Sutton Boys' Second Stage Maths Mock Paper 10

23

The two numbers are close together, so each must be near the square root of 552.

20 x 21 = 420, which is too small, and 25 x 26 = 650, which is too big. Try in between: 23 x 24 = 552.

So the smaller number is 23.

Halving 552 to get 276 does not help, because the two numbers are multiplied, not added.

30

• 1 mark for showing that the final price is 96 per cent of the original price.
• 1 mark for 30 pounds.

A rise of 20 per cent multiplies the price by 1.2, and a fall of 20 per cent then multiplies it by 0.8.

1.2 x 0.8 = 0.96, so the final price is 96 per cent of the original. The two changes do not cancel out, because the 20 per cent is taken off a bigger number than it was added to.

So 28.80 pounds is 96 per cent of the original price. One per cent is 28.80 / 96 = 0.30 pounds, and 100 per cent is 30 pounds.

Check: 30 + 20 per cent = 36, and 36 - 20 per cent of 36 = 36 - 7.20 = 28.80.

Answering 28.80 because the two changes look as though they undo each other is exactly what this question is built to catch.

2250 cm3

• 1 mark for a base measuring 30 cm by 15 cm.
• 1 mark for a height of 5 cm.
• 1 mark for 2250 cubic centimetres.

A 5 cm square is removed from both ends of the 40 cm side, so the base of the box is 40 - 5 - 5 = 30 cm long.

The same happens to the 25 cm side, so the base is 25 - 5 - 5 = 15 cm wide.

The flaps that fold up are 5 cm tall, so the height of the box is 5 cm.

Volume = 30 x 15 x 5 = 450 x 5 = 2250 cubic centimetres.

Taking only 5 cm off each side, giving 35 cm by 20 cm, is the standard error. Each side loses a square at both of its ends.

360 km

Multiply the map distance by 25: 14.4 x 25.

A quick route is 14.4 x 100 = 1440, then divide by 4 to get 360.

So the towns are 360 km apart.

7.25

• 1 mark for multiplying each mark by the number of pupils who scored it.
• 1 mark for a total of 145 marks.
• 1 mark for 7.25.

Work out the marks scored at each row: 5 x 2 = 10, 6 x 4 = 24, 7 x 5 = 35, 8 x 6 = 48, 9 x 2 = 18 and 10 x 1 = 10.

Add them: 10 + 24 = 34, 34 + 35 = 69, 69 + 48 = 117, 117 + 18 = 135 and 135 + 10 = 145 marks in total.

Check the number of pupils: 2 + 4 + 5 + 6 + 2 + 1 = 20, as the question says.

The mean is 145 / 20 = 7.25.

Adding the six different marks and dividing by 6 gives 7.5, and ignores the fact that far more pupils scored 8 than scored 10.

48

If three quarters is 54, one quarter is 54 / 3 = 18, so the number itself is 4 x 18 = 72.

Two thirds of 72 is 72 / 3 x 2 = 24 x 2 = 48.

Taking two thirds of 54 gives 36 and skips the step of finding the number first.

1/2

• 1 mark for 15 red beads and 9 green beads at the start.
• 1 mark for 1/2.

Five eighths of 24 is 15, so there are 15 red beads and 24 - 15 = 9 green beads.

Adding 6 green beads makes 15 green beads, and the bag now holds 24 + 6 = 30 beads.

The probability of green is 15/30, which simplifies to 1/2.

Adding 6 to the bottom of the fraction only, giving 9/30, forgets that the new beads are green and so change the top as well.

33

• 1 mark for 378 pounds for 27 shirts.
• 1 mark for 345 pounds for 30 shirts.
• 1 mark for a saving of 33 pounds.

27 shirts at 14 pounds: 27 x 14 = 27 x 10 + 27 x 4 = 270 + 108 = 378 pounds.

30 shirts at 11.50 pounds: 30 x 11.50 = 345 pounds.

The saving is 378 - 345 = 33 pounds, so buying three shirts more actually costs less.

It is worth pausing on that. Buying more items for less money feels wrong, which is why questions like this catch people out.

Working out 27 x 11.50 is not allowed here: the lower price only applies to orders of 30 or more.

9 cm

The area of a triangle is half the base multiplied by the perpendicular height.

Half of 14 is 7, so 7 multiplied by the height is 63.

The height is 63 / 7 = 9 cm. Forgetting the half gives 4.5 cm.

8.64 litres

• 1 mark for 4320 drips in 24 hours.
• 1 mark for 8.64 litres.

In one hour there are 3 x 60 = 180 drips, so in 24 hours there are 180 x 24 = 4320 drips.

Each drip is 2 ml, so the tap loses 4320 x 2 = 8640 ml.

8640 ml divided by 1000 is 8.64 litres.

Leaving the answer as 8640 answers in the wrong unit, and forgetting to multiply by 60 gives a figure 60 times too small.

0.9 kg

Work out each step: 0.8, 0.9, 0.6, 0.5 and 0.4 kg.

The largest of these is 0.9 kg, between the second and third months.

The overall gain of 3.2 kg answers a different question: the graph is asking about the steepest single step, not the whole rise.

