T Madas T Madas Two numbers whose difference

Two numbers, whose difference is 18, add up to 44. What are the two numbers? 1 2 x – y = 18 x + y = 44 2 x x = 62 = 31 Sub x = 31 into either of the original equations 2 x + y = 44 31 + y = 44 y = 13 Check if the worded problem is satisfied © T Madas

A family of 2 adults and 3 children paid £ 12 for their cinema tickets. Another group of 3 adults and 5 children paid £ 19 for their tickets, in the same cinema. How much does each adult ticket cost? Since we are only required for the value of a, we will eliminate c 1 2 a + 3 c = 12 x(-3) 2 3 a + 5 c = 19 x 5 1 2 10 a + 15 c = 60 -9 a – 15 c = -57 a = 3 Adult tickets cost £ 3 each © T Madas

Coins © T Madas

I have 23 coins in my pocket. Some of them are 5 p coins and the rest are 10 p coins. In total I have £ 2. 05. How many of each type of coin do I have? Let x be: No of 5 p Let y be: No of 10 p x(-5) 1 x + y = 23 2 5 x + 10 y = 205 1 -5 x – 5 y = -115 2 5 x + 10 y = 205 5 y = 90 18 Sub y = 18 into either of the original equations 1 x + y = 23 x + 18 = 23 x =5 I have: • 18 ten pence pieces • 5 five pence pieces © T Madas

Leaving Present © T Madas

Teachers in the Maths Department want to buy a present for one of their colleagues who is leaving. If they contribute £ 10 each, they have a surplus of £ 10 If they contribute £ 5 each, they have a shortage of £ 45 How many teachers are contributing and what is the cost of the present they want to buy? Let x be the number of teachers Let c be the cost of the present 5 x = 55 11 100 = c c subtract 2 55 = c – 45 c 1 10 x = c + 10 2 5 x = c – 45 11 teachers were buying a present for £ 100 © T Madas

Cab Firms © T Madas

Tex Taxis Tariff: a £ 2 flat fee plus 10 pence per 100 metres Caleb Cabs Charges: a £ 5 flat fee plus 5 pence per 100 metres. What distance costs the same with both firms and what is that cost? Let d be the distance (multiple of 100 m) Let c be the cost of the fare (in pence) 5 d – 300 = 0 d = 60 800 = c c subtract 2 300 + 500 = c c 1 10 d + 200 = c 2 5 d + 500 = c A distance of 6 km (60 x 100) costs £ 8 (800 pence) with both firms. © T Madas

Today, Sam is sitting his last paper on a modular Exam. All modules count the same towards his final mark. If he scores 90% today, he will get an 80% final mark. If he scores 50% today, he will get a 72% final mark. How many papers in this subject? m + 50 = 72 n m + 90 = 80 n c Total added Mean = Total number m + 90 = 80 n c Let n be the total number of papers, Sam will have to sit in this subject. Let m be the total marks achieved from all his papers up until today. m + 50 = 72 n 40 = 8 n n=5 © T Madas

Two candles are of different heights. The blue candle is 4 cm higher than the green one. The blue candle burns down in 6 hours and the green one in 4 hours. When the blue candle has burned for 4 hours it is as long as the when the green candle has burned for 2½ hours. Find the original height of each candle. © T Madas

How do b and g relate to each other before the candles are lit up? If the blue candle burns down in 6 hours how much of the candle has burned down in 4 hours? b g How much of the blue candle is remaining after 4 hours? How much of the green candle is remaining after 2½ hours? © T Madas

How do b and g relate. How to do b and g relate to each other before each the other before the candles are lit up? If the blue candle burns down in 6 hours how much of the candle has burned down in 4 hours? How much of the blue candle is remaining aftercandle 4 is remaining after 4 hours? How much of the green candle is remaining after 2½ hours? © T Madas

How do b and g relate to each other before the candles are lit up? If the blue candle burns down in 6 hours how much of the candle has burned down in 4 hours? How much of the blue candle is remaining after 4 hours? How much of the green candle is remaining after 2½ hours? © T Madas

b 3 g = 8 3 8 b = 9 g 8 b = 9 g c 8 b = 8 g + 32 b= g+4 2 8 b = 9 g 3 = 8 g c 1 b 3 c 1 = x 2 1. 5 g 4 x 2 c 2 1 b 3 c 1 b=g+4 9 g = 8 g + 32 g = 36 The green candle had an initial height of 32 cm and the blue candle had an initial height of 36 cm. © T Madas

I bought 20 blank DVDs for £ 20. Each DVD-RW+ costs £ 3, each DVD-RW costs £ 1. 50 and each DVD-R costs £ 0. 50. How many DVDs did I buy from each type? Let a = number of DVD-RW+ b = number of DVD-RW c = number of DVD-R 1 a + b + c = 20 2 3 a + 3 2 b+ 1 c 2 = 20 Can we form another equation? © T Madas

