Year 7 Written Calculations Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Objectives: Be able to add, subtract, multiply and divide numbers of any size (including decimals). Extension: Solve problems involving missing digits within a multiplication/addition. Last modified: 12th April 2016

STARTER :: Long Multiplication Use any preferred method (but preferably long multiplication!) to multiply the following numbers. 284×32=𝟗𝟎𝟖𝟖 375×65=𝟐𝟒𝟑𝟕𝟓 11 1 2 =𝟏𝟐𝟑𝟐𝟏 739×384=𝟐𝟖𝟑𝟕𝟕𝟔 806×608=𝟒𝟗𝟎𝟎𝟒𝟖 ? ? ? ? ? This is known as Pascal’s Triangle. Each number is the sum of the two above it. Sadly we will no longer see Pascal’s Triangle for 115 onwards (why?) 11 0 = 11 1 = 11 2 = 11 3 = 11 4 = If you finish… Put the digits from each calculation into the triangle. What do you notice? 1 1 ? 1 ? 1 ? 2 1 1 ? 3 3 1 1 ? 4 6 4 1

3.2×0.04 32×4=128 3.2×0.04=0.128 Multiplying Decimals First ignore decimal points and multiply as whole numbers. 3.2×0.04=0.128 Correct by counting total number of decimal place jumps.

Further Example 84.1×0.05 841×5=4205 84.1×0.05=4.205

Test Your Understanding 0.3×0.2 ? 𝟑×𝟐=𝟔 𝟎.𝟑×𝟎.𝟐=𝟎.𝟎𝟔 3.21 2 𝟑𝟐𝟏×𝟑𝟐𝟏=𝟏𝟎𝟑𝟎𝟒𝟏 𝟑.𝟐𝟏×𝟑.𝟐𝟏=𝟏𝟎.𝟑𝟎𝟒𝟏 ?

11.3−8.74 11.30 −8.74 2.56 Decimal Addition/Subtraction Fill any ‘gaps’ with 0s. 11.30 −8.74 2.56 Ensure decimal points are lined up so that the digits in each column have the same place value.

Check Your Understanding 18.43+192.71 103.8−48.91 ? 𝟏𝟎𝟑.𝟖𝟎 −𝟒𝟖.𝟗𝟏 𝟓𝟒.𝟖𝟗 ? 𝟏𝟖.𝟒𝟑 +𝟏𝟗𝟐.𝟕𝟏 𝟐𝟏𝟏.𝟏𝟒

Exercise 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (On provided sheet) Calculate: 3.4×5=𝟏𝟕 4.56×0.2=𝟎.𝟗𝟏𝟐 0.04×0.003=𝟎.𝟎𝟎𝟎𝟏𝟐 48−13.63=𝟑𝟒.𝟑𝟕 5.48−1.584=𝟑.𝟖𝟗𝟔 42.7×0.56=𝟐𝟑.𝟗𝟏𝟐 11.3−7.444=𝟑.𝟖𝟓𝟔 45.1×0.0043=𝟎.𝟏𝟗𝟑𝟗𝟑 [JMC 2007 Q1] What is the value of 0.1+0.2+0.3×0.4? Solution: 0.42 [Kangaroo Pink 2012 Q1] What is the value of 11.11−1.111? Solution: 9.999 [JMC 2001 Q8] What is the difference between the largest and smallest of the following numbers? A 0.89 B 0.9 C 0.17 D 0.72 E 0.73 Solution: E [IMC 2015 Q1] What is the value of 1−0.2+0.03−0.004? Solution: 0.826 [IMC 2002 Q2] Which of the following has the greatest value? A 0.3×7 B 0.5×5 C 0.2×11 D 0.09×30 E 0.026×100 Solution: D [IMC 2003 Q5] What is the value of 2003 2 ? Solution: 4 012 009 [JMC 2012 Q11] In the following expression, each □ is to be replaced with either + or – in such a way that the result of the calculation is 100. 123 □ 45 □ 67 □ 89 The number of + signs used is 𝑝 and the number of – signs used is 𝑚. What is the value of 𝑝−𝑚? Solution: -1 1 5 ? ? ? ? 6 ? ? ? ? ? ? 7 2 ? ? 8 3 ? 4 ? ?