36

• 1 mark for testing all three ingredients rather than only one.
• 1 mark for identifying that the sugar and the eggs both allow only 3 batches.
• 1 mark for 36 muffins.

Flour: 1 kg is 1000 g, and 1000 / 250 = 4, so the flour alone would allow 4 batches.

Sugar: 500 / 150 = 3 with 50 g left over, so the sugar allows only 3 batches.

Eggs: 7 / 2 = 3 with 1 egg left over, so the eggs allow only 3 batches.

The ingredient that runs out first decides everything, so Nia can make 3 batches, which is 3 x 12 = 36 muffins.

Answering 48 uses the flour, which is the one ingredient she has plenty of. The first thing you check is not usually the thing that limits you.

4

Divide 2468 by 7. 7 x 300 = 2100, leaving 368. 7 x 50 = 350, leaving 18. 7 x 2 = 14, leaving 4.

So 2468 = 7 x 352 + 4.

The machine fills 352 bags and 4 marbles are left over. The question asks for the remainder, not the number of bags.

17

• 1 mark for identifying the eighth and ninth numbers as 64 and 81.
• 1 mark for 17.

These are the square numbers: 1 x 1, 2 x 2, 3 x 3 and so on.

The eighth number is 8 x 8 = 64 and the ninth is 9 x 9 = 81.

81 - 64 = 17.

Notice that the gaps between square numbers are 3, 5, 7, 9 and so on, so the gap from the eighth to the ninth is the eighth odd number after 1, which is 17. Either route works.

Reading the pattern as going up in 3s, 5s and 7s but losing count of which gap is wanted is the usual slip.

12.5

Halve repeatedly: 200, 100, 50, 25, 12.5.

25 is still a whole number, and halving it gives 12.5, which is not.

So the first term that is not a whole number is 12.5.

40

• 1 mark for a square of side 12 m, the highest common factor of 96 and 60.
• 1 mark for 8 squares along the long side and 5 along the short side.
• 1 mark for 40 plots.

The side of each square must divide exactly into both 96 and 60, and the largest number that does is 12.

You can check: the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, and of those the largest that also divides 96 is 12.

Along the 96 m side there are 96 / 12 = 8 squares, and along the 60 m side there are 60 / 12 = 5 squares.

So there are 8 x 5 = 40 plots.

Answering 12 gives the side length rather than the number of plots, and answering 13 comes from adding 8 and 5 instead of multiplying.

11

The seven numbers add up to 7 x 12 = 84.

Removing 18 leaves a total of 84 - 18 = 66 shared between six numbers.

66 / 6 = 11.

The mean falls because the number removed was above the old mean.

47

• 1 mark for establishing that the units digit is 3 more than the tens digit.
• 1 mark for 47.

Swapping the digits of a two-digit number changes it by 9 times the gap between the digits. Here the number grows by 27, so the gap is 27 / 9 = 3, with the units digit the larger one.

The digits add to 11 and differ by 3, so they are 7 and 4, with 4 as the tens digit.

Ravi's number is 47.

Check: swapping gives 74, and 74 - 47 = 27, and 4 + 7 = 11.

If you prefer, list the two-digit numbers whose digits add to 11: 29, 38, 47, 56, 65, 74, 83, 92, and test each one. Only 47 works.

90

15 per cent of 4000: 10 per cent is 400 and 5 per cent is 200, so 15 per cent is 600 cyclists.

15 per cent of those 600: 10 per cent is 60 and 5 per cent is 30, so the answer is 90 adults.

Taking 30 per cent of 4000 gives 1200 and is wrong, because the second 15 per cent is measured against the 600 cyclists, not against everyone surveyed.

5/12

• 1 mark for 36 equally likely pairs of rolls.
• 1 mark for counting 15 pairs that give a prime total.
• 1 mark for 5/12.

There are 6 results on the first roll and 6 on the second, so 6 x 6 = 36 equally likely pairs.

The possible totals run from 2 to 12. The prime totals among them are 2, 3, 5, 7 and 11. Remember that 1 is not prime and 9 is not prime, since 9 = 3 x 3.

Count the pairs for each prime total: 2 can happen 1 way, 3 can happen 2 ways, 5 can happen 4 ways, 7 can happen 6 ways and 11 can happen 2 ways.

1 + 2 + 4 + 6 + 2 = 15 pairs out of 36.

15/36 simplifies by dividing both by 3, giving 5/12.

Counting the five prime totals as five outcomes out of eleven possible totals gives 5/11, which is wrong because the totals are not equally likely.

2 hours 30 minutes

• 1 mark for 150 minutes.
• 1 mark for 2 hours 30 minutes.

37500 / 250: divide both numbers by 50 to get 750 / 5, which is 150.

So the pool takes 150 minutes to fill.

150 minutes is 2 hours and 30 minutes, because 120 minutes is 2 hours and 30 minutes are left.

Writing 2.5 hours as 2 hours 50 minutes is the conversion slip to watch for.

6

A cube built this way has the same number of small cubes along each of its three directions, so you need the number that multiplies by itself three times to give 216.

5 x 5 x 5 = 125, which is too small, and 6 x 6 x 6 = 216.

So there are 6 small cubes along each edge. Dividing 216 by 3 to get 72 treats the three directions as adding rather than multiplying.