**48cm**

Area of rectangle is 128cm²

So use algebra to solve (remember area of rectangle = LxW)

2x * x = 128 (where 2x = length and x = width. Use 2x given length is twice width)

3x=128

2x²=128

x²=64

**x=8**

So perimeter = (8*4 for the length + 8*2 for the width) = **48cm**

**10 Times**

Monday Spaghetti & Apple Pie

Day 1 Tuesday

Day 2 Wednesday Spaghetti

Day 3 Thursday Apple Pie

Day 4 Friday Spaghetti

Day 5 Saturday nothing

Day 6 Sunday Spaghetti & Apple Pie

Every 6th day she eats both. So in the next 60 days, she eats both 10 times

**B**

A+C+D = 70%. Therefore, 30% chose B.

If 25% = 320, what does 30% = ?

The approach is to use HCF of 25 and 30 which is 5%. Work out 5% first of 320

This equals 64

Then 64 * 6 (given that 30/5% = 6 times)

Total is 384

a) 0.09

b) 0.05

c) 0.375

If not sure, you can use “reverse bus stop”. So for example, calculate per below:

**17**

Star = 4 from middle row.

Smilie Face = 7 from first column

Heart = 6 from first row (16-4) = 12/2

So heart + Star + Smilie Face = 4+7+6

**24**

Do inverse which is 624/26 = 24

**a) 26**

**b) (32+9+6)=47 * (9-6) = 141 – (9-6) = 138**

**72cm**

(15 * 4) + (3*4) = 60 + 12

**a) i) 31°**

W = 180 – 107 – 42 = 31

180° in a straight line

a)ii) Reason is 180 in a straight line so its 180 less 107 less 42 to give the missing angle

**b) x = 52°; y= 47°**

For x…given JKL & MKN are straight lines, opposite angles in this set up are equal. Hence x = 52

For y….180 – 52 – 81 = 47°

**13/40**

- You have 1/4 shaded of the first square. 2/5ths shaded of the second square
- This is a question of being able to convert both fractions to the same denominator
- So this would be 20. 1/4 becomes 5/20 and 2/5ths is 8/20
- Then add both these together to get 13/40. Note its not 13/20 as you have to add the squares together which is unlike a normal fraction addition

**17**

130/8 = 16.25

Cannot buy 0.25 of a packet so need to buy 17 packets

**a) See below**

**b) 41 Tiles**

- Pattern 1 = 3Grey + 1White = 4 tiles
- Pattern 2 = 4Grey + 2 White = 6 tiles
- Pattern 3 = 5Grey + 3 White = 8 tiles
- So rule is (Pattern number + 1) + Pattern number = Total number of tiles
- (20+1)+20 = 41

**c) 50**

- The pattern starts at 4 tiles and goes up in 2s. But then you have to reduce it by 1 to get to the pattern number. So for Pattern 3, this is 8 tiles. 8 divided by 2 = 4. Less 1 = 3. Confirmed above that this is Pattern 3
- Therefore, if you have 102 tiles, then this is 51.
- 51 less 1 = 50

**25**

1 + 2 x 3 x 4 = 25

**B**

Number of grey tiles = 2*n where n=shape number

So for Shape 3, it would be 2*3 = 6 grey tiles

28 grey tiles = 2 * n so n = 14. Shape 14

For Shape 14, white tiles = (n+1) = 15

**B**

2Triangles = 1 Hex minus 1 Square = 6-4 =2

So a triangle = 1

**A**

Work backwards. 23 – 3 = 20. Divide by 4 and you get 5

**A**

1680 = 1680

Quick way is you are doubling the 24 to get to 48 on the other side. Then halving the 70 and getting 35 on the other side. So no calculations required in this question

**a) 160cm**

**b) 168cm²**

**462cm**

Large Triangle = 16cm * 3 (all sides same as its equilateral). So 48cm

Middle Triangle = 7cm * 3 triangles * 3 sides (7 small triangles of 1cm each on the horizontal base tells us its a 7x7x7 triangle). So 63cm

