SATs 2023 Key Stage 2 Mathematics Paper 1: Arithmetic

✏ Working:

The number is 707.

Hundreds: 7
Tens: 0
Units: 7

We need to take away 1 ten. Since we have 0 tens, we can think of the number as 70 tens and 7 units.

\( 70 \text{ tens} – 1 \text{ ten} = 69 \text{ tens} \)

So the result is 697.

  707
-  10
-----
  697

✏ Working:

  6 1 3 8
+   4 5 6
    1
---------
  6 5 9 4

1. Units: \(8 + 6 = 14\). Write 4, carry 1.

2. Tens: \(3 + 5 + 1 \text{ (carried)} = 9\).

3. Hundreds: \(1 + 4 = 5\).

4. Thousands: \(6 + 0 = 6\).

✏ Working:

   7 0 2
x      4
--------
 2 8 0 8

1. \(4 \times 2 = 8\).

2. \(4 \times 0 = 0\).

3. \(4 \times 7 = 28\).

✏ Working:

  8 0 0 5
+   4 0 8
     1
---------
  8 4 1 3

1. \(5 + 8 = 13\). (Write 3, carry 1)

2. \(0 + 0 + 1 = 1\).

3. \(0 + 4 = 4\).

4. \(8 + 0 = 8\).

✏ Working:

We know that \( 8 \times 3 = 24 \).

Since we are multiplying by 30 (which is 10 times bigger than 3), our answer must be 10 times bigger than 24.

\[ 24 \times 10 = 240 \]

✏ Working:

96 becomes 960.

\[ 96 \times 1 = 96 \]

\[ 96 \times 10 = 960 \]

✏ Working:

   7 . 8 0 0
+  6 . 9 5 3
   1
-------------
  1 4 . 7 5 3

1. Thousandths: \(0 + 3 = 3\)

2. Hundredths: \(0 + 5 = 5\)

3. Tenths: \(8 + 9 = 17\) (Write 7, carry 1)

4. Units: \(7 + 6 + 1 (\text{carried}) = 14\)

Don’t forget to put the decimal point in the answer!

✏ Working:

  7   11   
  8 2 1 7
- 5 4 6 3
---------
  2 7 5 4

1. Units: \(7 – 3 = 4\).

2. Tens: \(1 – 6\) we can’t do. Exchange from hundreds. The 2 becomes 1, and the 1 ten becomes 11 tens. \(11 – 6 = 5\).

3. Hundreds: \(1 – 4\) we can’t do. Exchange from thousands. The 8 becomes 7, and the 1 hundred becomes 11 hundreds. \(11 – 4 = 7\).

4. Thousands: \(7 – 5 = 2\).

✏ Working:

450 is 10 times bigger than 45.

So, the answer must be 10 times bigger than 5.

\[ 5 \times 10 = 50 \]

Check: \( 50 \times 9 = 450 \). Correct.

✏ Working:

    6 5
x     8
-------
  5 2 0
  4

1. \(8 \times 5 = 40\). Write 0, carry 4.

2. \(8 \times 6 = 48\).

3. \(48 + 4 (\text{carried}) = 52\).

✏ Working:

2,800 is 100 times bigger than 28.

So our answer must be 100 times bigger than 4.

\[ 4 \times 100 = 400 \]

Check: \( 400 \times 7 = 2,800 \). Correct.

✏ Working:

We can count up from 795 to 801.

795 + 5 = 800

800 + 1 = 801

Total difference = \( 5 + 1 = 6 \).

Alternatively:

\[ 801 – 795 = 6 \]

✏ Working:

2,700 is 100 times bigger than 27.

So the answer is 100 times bigger than 9.

\[ 9 \times 100 = 900 \]

✏ Working:

Top: \( 2 \times 5 = 10 \)

Bottom: \( 7 \times 9 = 63 \)

Fraction: \( \frac{10}{63} \)

✏ Working:

1. How many 9s in 7? 0. Carry the 7. (Now we have 74).

2. How many 9s in 74? \( 8 \times 9 = 72 \). So 8 remainder 2. Carry the 2. (Now we have 27).

3. How many 9s in 27? 3 exactly.

     0 8 3
   -------
 9 | 7 4 7
     7 2

✏ Working:

Multiply top and bottom of \( \frac{5}{8} \) by 2:

\[ \frac{5 \times 2}{8 \times 2} = \frac{10}{16} \]

Now add:

\[ \frac{3}{16} + \frac{10}{16} = \frac{3 + 10}{16} = \frac{13}{16} \]

✏ Working:

Start: 0.3

Move digits right: 0.03

(The 3 moves from tenths to hundredths column).

✏ Working:

Convert \( \frac{1}{3} \): Multiply by 6. \[ \frac{1 \times 6}{3 \times 6} = \frac{6}{18} \]

Convert \( \frac{2}{6} \): Multiply by 3. \[ \frac{2 \times 3}{6 \times 3} = \frac{6}{18} \]

The last fraction is already \( \frac{5}{18} \).

✏ Working:

\[ \frac{6}{18} + \frac{6}{18} + \frac{5}{18} = \frac{6+6+5}{18} = \frac{17}{18} \]

✏ Working:

       4   9   10
  2 9 . 5 0 0
- 1 6 . 1 2 5
-------------
  1 3 . 3 7 5

1. 0 – 5? Can’t do. Borrow from 5… make it 4, middle 0 becomes 9, end 0 becomes 10.

2. \( 10 – 5 = 5 \).

3. \( 9 – 2 = 7 \).

4. \( 4 – 1 = 3 \).

5. \( 9 – 6 = 3 \).

6. \( 2 – 1 = 1 \).

