SATs 2019 KS2 Mathematics Paper 1: Arithmetic

💡 Strategy:

We are adding 90 (9 tens) to 6,000 (6 thousands). There are no other digits in the tens column of 6,000.

🔧 Calculation:

We simply place the 90 into the last two digits.

\( 6,000 + 90 = 6,090 \)

💡 Strategy:

Line up the digits correctly. The 82 must line up with the 75 (tens and units).

🔧 Method:

1. Units: \( 5 + 2 = 7 \)
2. Tens: \( 7 + 8 = 15 \) (Write 5, carry 1)
3. Hundreds: \( 2 + 1 (\text{carry}) = 3 \)
4. Thousands: \( 8 + 0 = 8 \)

   8 2 7 5
 +     8 2
 ─────────
   8 3 5 7
     1
 

💡 Why we do this:

This question asks us to split (partition) the number 826 into Hundreds, Tens, and Units.

\( 826 = 8 \text{ Hundreds} + 2 \text{ Tens} + 6 \text{ Units} \)

🔧 Method:

• We have the Hundreds (800).
• We have the Units (6).
• We are missing the Tens.

The digit in the tens column is 2.

2 Tens = 20.

💡 Strategy:

To find a missing number in an addition, we do the inverse (opposite), which is subtraction.

\( 341 – 5 = \square \)

🔧 Method:

Count back 5 from 341.

• 341 – 1 = 340
• 340 – 4 = 336

Or use column subtraction:

     3 34 11
 -       5
 ─────────
     3  3  6
 

💡 Strategy:

It is easier to calculate if we put the larger number on top.

\( 41 \times 9 \)

🔧 Method:

1. Units: \( 1 \times 9 = 9 \)
2. Tens: \( 4 \times 9 = 36 \)

     4 1
 x     9
 ───────
   3 6 9
 

💡 Critical Rule:

When adding decimals, you MUST line up the decimal points.

5.87 has two decimal places. 3.123 has three. It helps to add a “ghost zero” to 5.87 so they are the same length: 5.870.

🔧 Method:

1. Thousandths: \( 0 + 3 = 3 \)
2. Hundredths: \( 7 + 2 = 9 \)
3. Tenths: \( 8 + 1 = 9 \)
4. Units: \( 5 + 3 = 8 \)

   5 . 8 7 0
 + 3 . 1 2 3
 ───────────
   8 . 9 9 3
 

💡 Strategy:

Ignore the zero for a moment. We know that:

\[ 18 \div 3 = 6 \]

🔧 Calculation:

Since the question is \( 180 \div 3 \) (which is 10 times bigger than 18), the answer must be 10 times bigger than 6.

\[ 6 \times 10 = 60 \]

💡 Strategy:

This relies on knowing your 12 times table.

How many 12s go into 120?

🔧 Method:

\[ 12 \times 10 = 120 \]

Therefore,

\[ 120 \div 12 = 10 \]

💡 Why we do this:

Multiplying by zero means “we have zero lots of something”.

No matter how big the number is, if you multiply it by zero, the answer is always zero.

💡 Strategy:

We are dividing a 2-digit number by a 1-digit number. Short division (bus stop) is efficient.

🔧 Method:

1. 9 ÷ 7: 7 goes into 9 1 time, with a remainder of 2.
2. Carry the 2 to the next digit to make 21.
3. 21 ÷ 7: 7 goes into 21 exactly 3 times.

     1 3
   ┌────
 7 │ 921
 

💡 Strategy:

This is a subtraction calculation. We can set it up in a column to subtract the ones and the tens separately.

🔧 Method:

1. Units: \( 7 – 5 = 2 \)
2. Tens: \( 8 – 6 = 2 \)

   8 7
 - 6 5
 ─────
   2 2
 

💡 Why we do this:

The question asks: “What do we take away from 602 to get 594?”

We can find the difference between the two numbers to find the answer: \[ 602 – 594 = \square \]

🔧 Method:

The numbers are close together, so we could count up from 594 to 602.

• 594 + 6 = 600
• 600 + 2 = 602
• Total difference = 6 + 2 = 8

Alternatively, use column subtraction with borrowing:

   56 90 12
 -    5  9  4
 ────────────
            8
 

💡 Strategy:

We are dividing a 4-digit number by a 2-digit number (11). Short division is efficient here because the 11 times table is straightforward.