Exercise 1 ? ? ? ? (On provided sheet) [JMC 1997 Q10] Each day throughout July 1995 I picked 300g of raspberries from my garden. What was the total weight of the raspberries I picked that month? A 10g B 9kg C 9.3kg D 10kg E 9300kg Solution: C [JMC 1998 Q7] The Mystery Prize at the Bank of England Christmas Party was a pile of crisp new £5 notes, numbered from 659384 up to 659500. What was the value of the prize? A £116 B £117 C £580 D £585 E £1420 Solution: D 11 [JMC 2000 Q10] Each Junior Mathematical Challenge answer sheet weighs 6 grams. If 140000 pupils enter the challenge this year, what will be the total weight of all their answer sheets? A 84kg B 840kg C 8 400kg D 84 000kg E 840 000kg Solution: B [JMC 2000 Q14] The DISPUTOR is similar to a calculator, but it behaves a little oddly. When you type in a number, the DISPUTOR doubles the number, then reverses the digits of this result, then adds 2 and displays the final result. I type in a whole number between 10 and 99 inclusive. Which of the following could be the final result displayed? A 39 B 41 C 42 D 43 E 45 Solution: E 9 ? ? 12 10 ? ?

Exercise 1 ? ? ? ? (On provided sheet) [Kangaroo Grey 2014 Q6] Which of the following calculations gives the largest result? A 44×777 B 55×666 C 77×444 D 88×333 E 99×222 Solution: B. We can ignore factors of 11 and 111 and just do 𝟒×𝟕, etc. [JMC 1999 Q21] Granny says “I am 84 years old – not counting my Sundays”. How old is she really? A 90 B 91 C 96 D 98 E 99 Solution: D 15 [JMC 1997 Q22] The Grand Old Duke of York, he had ten thousand men, he marched them up to the top of the hill, … By 2pm they were one third of the way up. By 4pm they were three quarters of the way up. When did they set out? A 12 noon B 12.24pm C 1.12pm D 1.36pm E 1.48pm Solution: B [TMC Final 2014 Q7] Place exactly three common mathematical operations (which need not all be different) between the digits below so that the result equals 100. You are not allowed to rearrange the order of the digits. 1 2 3 4 5 6 7 8 9For example, 1234 × 5 − 67 × 89. We know this example is wrong because the result is 207 Solution: 𝟏𝟐𝟑−𝟒𝟓−𝟔𝟕+𝟖𝟗 13 ? ? 16 14 ? ?

Exercise 1 ? ? (On provided sheet) 17 [JMO 1999 A7] Before the decimalisation of money in the UK, there were 12 pence (d) in 1 shilling (s) and 20 shillings in 1 pound (£). Thus 1 pound 3 shillings and 4 pence was written £1 3s 4d. What would have been the total cost of 7 items each costing £1 6s 8d? Write your answer in simplest £ s d form. Solution: £8 [JMC 2003 Q22] Two builders, Bob and Geri, buy bricks at the same price. Bob sells 10 for £6 and Geri sells 12 for £7. Supposing they sell equal numbers of bricks, what number has each sold when Bob has gained £4 more than Geri? A 42 B 60 C 72 D 120 E 240 Solution: E ? 18 ?

8÷0.2 =80÷2 =40 Dividing Decimals ? ? ? ? It is fine in practice to divide decimals by whole numbers. e.g. Dividing £1.68 between 4 people. However, we don’t in general like dividing by decimals. e.g. Dividing £3 between 5.6 people. We can’t do that! ? ? ? Consider that for example 8÷4=2 and 80÷40=2 What does this suggest we can do to both numbers in the division without affecting the result? =80÷2 =40 ?

1.47÷0.3 =14.7÷3 04.9 3 | 14.7 2.53÷0.011 =2530÷11 =230 More Examples Multiply each number by 10 until we’re dividing by a whole number. =14.7÷3 04.9 Ensure decimal point goes in same place in result. 3 | 14.7 2.53÷0.011 =2530÷11 =230

Test Your Understanding 174.9÷0.03=𝟓𝟖𝟑𝟎 6.055÷0.7=𝟖.𝟔𝟓 ? ?

Exercise 2 [JMC 2012 Q2] What is half of 1.01? Solution: 0.505 [Kangaroo Pink 2010 Q1] What is the result of dividing 20102010 by 2010? Solution: 10,001 [IMC 1999 Q5] 30÷0.2 equals Solution: 150 Calculate: a) 22.8÷0.5=𝟒𝟓.𝟔 b) 5.376÷0.06=𝟖𝟗.𝟔 c) 2825.2÷0.007=𝟒𝟎𝟑𝟔𝟎𝟎 d) 10.593÷1.1=𝟗.𝟔𝟑 e) 1.00001÷0.11=𝟗.𝟎𝟗𝟏 A death laser can fire every 0.004 seconds. How many times can it fire in 3 minutes? Solution: 45000 6 A rectangle has area 21.242cm2 and height 1.3cm. What is its length? 16.34cm What is its perimeter? 35.28cm Calculate: a) 0.09104÷0.16=𝟎.𝟑𝟖𝟏𝟓 b) 149.85÷1.5=𝟗𝟗.𝟗 c) 610.8÷0.12=𝟓𝟎𝟗𝟎 1 ? 2 ? ? ? 3 7 ? ? ? 4 ? ? ? ? ? ? 5 ?

