THE TABLE OF JOY - Flying Colours Mathsgames.flyingcoloursmaths.co.uk/toj_ebook.pdf · Table of Joy e-book – Colin Beveridge – Flying Colours Maths – colin@ﬂyingcoloursmaths.co.uk

Table of Joy e-book – Colin Beveridge – Flying Colours Maths – [email protected]

THE TABLE OF JOY - or -

How to score up to 22 more marksin your Maths GCSE paper*

*22 marks out of 100 in the 2007 EdExcel GCSE Mathematics A Paper 6 (Higher Calculator) could have been gained using this method.

http://creativecommons.org/licenses/by-nc-sa/2.0/uk/


Do you have problems with percentages?

Are you concerned about conversions?

Is trigonometry troubling you?

Can't do the questions on pages 3-4?

Then this e-book is for you.

Who is this book for?



My haggis recipe serves five people and calls for 225g of oatmeal. If I'm cooking for 14 people,

how much oatmeal do I need?

100 Japanese yen are worth 66.3p. I want to take £200 with me to Japan – how many yen is that?

In an exam, 24 of the 62 marks can be gained with the Table of Joy. What's that as a percentage?

My gas bill went up by £3.78 to £98.28. What was the percentage increase?

Test yourself



I mix orange and guava juice in a ratio of 7:3. How much orange juice do I need for two litres of

the mixture?

It takes me 40 minutes to drive 15 miles. What is my average speed?

The gravitational pull of the earth is inversely proportional to the square of your distance from its

centre. If you weigh 60kg when you're 6,000km away, how much would you weigh at 12,000km?

Test yourself



Answers:

630g¥30,165.91

38.7%4%

1,400ml22.5mph

15kg

Check your w

ork



The Table of Joy is a way to solve questions with proportions – including all of the ones above.

It used to be just called 'the table' but my student (Tain) thought that was boring.

It's the same thing as cross-multiplying (or the rule of three), but much less involved.

You still have to think a bit, though.

What is the Table of Joy?



Chapter 1

A worked examplewith lots of spaghetti



A recipe for pasta serves four people and uses 300g of spaghetti. If I'm cooking for

ten people, how much spaghetti will I need?

Start by drawing out a table and labelling the rows and columns.

People SpaghettiRecipe

My meal

Scaling – an example





Now fill in the numbers from the question. The recipe serves 4 with 300g of pasta.

People SpaghettiRecipe 4 300g

My meal






We know there are 10 people at my meal, but not how much pasta we need.


My meal 10 ?





My meal 10 ?

Now for the sum! First, we multiply the two numbers directly next to the question mark:

10 x 300g = 3,000g





My meal 10 ?

Then we divide the result (3,000g) by the number opposite the question mark:

3,000 ÷ 4 = 750g

So I need 750g to feed 10 people.





My meal 10 ?

So you multiply diagonally and divide by the number that's left over. You do the same thing for all of the examples here.




Chapter 2

Easy conversion tables

These all follow exactly the same rules – multiply diagonally and divide by what's left over.



You can do the same thing to convert units: What is 26.1 miles in kilometres?

You know that 5 miles is 8km:Miles Kilometres

Conversion 5 8Total 26.1 ?

26.1x 8 = 208.8208.8 ÷ 5 = 41.76km

Unit conversion



The exchange rate is £1 to A$1.94. I want A$200 – how much is that in pounds?

£ A$Conversion 1 1.94

Total ? 200

It works even if we move the question mark!

1 x 200 = 200200 ÷ 1.94 = £103.09

Currency conversion



There are 315 boys and 360 girls in a school. How many boys should be in a

stratified sample of 75 students?

Survey SchoolBoys ? 315Total 75 675

Again, it doesn't matter where the ? is.

315 x 75 = 2362523625 ÷ 675 = 35

Stratified surveys



What is the gradient of the line connecting (-2,5) to (10,-1)?

Gradient is how much y changes for every unit change in x.

x change y changePoints 12 -6

Gradient 1 ?

-6 x 1 = -6 -6 ÷ 12 = -0.5

Gradients



A, B, C and D are points on a

circle. The chords AB and CD

intersect at X. AX = 3cm, CX = 6cm and DX = 9cm.