Next size Triangle of which there are 3 = 3cm * 3 triangles * 3 sides = 27cm. There are 3 of them so 27cm

Smallest triangle = 36 small triangles all of 1cm per side. So 3cm *36 triangles * 3 sides = 324cm

Total of all of these is 48 + 63 + 27 + 324

**64cm**

Perimeter = 14+3+4+6+7+2+17+11

a) See below. **1.52**

Convert to the same metric. So in this case, its better to convert to decimals

**b) 0.3**

For this, you should use reverse bus stop. So 70 goes outside the bus stop with 21 inside.

Then add zeros on so that you can work out the answer to 2 decimal places

Then change the answer by the number of decimal places you added. So if you added 2 zeros, then your answer of eg. 70 would become 0.7

**1.08**

- Sometimes, these questions can be done in 2 parts. First 6/6 and then 0.48/6
- 6/6 = 1
- 0.48/6. Treat this is 48/6 = 8.
- Then move 8 2 decimal places to the right. So 0.08
- So answer is 1.08

**0.5 hours**

- In one hour, Sachin can clean 1/3rd of the house
- In one hour, Peter can clean 1/6th of the house
- Convert to the same denominator which is 6
- So Sachin = 2/6 and Peter = 1/6
- Together, it takes 3/6 divided by 6/6 = 0.5 hours

**15/26**

When doing this type of question, you can’t answer 7.5/13 obviously. So the solution is to double each fraction and “force” a gap

So this becomes 14/26 and 16/26

Answer is 15/26

**a) See below**

**b) 7 &10**

**a) 8**

Mode = most common score. Scores were 9,6,8,4,8

**b) 5**

Range = 9-4

**c) 7**

9+6+8+4+8 = 35. Mean is average 35/5

**d) 4**

6.5 * 6 (6 being number of tests) = 39

French score is 39-35 = 4

**e) 2/3**

9/10; 6/10; 8/10; 4/10; 8/10; 4/10

He passed 4 out of 6 tests so 4/6 or 2/3

**f) 35%**

History score before was 4/10 – 40%

Now he scores 15/20 = 75%

Improvement = 35%

**1/20; 1/5; 1/4; 6/20; 3/2**

- Use estimation if needed. So this would involve converting all of the fractions to the highest denominator which is 20
- 6/20; 4/20; 30/20; 1/20; 5/20
- This makes it easier to rank them

**6 DARK TOFFEES**

Approach is to draw out the breakdown per below and start filling in one section at a time

**225**

- In 4 nights, she ate an extra 25 + (25*2) + (25*3) so this is 150 more.
- Take this away from the total of 1050 gives 900
- So divide this by 4 and you get 225
- So day 1 225, day 2 250, day 3 275 and day 4 300
- Total reconciles back to 1050

**267,120**

**742**

**53424**

**26748**

- Add a zero on. So 267,120
- Its rearranging the formula so it becomes 742 = 26712/36
- Double the base answer of 26712 so it becomes 53424
- Add on 36 to the original answer. So its 26712 + 36 = 26748. (ie an extra 1 x 36 since number changes from 742 to 743)

**Area of 1 rectangle = 15 x 5 = 75cm²**

**4 children**

- 56 legs with 3:1 ratio dogs:children
- 12 dogs = 48 legs. 4 children = 8 legs. So thats 56 legs altogether
- For every 3 dogs there are 12 legs. For every 1 child, 2 legs. So leg ratio is 12:2 = 14 legs
- Then work up in 14s to get to 56 which takes you to 48:8
- So 8 legs = 4 children

**11**

- 24 people. 4 got off so this left 20. X people got on
- Assume zero got on for now at the last stop
- Second stop . 20 with 3 people getting on leaving 23
- If there were then 34 people, it means 11 people got on at the first stop

Work backwards for this section….