Missing Digit Puzzles 𝐽 𝐽 𝑀 𝑀 + 𝐶 𝐶 𝐽 𝑀 𝐶 ? ? ? ? This lesson you will be able to work in groups on some puzzles to do with missing digits within column addition/multiplication. [JMC 2007 Q18] The letters 𝐽, 𝑀, 𝐶 represent three different non-zero digits. What is the value of J+𝑀+𝐶 ? A 19 B 18 C 17 D 16 E 15 𝐽 𝐽 Clue: These two digits are the same. Thus J and M must add to 10. Clue: 𝐽 could only be 1 or 2, as greatest result is 99 + 99 + 99 = 297 ? ? 𝑀 𝑀 + 𝐶 𝐶 𝐽 𝑀 𝐶 Putting this together: 𝑱=𝟏, 𝑴=𝟗, 𝑪=𝟖 Answer is B Clue: We’re adding the same digits in each of the two columns, but get a different digit as the result. There therefore must have been a carry, and hence M is one more than C. ? ?

Missing Digit Puzzles Question 1 ? Get into groups of 3 or 4. Merits to the team with the most correct answers at the end. Effective Team Maths Challenge strategy: Have two people independently work on each problem, and each ‘initial’ the problem on a shared sheet if you have a solution. When both people have solved the problem, compare your two answers. [JMC 2014 Q12] In this subtraction, 𝑃, 𝑄, 𝑅, 𝑆 and 𝑇 represent single digits. What is the value of 𝑃+𝑄+𝑅+𝑆+𝑇? A 30 B 29 C 28 D 27 E 26 Solution: B ?

Missing Digit Puzzles Question 2 ? [JMC 2003 Q16] In this multiplication each letter stands for a different digit. Which letters stands for 3? A B C D E Solution: D ?

Missing Digit Puzzles Question 3 ? [JMC 2005 Q18] In the subtraction sum on the right 𝑎, 𝑏 and 𝑐 are digits, and 𝑎 is less than 𝑏. What is the value of 𝑐? A 3 B 4 C 5 D 6 E a number greater than 6 Solution: A ?

Missing Digit Puzzles Question 4 ? [TMC Regional 2011 Q10] 𝐽6𝐾4×7=𝐿9𝑀98 Each of 𝐽, 𝐾, 𝐿 and 𝑀 is a different digit. Find the values of 𝐽, 𝐾, 𝐿 and 𝑀. Solution: 𝑱=𝟓, 𝑲=𝟏, 𝑳=𝟑, 𝑴=𝟐 ?

Missing Digit Puzzles Question 5 ? [TMC Final 2010 Q3] Each of the letters used below is standing for a different single digit. Example: 𝐴𝐵+2=𝐴𝐶 [the answer could be 𝐴=1, 𝐵=3, 𝐶=5 because 13+2=15] If 𝐴𝐵𝐶𝐷×9=𝐷𝐶𝐵𝐴 find 𝐴, 𝐵, 𝐶 and 𝐷. Solution: 𝑨=𝟏, 𝑩=𝟎, 𝑪=𝟖, 𝑫=𝟗 ?

Missing Digit Puzzles Question 6 ? [Junior Kangaroo 2015 Q12] In the sum shown, different shapes represent different digits. What digit does the square represent? A 2 B 4 C 6 D 8 E 9 Solution: C ?

Missing Digit Puzzles Question 7 ? [JMC 2002 Q24] In the multiplication on the right, each letter represents a different digit and only the digits 1, 2, 3, 4, 5 are used. Which of the letters represents 2? A B C D E Solution: E ?

Missing Digit Puzzles Question 8 ? [JMC 2010 Q25] What is the value of 𝑃+𝑄+𝑅 in the multiplication on the right? A 13 B 12 C 11 D 10 E 9 Solution: A ?

Missing Digit Puzzles Question 9 ? [JMC 1999 Q25] The two-digit by two-digit multiplication on the right has lots of gaps, but most of them can be filled in by logic (not by guesswork). Which digit must go in position ∗ ? A 1 B 3 C 5 D 7 E 9 Solution: D ?

Missing Digit Puzzles Question 10 ? [JMO 2000 B4] How many different solutions are there to the letter sum on the right? Different letters stand for different digits, and no number begins with a zero. J M C + J M O S U M S Solution: 6 ?