Key Stage 2 Mathematics Paper 1: Arithmetic (2016)

Mental Method:

987 has 9 hundreds.

\( 9 \text{ hundreds} + 1 \text{ hundred} = 10 \text{ hundreds} \)

So, 987 becomes 1087.

Column Method (for verification):

Units: \( 4 + 6 = 10 \). Write 0, carry 1.

Tens: \( 0 + 4 + 1 \text{ (carried)} = 5 \).

Hundreds: \( 3 + 0 = 3 \).

\( 468 – 10 = 458 \)

\( 458 + 1 = 459 \)

Start at 468. Count back 9.

467, 466, 465, 464, 463, 462, 461, 460, 459.

Units: \( 6 + 5 = 11 \). Write 1, carry 1.

Tens: \( 3 + 8 + 1 \text{ (carried)} = 12 \). Write 2, carry 1.

Hundreds: \( 9 + 2 + 1 \text{ (carried)} = 12 \). Write 12.

\( 95 = 50 + 45 \)

\( 50 \div 5 = 10 \)

\( 45 \div 5 = 9 \)

\( 10 + 9 = 19 \)

How many 5s in 9? 1 remainder 4.

How many 5s in 45? 9.

Units: \( 4+3=7 \)

Tens: \( 9+4=13 \) (carry 1)

Hundreds: \( 9+6+1=16 \) (carry 1)

Thousands: \( 9+7+1=17 \) (carry 1)

Ten Thousands: \( 8+1=9 \)

\( 3 \text{ tens} – 3 \text{ tens} = 0 \text{ tens} \)

The hundreds (4) and units (5) stay the same.

435 becomes 405.

Half of 96 is 48.

Half of 48 is 24.

4 goes into 9 twice (\(2 \times 4 = 8\)), remainder 1.

The 1 joins the 6 to make 16.

4 goes into 16 four times.

Units: \( 1 \times 8 = 8 \)

Tens: \( 7 \times 8 = 56 \). Since there are no hundreds to carry over to, we write 56.

\( 5 \times 7 = 35 \)

There is one zero in 50 and one zero in 70 (total 2 zeros).

So, multiply 35 by 100.

\( 35 \times 100 = 3,500 \)

\( 412 \times 1 = 412 \)

Add two zeros: \( 41,200 \)

Thousandths: \( 5 + 0 = 5 \)

Hundredths: \( 0 + 2 = 2 \)

Tenths: \( 0 + 1 = 1 \)

Units: \( 3 + 6 = 9 \)

Hundreds: 3 into 4 goes 1 time, with remainder 1. The remainder moves to the tens.

Tens: 3 into 18 goes 6 times exactly.

Units: 3 into 6 goes 2 times exactly.

\( 0 + 4 = 4 \)

\( 8 + 1 = 9 \)

\( 9 + 3 = 12 \) (write 2, carry 1)

Don’t forget the decimal point!

\( 5 + 6 + 1 \text{ (carried)} = 12 \) (write 2, carry 1)

\( 1 + 2 + 1 \text{ (carried)} = 4 \)

\( 8 – 0 = 8 \)

\( 4 – 3 = 1 \)

Decimal point

\( 5 – 2 = 3 \)

\( 12 – 7 = 5 \)

\( 122,456 – 12,000 = 110,456 \)

\( 110,456 + 1 = 110,457 \)

This requires significant borrowing (exchange).

\( 3^2 = 3 \times 3 = 9 \)

Now add the 10:

\( 9 + 10 = 19 \)

Start: \( 0.9 \) (9 tenths)

Move 9 one place right: \( 0.09 \) (9 hundredths)

We cannot do \( 0 – 5 \). We must borrow across from the 4.

Change 4 to 3. The 0 becomes 10 (tenths), then borrow again to make 9 (tenths) and the final 0 becomes 10 (hundredths).

\( 10 – 5 = 5 \)

\( 9 – 1 = 8 \)

\( 3 – 1 = 2 \)

12 into 13 goes 1 time, remainder 1.

The 1 joins the 2 to make 12. 12 into 12 goes 1 time exactly.