Then BC + A = BC + 4 = 37 so BC = 37 –> **B = 3, C = 7**. **A=4**

Checking, 43 + 7 = 50.

**42**

- Work through the multiples and add on the remainder where necessary
- 4 – So multiples with 2 added on would be 6,10,14,18,22,26,30,34,38,42
- 5 – So multiples with 2 added on would be 7,12,17,22,27,32,37,42
- 6 – 6,12,18,24,30,36,42,48

**Sam 9 years**

**Ben 3 years**

**Frederick 16 years**

**Mylo 11 years**

- Always write it down numerically as much as possible. This gives a clearer view on the workings required and/or the answer
- Sam + Ben = 12
- Frederick + Sam + Ben = 28
- Therefore, Frederick must be 16
- Mylo + Sam = 20
- Total of all brothers = 39
- Therefore, remaining ages total is 23.

**a) 51,000**

with 2 zeros added onto the equation, you just add 2 zeros to your answer

**b) 255**

halve the answer

**c) 30**

510/34 would give 15

Therefore 510/17 would give 30

**1 and 4**

**48**

- Do your 3 x 4 times table to find a number in between 41 and 51
- So 12,24,36,48
- Answer is 48

**8**

- Convert £ to pence
- 300/36 = 8.33
- So 8 full bars

**6 weeks**

- Per week she gets (79p-44p) profit. So this is 35p per download
- 150 downloads per day = 150x35p = £52.50 per day
- £2000/£52.50 = 38.1 days
- So this is 6 weeks as this is 42 days as she gets paid at the end of each week so you can’t use part a week

**6.25**

probability (3,7) = 1/2

probability (1,2,3,4) = 1/4

To get 7, you would need 3 on the coin and 4 on the dice. So thats 1/2 * 1/4 = 1/8 times you get a total of 7. So 1 out of 8 times

6 times you would expect to get the total 7. As you would get 7 every 8th time. So then after 48 times you get 7 6 times

The 2 remaining roles you have a 0.25 chance of getting 6 .

So 6.25

**96 tiles**

3metres long and 4m wide = 12m² area

You will get 12 tiles in the width from 25cm tiles (3metres = 300cm. 300cm/25 = 12 tiles)

Across you will get 8 tiles in the length from 50cm tiles ( 4metres = 400cm/50cm – 8 tiles)

12 x 8 = 96 tiles

**a) 64cm²**

Perimeter = 32. Therefore, each side is 8cm

Area = 8 x 8 = 64

b) 48cm²

**a) £27**

Cost = ((8×3) + (3×1)) = 24+3 = 27

**b) 2 adults. 3 children**

Use £25 and work backwards taking off children until you get to a multiple of 8

So £16 would be the number

£25-£16 = £9. So thats 3 children

Leaving £16 so its 2 adults

k = 180 -50-95 = **35°**

**39**

- Catherine gets 22 sweets
- Alison gets 11 sweets
- Bhavini gets 6 sweets

**3**

- Work backwards
- 38 + 7 =45
- 45 /15 = 3

**25,100 millimetres**

- 25.1metres = 25.1 x 100 = 2,510 centimetres
- 2,510 x 10 = 25,100 millimetres

**All the common factors of 72**

**100°**

- At 4:40, hour hand will be 2/3rds of the way between 4 and 5
- Minute handle at 40
- So each hour or number on the clock is 30°
- From 4:40 to 5:00 the degree angle is 10°
- From 5:00 to 8 (Which is for 40 mins), the angle is 3 x 30 – 90°
- Add 10 degrees per note 4 and you get 100

**20 students**

- Work on basis 6 students = 30°
- Therefore, 2 students = 10°
- So, 100° = 20 students

**32 minutes**

- Convert time to minutes
- So 75 minutes – 43 minutes

**Approximately 40°**

An acute angle

**a) 52**

5 x 10 + 2

**b) 127**

5 x 25 + 2

**c) 67**

Work backwards. 337 – 2 = 335

335/5 = 67

**d) Van = 2.5 metres; Car = 1.8 metres**

Additional 5 metres = 2 vans

Van = 2.5m

If a van is 2.5metres, then using the first jump of 14 metres, we take 2 vans off this which is 5 metres leaving 9 metres.