✏ Working:

     5 0 8
x      7 4
----------
   2 0 3 2  (508 x 4)
 3 5 5 6 0  (508 x 70)
----------
 3 7 5 9 2

Row 1 (\( \times 4 \)):

\( 4 \times 8 = 32 \) (write 2, carry 3)

\( 4 \times 0 = 0 \), plus 3 = 3

\( 4 \times 5 = 20 \)

Row 2 (\( \times 70 \)): (Write 0 first)

\( 7 \times 8 = 56 \) (write 6, carry 5)

\( 7 \times 0 = 0 \), plus 5 = 5

\( 7 \times 5 = 35 \)

Total: Add rows together.

✏ Working:

\[ \frac{1}{8 \times 3} = \frac{1}{24} \]

✏ Working:

\[ \frac{2}{7} + \frac{5}{7} = \frac{7}{7} \]

We know that \( \frac{7}{7} \) is equal to 1 whole.

Now add the initial 1:

\[ 1 + 1 = 2 \]

✏ Working:

First, calculate \( 48 \div 6 = 8 \).

Now substitute this back into the problem:

\[ 70 + 8 = 78 \]

✏ Working:

   3 2
x  1 2
------
   6 4  (32 x 2)
 3 2 0  (32 x 10)
------
 3 8 4

We found \( 32 \times 12 = 384 \).

Now put the decimal point back (one place from the right).

\[ 38.4 \]

✏ Working:

       1 3
    -------
 47 | 6 1 1
    - 4 7 0   (10 x 47)
    -------
      1 4 1
    - 1 4 1   ( 3 x 47)
    -------
          0

1. How many 47s in 61? \( 1 \). (\( 1 \times 47 = 47 \)). Remainder is \( 61 – 47 = 14 \).

2. Bring down the 1 to make 141.

3. How many 47s in 141? Let’s check our list… \( 3 \times 47 = 141 \). Exactly 3.

Total: 13.

✏ Working:

     1 1 4 9 r1
   ---------
 5 | 5 7 4 6
       2 4

1. 5 into 5 goes 1.

2. 5 into 7 goes 1, remainder 2. (Carry 2 to make 24).

3. 5 into 24 goes 4, remainder 4. (Carry 4 to make 46).

4. 5 into 46 goes 9, remainder 1.

Answer: 1,149 remainder 1.

✏ Working:

Find 50%:

Half of 700 = 350

Find 1%:

\( 700 \div 100 = 7 \)

Find 2%:

\( 7 \times 2 = 14 \)

Total (52%):

\( 350 + 14 = 364 \)

✏ Working:

\[ \frac{1}{3 \times 6} = \frac{1}{18} \]

✏ Working:

     5 2 2 7
x        4 3
------------
   1 5 6 8 1  (5227 x 3)
 2 0 9 0 8 0  (5227 x 40)
------------
 2 2 4 7 6 1

Row 1 (\( \times 3 \)):

\( 3 \times 7 = 21 \)

\( 3 \times 2 = 6 \) (+2 carried) = 8

\( 3 \times 2 = 6 \)

\( 3 \times 5 = 15 \)

Row 2 (\( \times 40 \)):

\( 4 \times 7 = 28 \)

\( 4 \times 2 = 8 \) (+2 carried) = 10

\( 4 \times 2 = 8 \) (+1 carried) = 9

\( 4 \times 5 = 20 \)

✏ Working:

1. Find 100%: \( 180 \)

2. Find 10%: \( 180 \div 10 = 18 \)

3. Find 5%: Half of 10%. Half of 18 is 9.

4. Subtract: \( 180 – 9 = 171 \).

✏ Working:

   3 7
x    4
------
 1 4 8
   2

\( 4 \times 7 = 28 \) (carry 2)

\( 4 \times 3 = 12 \), plus 2 = 14.

So \( 37 \times 4 = 148 \).

Now divide by 10 (move decimal point one place left):

\[ 14.8 \]

✏ Working:

We need to find the difference between 10 tenths and 7 tenths.

\[ 10 – 7 = 3 \]

So the missing fraction is \( \frac{3}{10} \).

✏ Working:

       1 7 2
    ---------
 26 | 4 4 7 2
    - 2 6 0 0   (100 x 26)
    ---------
      1 8 7 2
    - 1 8 2 0   ( 70 x 26)
    ---------
          5 2
    -     5 2   (  2 x 26)
    ---------
            0

1. 44 hundreds \(\div\) 26? 100 times 26 is 2600. Remaining: 1872.

2. 187 tens \(\div\) 26? Try \( 7 \times 26 \). \( 7 \times 20 = 140, 7 \times 6 = 42 \), so \( 182 \). Remaining: 52.

3. 52 units \(\div\) 26? Exactly 2.

Total: \( 100 + 70 + 2 = 172 \).

✏ Working:

1. Convert \( \frac{5}{6} \) to twelfths: Multiply by 2. \( \frac{10}{12} \).

So \( 2\frac{5}{6} = 2\frac{10}{12} \).

2. Convert \( \frac{3}{4} \) to twelfths: Multiply by 3. \( \frac{9}{12} \).

3. Subtract: \( 2\frac{10}{12} – \frac{9}{12} = 2\frac{1}{12} \).

✏ Working:

1. 10% of 750 = 75

2. 40% = \( 75 \times 4 = 300 \)

3. 1% of 750 = 7.5

4. 2% = \( 7.5 \times 2 = 15 \)

5. 38% = \( 300 – 15 = 285 \)

✏ Working:

1. Divide by 3: \( 900 \div 3 = 300 \).

2. Multiply by 2: \( 300 \times 2 = 600 \).