What is BX?

A circle theorem



Just copy the picture into the table!The ones on the left go in the left column,

the ones on the right to the right.

left righttop 3 6

bottom 9 ?

6 x 9 = 5454 ÷ 3=18cm

A circle theorem



Chapter 3

Slightly trickier tables

The table doesn't think for you. Sometimes you need to add a value in by yourself



Sixty people were surveyed and 3 of them liked Big Brother. How big should the angle

in that section of the pie chart be?

We know there are 360º in a circle:Degrees People

BB fans ? 3Total 360 60

5)360 x 3 = 1080 → 1080 ÷ 60 = 18º

Pie charts



My train ride home covers 40 miles in 80 minutes. What is its average speed?

Speed in mph is the number of miles travelled in one hour (60 minutes)

Minutes MilesJourney 80 40

One hour 60 ?

60 x 40 = 2400 → 2400 ÷ 80 = 30mph

Speed, distance, time



Silver has a density of 10.5g/cm3. What is the volume of 100g of silver?

One cm3 of silver has a mass of 10.5g.grams cm3

Total 100 ?One gram 10.5 1

100 x 1 = 100 100 ÷ 10.5 = 9.52 cm3

Density



Six sweets cost 37p. A hundred sweets cost £6.10. Which is better value for money?

Let's work out what 100 sweets should cost:Sweets £

Six 6 0.37Hundred 100 ?

100 x 0.37 = 37→37 ÷ 6 = £6.17That would be more than the 100 sweet

deal, so the 100 sweet deal is better value.

Value deals



A cliff is 50m high. Its angle of elevation from Batman's boat is 27º. How long must his batrope be to land on top of the cliff?

Trigonometry 1: sides



We don't care about A, so we use SOH. We write the S and O across the top and the H

below the O. We fill the gap with 1.

trig sidesSO S = sin 27 O = 50mH 1 H=?

1 x 50 = 5050 ÷ sin 27 = 110.13m

Trigonometry 1: sides



Two angles of a triangle are 60º and 45º. The side opposite the 60º angle is 7m. How

long is the side opposite the 45º angle?

Sine rule 1: sides



We put the sines in one column, and the side lengths in the other:

sines sidesA sin 60 7mB sin 45 ?

7 x sin 45 = 4.95 4.95 ÷ sin 60 = 5.72m

Sine rule 1: sides



Seth and Leo split some sweets in a ratio of 5:3. If there were 72 sweets, how many

does Leo get?

Be careful to use the total of the ratio!Leo Total

Ratio 3 8Sweets ? 72

3 x 72 = 216 216 ÷ 8 = 27

Ratios



In a fairground game, I hit the wooden rat 4 times in 25 attempts. What is my

estimated probability of hitting the rat?

Probabilities are always out of 1.Hits Attempts

Observed 4 25Probability ? 1

4 x 1 = 44 ÷ 25 = 0.16

Probability



A spinner lands on red with a probability of 0.4. How many times would you expect to get red if you span the spinner 200 times?

Green SpinsEstimated ? 200Probability 0.4 1

0.4 x 200 = 800.80 ÷ 1 = 80

Probability



Chapter 4

Percentages

You usually need to think a bit about percentages, too. The Table of Joy takes some of the work out

of it, though.



83.6% of statistics are made up on the spot. A recent survey found that there are 16

million statistics. How many are made up?

The whole thing is always 100%.Made up Total

Statistics ? 16mPercent 83.6 100

16m x 83.6 = 1338m1338m ÷ 100 = 13.4m

Percentages



The population of Euclidia last year was 43 million. This year it is 48 million. What

was the percentage increase?

The original value is always 100%. The increase is 48m – 43m = 5m.

Last year IncreasePopulation 43m 5m

Percent 100 ?

5 x 100 = 500 → 500 ÷ 43 = 11.6%

Percentages



My computer cost £525 in a 25% off sale. What is its regular price?

In the sale it costs 75% of its usual price.

Sale NormalPrice 525 ?