Per car, this will be 9 metres/ 5 cars = 1.8m

**3**

- Convert to 10ths and then back again
- So that would be 6/10 + 3/10
- This would convert back to 3/5 + 3/10

**7**

- Work backwards.
- 16 + 5 = 21
- 21/3 = 7

**33**

- Use division so you get 6240/192 = 32.5
- So we need 33 trips

**3/12; 2/8; 1/4**

- Best way to compare fractions is to convert to the Lowest Common Denominator
- But this is tricky with tight timelines
- Therefore look for a common denominator where possible
- 3/12; 2/8 & 1/4

**321**

**279**

**20**

- It is one extra “21” So 321
- It is one less “21”. So 279
- It is 20

**42857**

It is possible to make the following equation:

3(100000 + x) = 10x+1

(Why? Well, adding 100000 puts a 1 at the front of a five-digit number, and multiplying by 10 and adding 1 puts a 1 at the end of a number)

Solving this gives:

10x+1 = 3(100000 + x)

10x+1 = 300000 + 3x

10x = 299999 + 3x

7x = 299999

x = 299999/7 = 42857

The answer is 42857 (142857 is three times smaller than 428571)

**a) 127cm**

Convert to cm. 214-87

**b) 25 bags**

290/12 = 24.2

Round up as you need to cover the “0.2” requirement. So 25 bags

**D**

250cm = 2.5m

**A=1, B= 4, C=8**

There is a bit of trial and error involved here.

When you work through some permutations, you know C can’t be 0-3. This is because you do need a number that drives some form of carry. This impacts the tens and hundreds columns

So its 4-9

b) 1089For 4-digit numbers, let a, b, c, and d be the digits, so 9*(1000*a+100*b+10*c+d) = 1000*d + 100*c + 10*b + a 8999*a + 890*b = 10*c + 991*d This implies that d = 10 - a, by looking at last digits, which we substitute, and get 9990*a + 890*b = 10*c + 9910 When we divide by 10, we get 999*a + 89*b = c + 991 This implies that a = 1, so 89*b = c - 8 This implies that b = 0 and c = 8, and there is only one solution 9*1089 = 9801.

**a) 12 seconds**

- After 2 seconds, Gary is at 20metres which is when Andy starts running
- So Andy has 20 metres to make up
- It will take him 10 seconds to close this gap as he makes up 2 metres per second
- Add the initial 2 second start that Gary had over Andy

**b) 120 metres**

- Distance = Speed x Time
- Therefore = 12m/s x 10 seconds = 120 metres

**a) 49cm**

Length of cane that snapped off is 2.05m = 205cm minus 156cm

Converting all to cm makes the question much easier

**b) 14**

Would need 14 = 168m²

13 would give 156m² so not quite enough

**D**

- If the pen costs 50p more, then the way to start with this question is to assume they cost the same
- This would be 55p each
- Then take the difference of 50p and split it into 25p and 25p
- Add 25p to the pen and take 25p from the pencil
- So the pen would be 55p + 25p = 80p

**C**

- Logical Thinking. 17cm

**a) 45**

Fail = 15 boys = 1/4 of the pie or 90 degrees

Therefore, if 1/4 = 15, 3/4 = 45

**b) Charlie is incorrect because you do not know how many boys and how many girls there are in Year 7. This is required to work out the answer. There may for example be 24 boys and only 5 girls. This would mean 6 boys fail the test and less than 5 girls fail the test. **

