Sutton Boys' Second Stage Maths Mock Paper 4

7650

Multiplying by 25 is the same as multiplying by 100 and then dividing by 4.

306 x 100 = 30 600, and 30 600 divided by 4 = 7650.

The long way works too: 306 x 20 = 6120 and 306 x 5 = 1530, and 6120 + 1530 = 7650.

88 degrees

• 1 mark for 176 degrees shared between the two equal angles.
• 1 mark for 88 degrees.

The four angles inside any quadrilateral add up to 360 degrees.

The two given angles come to 84 + 100 = 184 degrees.

That leaves 360 - 184 = 176 degrees for the other two angles together.

They are equal, so each one is 176 divided by 2 = 88 degrees.

Using 180 degrees instead of 360 is the usual slip. That is the angle sum of a triangle, not a quadrilateral.

486

Each term is three times the one before it: 2, 6, 18, 54.

Fifth term: 54 x 3 = 162. Sixth term: 162 x 3 = 486.

Looking for a fixed amount to add does not work here, because the gaps grow.

25

• 1 mark for multiplying each number of pets by its frequency, giving 0, 7, 8, 6 and 4.
• 1 mark for the total 25 pets.

Each row tells you how many pupils, not how many pets, so multiply the two columns together.

6 pupils with no pets: 0 pets. 7 pupils with one pet: 7 pets. 4 pupils with two pets: 8 pets. 2 pupils with three pets: 6 pets. 1 pupil with four pets: 4 pets.

0 + 7 + 8 + 6 + 4 = 25 pets.

Adding the pet numbers 0 + 1 + 2 + 3 + 4 to get 10 ignores how many pupils there are in each row.

1

The mode is the value that comes up most often.

More pupils own one pet, 7 of them, than own any other number, so the mode is 1 pet.

Answering 7 gives the frequency rather than the value it belongs to.

7/20

• 1 mark for counting 7 pupils with two or more pets.
• 1 mark for the probability 7/20.

At least two pets means two, three or four pets, so add those rows: 4 + 2 + 1 = 7 pupils.

There are 20 pupils altogether, so the probability is 7/20.

7 and 20 share no factor, so it is already in its simplest form.

Leaving out the pupils with three or four pets, and answering 4/20, is the trap in the phrase at least.

144 pounds

• 1 mark for seeing that the difference of 24 pounds is 2 parts.
• 1 mark for one part equal to 12 pounds.
• 1 mark for the total 144 pounds.

Cleo has 5 parts and Amir has 3, so Cleo has 5 - 3 = 2 parts more.

Those 2 parts are worth 24 pounds, so one part is 24 divided by 2 = 12 pounds.

Altogether there are 3 + 4 + 5 = 12 parts, so the total is 12 x 12 = 144 pounds.

Check: Amir has 36 pounds, Bea has 48 pounds and Cleo has 60 pounds. 60 - 36 = 24 and the three add to 144.

Treating the 24 pounds as one part gives 288 pounds, which is the mistake to watch for.

1150 ml

Change the litres into millilitres first: 1.5 litres is 1500 ml.

1500 - 350 = 1150 ml.

660 pounds

• 1 mark for a correct method, such as 1080 pounds spent and 1740 pounds taken, or a profit of 2 pounds 75 on each copy.
• 1 mark for the profit 660 pounds.

The quick route is to find the profit on one copy: 7.25 - 4.50 = 2.75 pounds.

Then 240 x 2.75. Split it as 240 x 2 = 480 and 240 x 0.75 = 180, giving 660 pounds.

The long route agrees: the shop spends 240 x 4.50 = 1080 pounds and takes 240 x 7.25 = 1740 pounds, and 1740 - 1080 = 660 pounds.

Answering 1740 gives the money taken, not the profit.

5 faces, 8 edges and 5 vertices

• 1 mark for all three numbers correct: 5 faces, 8 edges and 5 vertices. Award nothing for a partly correct set.

The base is one square face and there are four triangles leaning in to meet at the top, so 1 + 4 = 5 faces.

The base square has 4 edges, and there are 4 more edges running up to the top point, so 8 edges.

The corners are the 4 corners of the base plus the point at the top, so 5 vertices.

Confusing this with a triangular-based pyramid, which has 4 faces, 6 edges and 4 vertices, is the usual slip.

72

• 1 mark for seeing that the 18 fish are one quarter of the tank.
• 1 mark for the total 72 fish.

If three quarters are neon tetras, the fish that are not make up the remaining one quarter.

So one quarter of the tank is 18 fish.

The whole tank is four of those quarters: 4 x 18 = 72 fish.

Working out three quarters of 18, or answering 24, misreads which part the 18 belongs to.

(11, 3) and an area of 54 square units

• 1 mark for the fourth corner (11, 3).
• 1 mark for the two side lengths 6 units and 9 units.
• 1 mark for the area 54 square units.

A and B share the same first coordinate, 2, so AB is a vertical side. B and C share the same second coordinate, 9, so BC is horizontal.

