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Percentages Examples Percent means “out of one hundred”. 1%= In a survey of 160 people, 75% said they had a computer at home. How many people was this? 1 100 Some percentages can easily be turned into fractions: 5% = 25% = 1 20 1 4 10% = 331/3% = 1 10 1 3 20% = 50% = 75% = 662/3% = 80% = 3 10 2 3 4 5 40% = 70% = 90% = 2 5 7 10 9 10 60% = 75% = 3 4 of 160 = (160 4) x 3 = 120 people. 1 5 1 2 And we can also say that: 30% = 3 4 3 5 3 4 100% = 1 Jim earns £24000 per year and gets a 4% pay rise. How much more money is this? 4% of £24000 = 4 100 x 24000 = £960 more Write 18 out of 20 as a percentage. 18 20 x 100 = 90% Fractions Examples 8 are shaded 12 Fractions are made when objects or amounts are split up evenly. Fractions are made up of two parts: 2 the NUMERATOR (how many) 3 the DENOMINATOR (the type) Fractions which describe the same amount using different numerators and denominators are called EQUIVALENT. To find a fraction of an amount: 1. Divide by the denominator 2. Multiply the answer by the numerator 4 12 8 12 4 4 = 2 3 20 50 10 10 = 2 5 aren’t shaded Divide top and bottom to simplify equivalent fractions. Of the 900 people at a concert, 3 4 aged 16 or under. How many is this? 900 4 = 225 225 x 3 = 675 So 675 people are aged 16 or under. are Estimating Examples Height and length mm, cm, m, km 10 cm 15 mm Area – covering of a surface mm2, cm2, m2, hectares The area of a football pitch is 7500 m2 Volume – measurement of the shape enclosed by a shape mm3, cm3, m3 , ml or litres Weight grams and kg Estimate the volume of liquid in the flask. Answer: 100ml Proportion Example If 5 bananas cost 80 If you know the cost of several items, you pence, then what do 3 can easily find the cost of one item. bananas cost? The cost of one item is usually called the “cost per item”. Record appropriate headings with the unknown on the right. Find the value per item first then multiply to find the required value If rounding is required, do not round until the last stage. Bananas 5 Cost 80 80 5 1 3 3 x 16 16 48 Divide to find the cost of one Multiply to find the cost of the required amount 3 bananas cost 48 pence Bar Graphs Examples Bar graphs are used to compare values or amounts. Use correct scales 10 Number of Pupils Use a pencil and a ruler Shoe sizes in class 1S6 8 6 4 2 0 3.5 4 4.5 5 5.5 6 Shoe Size Label the axes Shoe sizes in one class Shoe sizes in classes 1S6 and 2S6 Each bar is in the middle of its label 10 Number of Pupils Give the graph a title 8 6 1S6 4 2S6 2 0 3.5 4 4.5 5 5.5 6 Shoe Size Shoe sizes in two classes Line Graphs Examples Line graphs are often used to show trends (how amounts change over time). Use correct scales 25 Temperature (C) Use a pencil and a ruler Temperature over 1 week 20 15 10 5 0 Mon Tues Wed Thurs Fri Sat Sun Day Label the axes Temperature in July Temperature over 1 week Give the graph a title Join each point Temperature (C) Plot each point carefully 25 20 15 July 10 January 5 0 -5 Mon Tues Wed Thurs Fri Sat Sun Day Temperatures in January & July Pie Charts Examples Pie charts are circular graphs which are used to compare information given in percentages. Transport to School Bus Use a pencil, ruler, compasses and protractor Measure each slice accurately Label the slices Give the graph a title Walk Car Bike If 80 pupils were surveyed: Bus = 90 = 25% 25% of 80 = 80 4 = 20 pupils 48 walk to school /80 of 360 = (360 80) x 48 = 216 48 Coordinates The position of an objector point can be described by using a coordinate grid system. Examples 6 5 A 4 3 Use a pencil and a ruler Use correct scales Label the axes Plot each point carefully Write using brackets and a comma, e.g. (1,2) 2 1 -6 -5 -4 C -3 -2 0 -1 -1 0 B 1 2 3 4 5 6 -2 -3 -4 First number is the x-axis (along/back) Second number is the y-axis (up/down) A = (2,4) B = (0,2) C = (-4,-2) Subtraction We can subtract numbers using a number of different methods: 41 – 27 = ? 