🔧 Method:

1. 12 ÷ 11: 11 goes into 12 once, with a remainder of 1. Carry the 1.
2. 11 ÷ 11: The number becomes 11. 11 goes into 11 exactly once.
3. 0 ÷ 11: 11 goes into 0 zero times.

     0 1 1 0
   ┌────────
 11│ 1 211 0
 

💡 Why we do this:

When multiplying a decimal number by 10, all digits move one place to the left. The number gets 10 times bigger.

🔧 How to do it:

• The 2 tens become 2 hundreds.
• The 5 units become 5 tens.
• The 3 tenths become 3 units.
• The 4 hundredths become 4 tenths.

Visually, the decimal point appears to move one place to the right:

\( 25.34 \rightarrow 253.4 \)

💡 Rule:

We must follow the order of operations:

1. Brackets (do this first!)
2. Orders (indices)
3. Division / Multiplication
4. Addition / Subtraction

🔧 Calculation:

First, calculate inside the brackets: \( 30 – 24 = 6 \)

The question becomes: \[ 60 \div 6 \]

🔧 Calculation:

\( 60 \div 6 = 10 \)

💡 Why we do this:

The small number (index or exponent) tells us how many times to multiply the base number by itself.

\( 3^3 \) means “3 cubed”, or \( 3 \times 3 \times 3 \).

Common Mistake: Do not calculate \( 3 \times 3 = 9 \). This is incorrect.

🔧 How to do it:

1. First, multiply the first two numbers: \( 3 \times 3 = 9 \)
2. Then, multiply the result by the third number: \( 9 \times 3 = 27 \)

💡 Strategy:

Multiplying by 1000 moves all digits 3 places to the left.

Alternatively, when multiplying a whole number by 1000, we simply append three zeros to the end.

🔧 Method:

Start with: \( 101 \)

Add three zeros: \( 101,000 \)

💡 Strategy:

It is easier to find 10% first and then double it to find 20%.

To find 10%, we divide by 10 (move decimal point one place left/remove one zero).

\( 10\% \text{ of } 3000 = 300 \)

🔧 Calculation:

If \( 10\% = 300 \), then \( 20\% \) is double that.

\( 300 \times 2 = 600 \)

💡 Critical Rule:

The number 7 is the same as 7.00. We MUST use these placeholder zeros so that we can subtract the decimals correctly.

🔧 Method:

We cannot do \( 0 – 5 \) or \( 0 – 2 \), so we must exchange from the units column.

   67 . 90 10
 -    2 . 2  5
 ──────────────
      4 . 7  5
 

💡 Strategy:

Dividing by 100 makes the number smaller. We move the digits 2 places to the right.

🔧 Method:

Start with 0.9.

• Divide by 10 (move 1 place): 0.09
• Divide by 100 (move 2 places): 0.009

We fill the empty spaces with zeros.

💡 Critical Rule:

To subtract a decimal from a whole number, turn the whole number into a decimal by adding a decimal point and a zero.

\( 9 \) becomes \( 9.0 \).

🔧 Method:

We align the decimal points.

1. Tenths: \( 0 – 9 \). We cannot do this. Borrow from the units.
2. The 9 units becomes 8. The 0 tenths becomes 10.
3. \( 10 – 9 = 1 \).
4. Units: \( 8 – 1 = 7 \).

   89 . 10
 -    1 .  9
 ───────────
      7 .  1
 

💡 Strategy:

We have a mixed number \( 1\frac{3}{7} \). It is easier to subtract if we convert this entire number into an improper fraction first.

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

So, \( 1\frac{3}{7} = \frac{7}{7} + \frac{3}{7} = \frac{10}{7} \).

🔧 Calculation:

Now the calculation is:

\[ \frac{10}{7} – \frac{4}{7} \]

Since the denominators are the same, we simply subtract the numerators:

\[ 10 – 4 = 6 \]

The denominator stays as 7.

💡 Why we do this:

We are multiplying by a 2-digit number (27). We will split this into two parts:

1. \( 836 \times 7 \)
2. \( 836 \times 20 \)

🔧 Row 1: \( 836 \times 7 \)

• \( 6 \times 7 = 42 \) (write 2, carry 4)
• \( 3 \times 7 = 21 + 4 = 25 \) (write 5, carry 2)
• \( 8 \times 7 = 56 + 2 = 58 \)

🔧 Row 2: \( 836 \times 20 \)

First, put a 0 in the units column (the magic zero).