The fourth corner sits below C and level with A, so it takes the 11 from C and the 3 from A, giving (11, 3).

Side AB is 9 - 3 = 6 units long. Side BC is 11 - 2 = 9 units long.

Area: 6 x 9 = 54 square units.

Writing the fourth corner as (3, 11) swaps the coordinates over, which is the easiest mark to lose here.

0.12

Ignore the decimal points for a moment: 4 x 3 = 12.

There is one decimal place in 0.4 and one in 0.3, so the answer needs two decimal places: 0.12.

Answering 1.2 is the classic error. Notice that the answer must be smaller than either number, since you are taking a fraction of a fraction.

5 minutes 15 seconds

• 1 mark for a correct method, such as changing the time to 3150 seconds or to 52.5 minutes before dividing by 10.
• 1 mark for 5 minutes 15 seconds.

Change the whole time into seconds: 52 x 60 = 3120, plus 30, gives 3150 seconds.

Divide by 10 kilometres: 3150 divided by 10 = 315 seconds.

315 seconds is 5 minutes with 15 seconds left over, because 5 x 60 = 300.

So each kilometre takes 5 minutes 15 seconds. Writing 5.25 minutes is correct as a decimal but does not answer in minutes and seconds.

70000

Read the columns from the right: 8 units, 0 tens, 9 hundreds, 2 thousands, 7 ten thousands, 4 hundred thousands, 3 millions.

So the 7 stands for 7 ten thousands, which is 70 000.

Answering 7, or seven thousand, misses the column the digit sits in.

0.63

• 1 mark for changing five eighths to 0.625, or for changing 0.63 to a fraction such as 63/100 and comparing fairly.
• 1 mark for stating that 0.63 is larger, supported by that comparison.

You cannot compare a fraction with a decimal until they are in the same form.

Five eighths means 5 divided by 8. Half of 8 is 0.5, a quarter is 0.125, so five eighths is 0.5 + 0.125 = 0.625.

0.63 is bigger than 0.625, so 0.63 is the larger number.

Assuming five eighths is 0.58 by copying the digits across is the trap here.

1/2

• Accept 1/2 or one half. 3/6 is not in its simplest form and scores nothing.

Count the As: there are 3 of them among the 6 sections.

So the probability is 3/6, which simplifies to 1/2.

Answering 1/3, because there are three different letters, ignores how many sections each letter takes up.

23 and 24

• 1 mark for a sensible search, such as trying 20 x 21 = 420 and 25 x 26 = 650 to close in on the answer.
• 1 mark for both numbers, 23 and 24.

Consecutive means next to each other, so the two numbers are very close in size and each is a little either side of the number that multiplies by itself to give 552.

Try 20 x 21 = 420, too small. Try 25 x 26 = 650, too big. So the answer sits between them.

Try 23 x 24 = 552. That works.

Check: 23 x 24 is 23 x 20 = 460 plus 23 x 4 = 92, and 460 + 92 = 552.

400

• 1 mark for the Monday to Friday total, 1700.
• 1 mark for the six-day total, 2100.
• 1 mark for 400 visitors on Saturday.

Add the five days: 320 + 285 = 605, then 605 + 410 = 1015, then 1015 + 296 = 1311, then 1311 + 389 = 1700 visitors.

If the mean over six days is 350, the six days add to 6 x 350 = 2100 visitors.

Saturday is the difference: 2100 - 1700 = 400 visitors.

Multiplying 350 by 5 instead of 6 is the trap, because the mean is given over six days, not five.

8.5 cm

A regular pentagon has 5 sides, all the same length.

42.5 divided by 5 = 8.5 cm.

Check: 5 x 8.5 = 42.5 cm.

78

• 1 mark for spotting that the amounts added grow by one each time: 2, then 3, then 4, and so on.
• 1 mark for continuing the pattern correctly at least as far as a tower 8 cubes high, which uses 36 cubes.
• 1 mark for 78 cubes.

Look at the jumps: 1 to 3 is 2 cubes, 3 to 6 is 3 cubes, 6 to 10 is 4 cubes. Each new layer adds one more cube than the layer before.

So the totals run 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78.

A tower 12 cubes high uses 78 cubes.

There is a shortcut: the total is the number of layers times one more than that, halved. Here 12 x 13 = 156, and half of 156 is 78.

Multiplying 10 by 3 to jump from a tower of 4 to a tower of 12 is the tempting mistake, because the pattern is not one of equal steps.

10:00

• 1 mark for looking for the smallest number that 12, 18 and 30 all divide into.
• 1 mark for 180 minutes, or 3 hours.
• 1 mark for the time 10:00.

The three routes meet again after a number of minutes that 12, 18 and 30 all go into exactly.

Multiples of 30 are 30, 60, 90, 120, 150, 180. Of these, 180 is the first that both 12 and 18 also divide: 180 divided by 12 = 15 and 180 divided by 18 = 10.

180 minutes is 3 hours.

3 hours after 07:00 is 10:00.

Multiplying 12 x 18 x 30 gives a time when they do all meet, but a much later one, so it does not answer the question.