27 Counting on Breaking up the number being subtracted 30 40 41 +10 +1 =14 +3 41 – 20 = 21 41 – 27 = 21 – 7 = 14 3 Decomposition Using a number line - -4 -3 -2 -1 4 2 1 1 0 1 1 7 4 2 3 – 5 = -2 3 4 5 Equations Examples An equation shows you that two things are Solve: a) x + 5 = 9 equal, for example b) 2y = 10 2+3=5 c) 3z – 2 = 7 5x – 13 = 22 a) x+5 =9 are both equations. Change SIDE, Terms in equations that we don’t know are given a letter instead of a number (e.g. x). b) Finding the value of the unknown is called solving the equation. c) x =9–5 x =4 change SIGN! 2y = 10 Y = 10 2 Y =5 Divide by the number of “y”s! 3z – 2 3z 3z z z = = = = = 7 7+2 9 9 3 3 Change SIDE, change SIGN! Divide by the number of “z”s! Using Formulae Examples A formula is a set of instructions which tell you how to carry out a calculation A bus travels a distance of 250km in 4 hours. What was the average speed of the journey? Formulae can be written using words or shortened to just letters: Speed = Distance Time S=DT Speed = Distance Time = 250 4 = 62.5 Average speed = 62.5 km/hour Convert 100C into F by using the formula When using formulae: Temp(F) = (Temp(C) x 1.8) + 32 always show your working be careful to put the correct Temp(F) = (100 x 1.8) + 32 = 180 + 32 number in the correct place = 212 answer the question with a statement and use the correct units 100C = 212F Time Calculations Examples 1 minute = 60 seconds 1 hour = 60 minutes 1 day = 24 hours 1 week = 7 days 1 year = 365 days (366 in a leap year) 1 year = 52 weeks a.m. before midday p.m. after midday 1200 = midday, 0000 = midnight Change 2.37 p.m. to 24 hour time After midday, so add 12 to the hours 2.37 p.m. = 1437 How many minutes in 4½ hours? 4.5 x 60 = 270 minutes How long is it from 0755 until 0948? 0755 0800 5 mins Change hours to minutes: x by 60 Change minutes to hours: by 60 0900 1 hour 0948 48 mins Total = 1 hour 52 minutes Ratio Examples Ratios are used to compare the amounts of different quantities Ratios can be simplified like fractions We can carry out ratio calculations by using a table Use appropriate headings Ratio of: fish : sharks = 2:3 sharks : fish = 3:2 As 6 8 = 3 4 , we can simplify 6:8 to 3:4 In a cat and dog home, the ratio of cats to dogs is 4:7. If there are 35 dogs, how many cats are there? Dogs Cats 7 4 Put the unknown on the right x5 35 20 Since 35 = 7 x 5, Cats = 4 x 5 = 20 Rounding We round numbers when we only need a rough idea of an amount Examples A bag of flour weighs 1483g. What is this to the nearest a) 100g b) 10g? We look at the number to the RIGHT of Answers: the one we want. To round to: 100’s, look at 10’s (1874 1900) 10’s, look at units (1874 1870) units, look at 1 d.p. (3.196 3) 1 d.p., look at 2 d.p. (3.196 3.2) 2 d.p., look at 3 d.p. (3.196 3.20) a) 1500g (8 = round UP) b) 1480g (3 = stays the same) In the high jump, Jill clears 2.471m. Round this to a) 1 d.p. b) 2 d.p. a) Look at the 2nd d.p. 2.471: 7 round UP Answer: 2.5m If the number to the right is 0, 1, 2, 3, or 4: we leave our number as it is 5, 6, 7, 8, or 9: we round our number UP by 1 b) Look at the 3rd d.p. 2.471: 1 stays the same Answer: 2.47m Symmetry Examples Line symmetry a line of symmetry through a shape means one half is an exact mirror image of the other half 4 lines Turn (or rotational) symmetry if we can spin a shape and it fits back inside its outline before we do a full turn, the shape has turn symmetry /2 - turn 1 Translation (sliding) symmetry if we can cover an area with exact copies of a shape, without leaving any gaps, then the shape has translation symmetry 2 lines 1 line /4 - turn 1 /8 - turn 1 Standard Form Examples Standard Form is a useful way of writing very Saturn is 884 million large or very small numbers as : miles from the Sun. a number between 1 and 10 Write this distance in standard form. multiplied by a power of 10 884 million = 884000000 For example, = 8.84 x 100000000 75000 = 7.5 x 10000 = 7.5 x 104 = 8.84 x 108 miles 0.000021 = 2.1 x 0.00001 = 2.1 x 10-5 A proton has a diameter of 1.5 x 10-13 The power of 10 (or index) shows how many cm. Write this number places the decimal point has moved: to the LEFT when positive (BIG numbers) in the normal way. to the RIGHT when negative (SMALL numbers) 1.5 x 10-13 = 0.00000000000015cm Significant Figures Examples Like rounding, significant figures are a way to give a rough estimate of an amount. As we increase the significant figures, the measurement becomes more accurate. To round whole numbers: count in the number of sig figs from the LEFT. Round this number up or down change the remaining numbers to zero To round decimals: start with the first non-zero number count in the number of sig figs from the LEFT. round this number up or down miss out any numbers left over There were 36915 people at the 1995 Scottish Cup Final. This is: 40000 to 1 s.f. 37000 to 2 s.f. 36900 to 3 s.f. 36920 to 4 s.f. Decimals 63.5 has 3 s.f. 6.35 has 3 s.f. 6.305 has 4 s.f. 6.350 has 4 s.f. 0.592 has 3 s.f. 0.059 has 2 s.f. 0.0590 has 3 s.f. 0.0059 has 2 s.f.