I bought 20 blank DVDs for £ 20. Each DVD-RW+ costs £ 3, each DVD-RW costs £ 1. 50 and each DVD-R costs £ 0. 50. How many DVDs did I buy from each type? Fact: To solve simultaneous equations we need at least the same number of (independent) equations as the number of the unknowns 1 a + b + c = 20 2 3 a + 3 2 b+ 1 c 2 = 20 Can we form another equation? © T Madas

I bought 20 blank DVDs for £ 20. Each DVD-RW+ costs £ 3, each DVD-RW costs £ 1. 50 and each DVD-R costs £ 0. 50. How many DVDs did I buy from each type? Here: 2 Equations and 3 unknowns (usually unsolvable) However what helps here: We are looking for whole numbers less than 20 1 a + b + c = 20 2 3 a + 3 2 b+ 1 c 2 = 20 Can we form another equation? © T Madas

I bought 20 blank DVDs for £ 20. Each DVD-RW+ costs £ 3, each DVD-RW costs £ 1. 50 and each DVD-R costs £ 0. 50. How many DVDs did I buy from each type? 1 a + b + c = 20 2 3 a + 3 2 b+ 1 c 2 = 20 © T Madas

I bought 20 blank DVDs for £ 20. Each DVD-RW+ costs £ 3, each DVD-RW costs £ 1. 50 and each DVD-R costs £ 0. 50. How many DVDs did I buy from each type? 1 a + b + c = 20 2 3 a + 3 2 b+ 1 c 2 = 20 [x 2] 3 6 a + 3 b + c = 40 Note that this is NOT a new equation. It is the same equation as (2) We say that it is not an independent equation. © T Madas

I bought 20 blank DVDs for £ 20. Each DVD-RW+ costs £ 3, each DVD-RW costs £ 1. 50 and each DVD-R costs £ 0. 50. How many DVDs did I buy from each type? 1 a + b + c = 20 2 3 a + 3 2 b+ 1 c 2 = 20 [x 2] 3 6 a + 3 b + c = 40 1 a + b + c = 20 5 a + 2 b if a = 1 then b = 7. 5 = 20 2 3 4 5 5 2. 5 0 -5 © T Madas

I bought 20 blank DVDs for £ 20. Each DVD-RW+ costs £ 3, each DVD-RW costs £ 1. 50 and each DVD-R costs £ 0. 50. How many DVDs did I buy from each type? 1 a + b + c = 20 2 3 a + 3 2 b+ 1 c 2 = 20 [x 2] 3 6 a + 3 b + c = 40 1 a + b + c = 20 5 a + 2 b = 20 so a = 2 (number of DVD-RW+) b = 5 (number of DVD-RW) c = 13 (number of DVD-R) © T Madas

It is known that a box contains 10 coins with a combined weight of 116 grams. Some are gold, some are silver and some are bronze, but we cannot see inside the box. Each gold coin weighs 23 grams, each silver coin weighs 13 grams and each bronze coin weighs 7 grams. How many of each type of coin are there? Let x = number of gold coins y = number of silver coins z = number of bronze coins 1 x + y+ z = 10 2 23 x + 13 y + 7 z = 116 Can we form another equation? © T Madas

It is known that a box contains 10 coins with a combined weight of 116 grams. Some are gold, some are silver and some are bronze, but we cannot see inside the box. Each gold coin weighs 23 grams, each silver coin weighs 13 grams and each bronze coin weighs 7 grams. How many of each type of coin are there? Fact: Let x = number of gold coins To solve simultaneous equations we need at least y = number of silver coins the same number of (independent) equations the number z = numberas of bronze coins of the unknowns 1 x + y+ z = 10 2 23 x + 13 y + 7 z = 116 Here we have 2 Equations and 3 equation? unknowns (usually unsolvable) Can we form another However there is a constraint: We are looking for whole numbers less than 10 © T Madas

It is known that a box contains 10 coins with a combined weight of 116 grams. Some are gold, some are silver and some are bronze, but we cannot see inside the box. Each gold coin weighs 23 grams, each silver coin weighs 13 grams and each bronze coin weighs 7 grams. How many of each type of coin are there? 1 x + y+ z = 10 2 23 x + 13 y + 7 z = 116 © T Madas

It is known that a box contains 10 coins with a combined weight of 116 grams. Some are gold, some are silver and some are bronze, but we cannot see inside the box. Each gold coin weighs 23 grams, each silver coin weighs 13 grams and each bronze coin weighs 7 grams. How many of each type of coin are there? 2 23 x + 13 y + 7 z = 116 1 x + y + z = 10 y+ z = 10 © T Madas

It is known that a box contains 10 coins with a combined weight of 116 grams. Some are gold, some are silver and some are bronze, but we cannot see inside the box. Each gold coin weighs 23 grams, each silver coin weighs 13 grams and each bronze coin weighs 7 grams. How many of each type of coin are there? 2 23 x + 13 y + 7 z = 116 1 x y 7 x + 70 7 y + 7 z = 10 16 x + 6 y = 46 8 x + 3 y = 23 1 = number of gold coins Let x 5 = number of silver coins y z 4 = number of bronze coins 1 2 3 4 etc 5 2. 333… -0. 333… -3 etc © T Madas