Start with Sneezy as this is given as 24 and work from there

Grumpy gets 1 more so is on 25 etc etc

**16:27**

- Logical thinking. Go in multiples of 10 until 6 is also a multiple of that number
- So this gives 30 minutes
- Next time both trains leave the platform at the same time is 16:27

- The rule of thumb to see if a number is a multiple of 3 is to add up all of the digits
- So in this case, 5+3+7 = 15
- If all of the digits added divide by 3, then thats the number
- In this case we were going for the largest number. Smaller options would have been 534 and 531

**5 Pens 4 Pencils**

- Whatever you buy in pens, just end in a 5 or 0. So the only number this applies to between 0 and 55 is 35
- So its 5 Pens. This leaves 20p
- 20p is for pencils @ 5p each
- So 4 pencils

**QUESTION 3….After 10 days**

**16**

- We know that 7 years ago, H was 4 years old which is 11-7
- Mothers age would be 4 x 12 = 48
- Therefore, 5 years on, would mean H would be 11+5
- So 16 years old

**19**

75/4 = 18.75

Answer is 19 not 18. Because you need 19 boxes to ensure all 75 cakes are stored

See below

Just work through each child at a time starting with the most obvious…..the Winner

Then going from there. Only tricky one is A & E. Which one comes first? Well its A because of the statement that A x D = F. This can only be 2 x 3

See below

**Because you can make coffee whilst the toast is in the toaster**

**Shortest time is 4 minutes**

Work on the basis that everyone starts at £4. 4 people and they shared the bill equally. So £16 total bill. Then go from there and adjust each individual as required

Need to work through each person to understand the possible options:

Thomas – Brasted, Westerham or Dunton

Henry – Knockholt or Dunton

Edward – Dunton or Sevenoaks

Gordon – Brasted or Knockholt

See below

The trick here is to substitute the actual scenarios with a given value. For example, assume the value costs £4.

Then use this figure in every single option to see which one works

**48cm**

- You have to think about times tables;
- So the shortest length will be the LCM of 6 & 8;
- This is 48cm

**35**

- Use the power of estimation. This is a crucial technique in maths
- As you are doing questions, try and estimate the answer.
- When you calculate your answer, you will have an idea if you are close to the answer
- In the above, round the numbers. So it becomes 7 * 5 (Given square root of 26 is approx 5)
- Answer will be 35

**4**

- Definately a logical thinking question
- What you do is cross cancel 56, 55 & 54 because they are on both sides of the brackets
- Therefore, the revised question becomes 57-53

**a) 16**

**b) 24**

(However many people there are, permutations are each number x each number. So for example if there are 3 people, possible permuations are 1x2x3 =6. For 4 people, its 1x2x3x4)

**a) 5**

**b) 10**

**c) 1 & 8**

Detail to be added

**a) i See below**

**a) ii See below**

**b) 13,20,24**

- 4=2² with difference as 2

**c) 0,1,4 & 9**

- 0²=0; 1²=1; 2²=4; 3²=9

**d) See below**

**T=36,S=54,R=108**

- Use a ratio style approach
- R=2xS. S=1.5xT
- So T=1; S=1.5; R=3
- Total parts = 5.5
- 198/5.5 = 36 (Use bus stop but you can do 55 into 1980 so that you do not have to deal with the decimal
- So T=(1*36); S=(1.5*36=54); R=((3×36)=108)

**a) See below**

**b) From the table:**

**Cone, Vanilla, Flakes****Tub, Mint, Sprinkles**

**a) 48cm**

See labelling below

**b) 108cm²**

Area = WxL. 6×18

**c) 18**

In 18cm across, she can fit 6 (18/3). Given its 6cm tall, she can fit 3 high. So in total 3 rows each with 6 in each row 18

**12**

Consider the various options to multiply to get to 72.

