -2 -3 Enlarge the shape by a scale factor of -1 about the point (0,2) -4

y 4 Enlargement 3 2 1 x -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Object -1 -2 -3 Enlarge the shape by a scale factor of -0.5 about the point (0,2) -4

Trigonometry 4 x 60 30 12 x x = 13.96 ? ? x = 3.46 65 x x = 6.99 ? 15

Trigonometry ? = 45 b c 1 1 1 a 2 3 d 6 3 8 = 33.69 ? = 70.53 ? = 48.59 ?

Trigonometry What is the cosine of the angle between the internal diagonal of a cube and the bottom face of the cube? Answer = 2 3?

Solving Equations Solve the following equations for x. ? ? ? ? ? ? ? x(2x + 1)(x 2) = 0 x2 = 4 x2 = 3x x2 + 5x 6 = 0 x3 = x x2 + 32 = 12x 25x2 4 = 0 x = 0 or -0.5 or 2 x = 2 or -2 x = 0 or 3 x = -6 or 1 x = -1, 0 or 1 x = 4 or 8 x = 2/5 or -2/5

Solving Equations Determine x Answer: By Pythagoras, x2 + (2x+2)2 = (3x-2)2 Expanding and simplifying, we get 4x2 20x = 0 Solving, x = 5 (we reject the 0 solution). 3x - 2 x ? 2x + 2

Similarity 3 x = 16? 5 x 10

Similarity A square is inscribed in a right-angled triangle with length 4 and height 3. Find the width of the square. Length of square = 12 7? 3 4

Loci A Spotted Studdert Sheep is known to be within 3km of A and 4km of B. What region could the sheep be in? 3km 4km A B

Loci Now the sheep is also known to be of equal distance from A and B. Where can it be? 3km 4km A B

Loci Now the sheep is within 3km of A, but at least 4km away from B. Where could it be? 3km 4km A B

Loci I m equidistance from two lines AB and AC. Where could I be? B A C

Loci I m the indicated distance away from the walls of a building. Where could I be? Circular corners. Straight corners.

Loci A My sheep is attached to a fixed point A on a square building, of 10m x 10m, by a piece of rope 20m in length. Both the sheep and rope are fire resistant. What region can he reach? 20m 10m

Dimensional Analysis Click your choice. (all variables are lengths) Expression Length Area Volume None of these ? ? ? ? ? ? 2rh ? ? r + 4h (r+h)2 3b ? ? ? ? ? ? ? ? ? ? ? b3 + rh ? r2(h + r) bhr_ (b+h) ? ? ? ?

Ratio Method 2: 3 4 of the tank is black and 1 leaves us with 2 2 3=1 Now the 1 3 of the tank wiped out is replenished with black fish, so that s 5 fish). This ratio is 5:1. Method 1: Suppose a full tank has 12 fish. Then 9 fish are black and 3 yellow. The plague leaves 6 black fish and 2 white. Then if we fill up the rest of the tank with black fish, we have 10 black fish and 2 yellow. This ratio is 5:1. 4 is yellow. The plague 3 of the fish, so the tank is now 3 2 full of black fish and 1 4 4 2 3=1 6 of yellow fish. 6 black fish (and still 1 6 yellow My fish tank has black and yellow fish in the ratio 3:1. A fish plague, Sanjotitus, wipes out a third of my fish. I then restock my fish tank with just black fish, so that I have the same number of fish as before. What s the new ratio of black to yellow fish? Answer = 5 : 1 ?

Proportion Given that y is proportional to x, find the missing values. ? x y 16 10 8 5 24 15 ?

Inverse proportion Given that L is inversely proportional to x, fill in the missing values in this table. ? x L 16 5 50 2.83 0.25 40 ?

Inequalities Solve the following. 2x > x - 6 x > - 6 ? -x + 1 6 x -5 ? -1 x < 3 ? 1 2x + 3 < 9 ? ? 2 ? 2 1

Inequalities on a number line. 2 x < 4 x < -1 or x > 4 0 1 2 3 4 5 -1 0 1 2 3 4 ? ?

Inequalities on a number line. Draw the range of x on the number line given that both of these inequalities hold. 2 x < 5 x < 3 or x > 4 0 1 2 3 4 5 ?

8 -4 < y -2 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6

8 y x + 1 and x 6 and y > 2 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6

Inequalities When would ?? ? ?? When all of x, y and z are negative, or one of x, y and z are negative. ?

Arcs and Sectors ? Sector area = 10.91 Area = 20 50 ? Arc length = 4.36 5 135 (Hint: Plug values into your formula and rearrange) ? Sector area = 4.04cm2 ? 105 Radius = 4.12 ? Arc length = 3.85cm 2.1cm

Arcs and Sectors Q The shape PQR is a minor sector. The area of a sector is 100cm2. The length of the arc QR is 20cm. a) Determine the length PQ. P ? Answer: 10cm R b) Determine the angle QPR Answer: 114.6 ?