12 into 0 goes 0 times.

\( 1 \times 6 = 6 \)

\( 7 \times 6 = 42 \)

So, \( 71 \times 6 = 426 \)

\( 1 \times 4 = 4 \)

\( 7 \times 4 = 28 \)

So, \( 71 \times 40 = 2,840 \)

\( 4 + 5 = 9 \)

So, \( \frac{9}{7} \)

How many 7s in 9? 1 whole with 2 remaining.

\( 1 \frac{2}{7} \)

Both \( \frac{9}{7} \) and \( 1 \frac{2}{7} \) are correct.

\( 10\% \text{ of } 1,800 = 180 \)

20% is double 10%.

\( 180 \times 2 = 360 \)

Multiply 15 by 1: \( 15 \times 1 = 15 \)

Multiply 15 by 60: \( 15 \times 60 = 900 \) (Start with a zero!)

\( \frac{3 \times 2}{10 \times 2} = \frac{6}{20} \)

\( \frac{6}{20} – \frac{1}{20} = \frac{5}{20} \)

Both 5 and 20 can be divided by 5.

\( \frac{5 \div 5}{20 \div 5} = \frac{1}{4} \)

1 × 29 = 29

2 × 29 = 58

3 × 29 = 87 (Too big for the first step)

First part: 29 into 72.

It goes 2 times (\( 2 \times 29 = 58 \)).

Remainder: \( 72 – 58 = 14 \).

Bring down the 5 to make 145.

Second part: 29 into 145.

Let’s estimate: 29 is nearly 30. \( 145 \div 30 \) is about 5.

Test \( 5 \times 29 \):

\( 5 \times 20 = 100 \)

\( 5 \times 9 = 45 \)

\( 100 + 45 = 145 \). It fits exactly!

Find 10%: \( 440 \div 10 = 44 \)

Find 5%: Half of 10%. \( 44 \div 2 = 22 \)

Add them: \( 44 + 22 = 66 \)

\( 6574 \times 1 = 6574 \)

\( 4 \times 3 = 12 \) (carry 1)

\( 7 \times 3 = 21 \). Add 1: 22 (carry 2)

\( 5 \times 3 = 15 \). Add 2: 17 (carry 1)

\( 6 \times 3 = 18 \). Add 1: 19

Row 2: 197220

\( \frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10} \)

So the question becomes: \( 1\frac{8}{10} + \frac{3}{10} \)

\( \frac{8}{10} + \frac{3}{10} = \frac{11}{10} \)

\( \frac{11}{10} \) is an improper fraction. It is equal to \( 1\frac{1}{10} \).

We had 1 whole number at the start.

\( 1 + 1\frac{1}{10} = 2\frac{1}{10} \)

\( 1 \times 43 = 43 \)

\( 2 \times 43 = 86 \)

\( 3 \times 43 = 129 \) (Too big for the first part)

43 into 111: Goes 2 times (86).

Remainder: \( 111 – 86 = 25 \).

Bring down 8 to make 258.

43 into 258: Let’s estimate using 40.

\( 258 \div 40 \approx 6 \).

Check: \( 6 \times 43 = (6 \times 40) + (6 \times 3) = 240 + 18 = 258 \). Exact!

\( 3 \text{ fifths} \div 3 = 1 \text{ fifth} \)

Keep \( \frac{3}{5} \)

Change \( \div \) to \( \times \)

Flip \( 3 \) (which is \( \frac{3}{1} \)) to \( \frac{1}{3} \)

\( \frac{3}{5} \times \frac{1}{3} = \frac{3}{15} \)

Simplify: \( \frac{3}{15} = \frac{1}{5} \)

\( 140 \div 5 \)

\( 100 \div 5 = 20 \)

\( 40 \div 5 = 8 \)

\( 20 + 8 = 28 \)

\( 28 \times 2 = 56 \)

\( 1\frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \)

\( \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12} \)

\( \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \)

\( \frac{15}{12} – \frac{4}{12} = \frac{11}{12} \)

\( 42 \div 6 = 7 \)

Now replace the division part with 7.

\( 60 – 7 = 53 \)