2 x 36; 3 x 24; 9 x 8, 12 x 6

12 x 6 is the only option which meets the criteria

**a) See below**

Use the HCF Principle. This is the best approach. So HCF of 6 & 15 is 3. Therefore, work out what is needed for 3 people

Then multiply the result by 5 to get to what is needed for 15 people

**b) 60 people**

If 100g is 6 people, then 1kg is 10x 6 people

**a) 1,592**

1554 + 38 = 1592

**b) 777**

Half of the original answer (37 * 42)/2

**c) 3.7**

37/10

**See below. The clue as to how to complete it is given by the fact that the diagonals given add up to 15. So all horizontal, vertical and diagonals need to add up to 15**

**a i) 80**

**ii) 23**

Work backwards. 120/4 = 30. 30 minus 7 = 23

**b) x2 -6 OR -3 x2**

**3**

Work Backwards. 56 / 8 = 7.

Then 7 minus 4 = 3

**a) 2040**

Move the decimal 2 places to the right when multiplying

**b) 1050**

- You can do the inverse operation here and multiply 1000 x 1.05 to get to the answer
- If not sure, then do the calculation in 2 parts:
- 1000 x 1 = 1000
- 1000 x 0.05 = 50 (Remember 0.1 = 10% = 100. So 0.05 = 5% = 50)
- Then add both numbers together

**Start with 7 and the permutations for 7 and work from there**

**So (1,6), (2,5), (3,4) etc**

**a) 14**

9 in lower level, 4 in middle level, 1 at the top.

**b) 30**

- Observe the sequence. This question is simply sequences in pictures
- 1st tower is 1×1 on the base. 2nd is 2×2. 3rd is 3×3.
- 4th will be (4 x 4)+ (3×3) + (2×2) + (1×1) = 16 + 9 + 4 + 1 = 30

**c) 55**

Add on a 5 x 5 to the answer for b)

**d) 100**

Noting the comments for b), this therefore is a 10×10

**a) 24**

- Give that Cats = 48 = 120º
- Hamsters = 60º. Therefore, they are half of cats = 24

The true way to look at this question is to say if 120º = 48, then what is the HCF of 60 & 120. This is 60. Therefore, take the 120 and say how many 60s go into 120. This is 2. Therefore, 60 = 24

**b) 60**

- Cats 120º + Hamsters 60º + Other 30º = 210º
- Remainder is 360 – 210 = 150º
- Again, HCF principle. Given you know 60º = 24, what is 150º?
- HCF = 30. So 30 is half of 60 so it equals 12
- Then how many 30s into 150 = 5
- Therefore 5 * 12 =
**60**

- The easiest way to complete these questions is to think about the HCF multiple.
- Look at how many cupcakes you have ingredients for. Its 8. You need to complete for 12.
- HCF of 8 & 12 is 4. Therefore, if you work out the ingredients for 4 cupcakes, then you can multiply up by 3x to get to 12
- So for 4, its half of what is required for 8
- So butter would be 75g.
- x3 would give you butter for 12 cakes etc

**729, 927, 972, 792, 297**

Different combinations = number of digits multiplied. So 3x2x1. So 6 combinations. 5 other ones listed above

**15 & 25**

- The technique here is to start with 40 and divide by 2. So thats 20
- Then to give a difference of 10 between the 2 numbers, you have to deduct 5 from one of the 20s and add 5 onto the other 20
- So this gives 15 and 25

**6**

Where “U” stands for Unknown. So Unknown 1 is “U1” etc

U1+4+17 = U2+U1+5

Rearrange:

21 = u2+5

**u2=16**

Given this, 5+17+u3 = u2+u3+m

22 = 16 + m

**m = 6**

**Zara**

Follow it logically step by step:

Glen > Florence

Florence > Ahmed > Zara

Oliver < Glen > Zara

Yasmin < Glen

Ahmed > Oliver

Florence < Yasmin

If Florence is less than Yasmin and given Yasmin is greater than Glen and Glen is greater than Zara, Zara is the youngest