Percent 75 100

525 x 100 = 52,500 52,500 ÷ 75 = £700

Percentages



The price of my train ticket went up 5% to £22.05. What did it cost last year?

It now costs 105% of last year's price.

Last year This yearPrice ? 22.05

Percent 100 105

22.05 x 100 = 22052205÷105 = £21.00

Percentages



Chapter 5

Stuff with functions

Sometimes you need to do something with the answer you get. If you've labelled your columns

properly, it should be a doddle.



The Eiffel Tower is 324m tall. It is 7.9m from the corner to the centre. What's the angle between

its leg and the ground?

Photo by ConspiracyOfHappiness

Trigonometry 2



We don't care about the hypotenuse, so we need TOA.

trig sideTO T = tan y O = 324A 1 A = 7.9

324 x 1 = 324324 ÷ 7.9 = 41.0

But this is tan y! y = tan-1(41.0) = 88.6º

Trigonometry 2



Two sides of a triangle have lengths of 8cm and 15cm. The angle opposite the first is

25º; what is the angle opposite the other? Sine rule 2



Again, the sines go in one column and the side lengths in the other.

sine sideA sin 25 8B sin y 15

15 x sin 25 = 6.346.34 ÷ 8=0.79

But this is sin y! y = sin-1(0.79) = 52.4º

Sine rule 2



Kinetic energy is proportional to the square of speed. A pebble has KE of 0.5J when

thrown at 10m/s. What is its KE at 15m/s?

We need speed squared in the first column.Speed2 KE

Given 100 0.5New 225 ?

225 x 0.5 = 112.5112.5 ÷ 100 = 1.125J

Direct proportion



A pebble's KE is 0.5J at 10m/s. Write an equation for speed and KE.

We need speed squared in the first column.To find k, we let v2 equal 1.

Speed2 KEGiven 100 0.5

k 1 ?1 x 0.5 = 0.5 0→ .5 ÷ 100 = 0.005

so: E = 0.005v2

Direct proportion



A pebble has KE of 0.5J when thrown at 10m/s. What is its speed if its KE is 2J?

We need speed squared in the first column.Speed2 KE

Given 100 0.5New v2 2

100 x 2 = 200200 ÷ 0.5 = 400

But this is speed squared! v=√400 = 20m/s

Direct proportion



In an electrical circuit, the resistance is inversely proportional to the current. When

the resistance is 10Ω, the current is 5A. What is the current if the resistance is 4Ω?

Inv. prop., so put 1 over the second column.A 1/Ω

Given 10 1/5New ? 4

10 x 4 = 40 → 40÷(1/5) = 200A

Inverse proportion



Chapter 6

All done!

That's the Table of Joy! You can use it elsewhere, but those are enough to be going on with.

All that's left is to make sure you can use it to answer exam questions... the last test is overleaf!



My cake recipe makes 12 slices and calls for 450g of sugar. If I only want 8 slices, how much sugar

do I need?

£1 is worth 52 roubles. I smuggle 2,000 roubles back from Russia – how many pounds is that?

The intake at a school increased by 12.5% to 198. What was the previous intake?

I bought a £60 shirt for £48. What percentage did I save?

Test yourself



I have two classical CDs for every five rock CDs. If I have 105 CDs, how many are classical?

I drive 220 miles at 55mph. How long does it take?

The value of a golden orb is directly proportional to the cube of its radius. An orb with a radius of 20mm is worth £1,000,000. How much is an orb

with a radius of 10mm worth?

Test yourself



Answers:

300g£38.46

17620%30

4 hours£125,000

Check your w

ork



Hi! I'm Colin from Flying Colours Maths. The Table of

Joy started as a way to simplify percentages for my

students, and they soon realised it worked for all kinds

of problems. I hope it's as useful for you.Please drop me an e-mail if you have any questions about it, or any unusual uses!

The Table of Joy, by Colin Beveridge is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales License

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THE TABLE OF JOY - Flying Colours Mathsgames.flyingcoloursmaths.co.uk/toj_ebook.pdf · Table of Joy e-book – Colin Beveridge – Flying Colours Maths – colin@ﬂyingcoloursmaths.co.uk