• \( 6 \times 2 = 12 \) (write 2, carry 1)
• \( 3 \times 2 = 6 + 1 = 7 \)
• \( 8 \times 2 = 16 \)

        8 3 6
 x        2 7
 ────────────
    5 8 5 2  (836 x 7)
  1 6 7 2 0  (836 x 20)
 ────────────
  2 2 5 7 2
  1 1
 

💡 Strategy:

The denominators (5 and 4) are different. We cannot add them yet. We must find a common multiple.

Multiples of 5: 5, 10, 15, 20…

Multiples of 4: 4, 8, 12, 16, 20…

The Lowest Common Denominator is 20.

🔧 Calculation:

Convert \( \frac{1}{5} \) to twentieths: Multiply top and bottom by 4.

\[ \frac{1 \times 4}{5 \times 4} = \frac{4}{20} \]

Convert \( \frac{3}{4} \) to twentieths: Multiply top and bottom by 5.

\[ \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \]

\[ \frac{4}{20} + \frac{15}{20} = \frac{19}{20} \]

💡 Strategy:

Dividing by 37 is difficult mentally. Write out the first few multiples of 37 to help.

• \( 1 \times 37 = 37 \)
• \( 2 \times 37 = 74 \)
• \( 3 \times 37 = 111 \)

🔧 Method:

Part A: 37 into 88.

Looking at our list, 74 is the closest without going over. That is \( 2 \times 37 \).

Remainder: \( 88 – 74 = 14 \).

Bring down the 8 to make 148.

Part B: 37 into 148.

Let’s continue our list:

• \( 4 \times 37 = (2 \times 74) = 148 \)

Perfect! It goes in exactly 4 times.

        2 4
    ┌──────
 37 │ 8 8 8
    - 7 4
    ──────
      1 4 8
    - 1 4 8
    ──────
          0
 

💡 Strategy:

We can add the whole numbers and the fractions separately.

Wholes: \( 1 + 2 = 3 \)

🔧 Calculation:

We need to add \( \frac{1}{5} + \frac{1}{10} \).

Change \( \frac{1}{5} \) into tenths by doubling top and bottom: \( \frac{2}{10} \).

Now add: \( \frac{2}{10} + \frac{1}{10} = \frac{3}{10} \).

Combine the wholes and the fraction:

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

💡 Strategy:

We can break 35% down into chunks we can easily calculate: \( 30\% + 5\% \).

🔧 Calculation:

10% of 320 is \( 320 \div 10 = 32 \).

30%: This is 3 lots of 10%.

\( 32 \times 3 = 96 \)

5%: This is half of 10%.

Half of 32 is \( 16 \).

\( 96 (30\%) + 16 (5\%) = 112 \)

💡 Strategy:

Multiples of 9: 9, 18, 27, 36…

Multiples of 4: 4, 8, 12… 36…

The lowest common multiple is 36.

🔧 Calculation:

\( \frac{8}{9} \): Multiply top and bottom by 4 → \( \frac{32}{36} \)

\( \frac{1}{4} \): Multiply top and bottom by 9 → \( \frac{9}{36} \)

\[ \frac{32}{36} – \frac{9}{36} = \frac{23}{36} \]

💡 Strategy:

51% is very close to 50%. We can calculate \( 50\% + 1\% \).

50%: This is half. Half of 900 is \( 450 \).

1%: Divide by 100. \( 900 \div 100 = 9 \).

\( 450 + 9 = 459 \)

💡 Strategy:

We perform long multiplication, splitting 62 into 2 and 60.

• \( 8 \times 2 = 16 \) (write 6, carry 1)
• \( 6 \times 2 = 12 + 1 = 13 \) (write 3, carry 1)
• \( 4 \times 2 = 8 + 1 = 9 \)
• \( 3 \times 2 = 6 \)
• Row 1: 6936

Add the placeholder zero.

• \( 8 \times 6 = 48 \) (write 8, carry 4)
• \( 6 \times 6 = 36 + 4 = 40 \) (write 0, carry 4)
• \( 4 \times 6 = 24 + 4 = 28 \) (write 8, carry 2)
• \( 3 \times 6 = 18 + 2 = 20 \)
• Row 2: 208080

        3 4 6 8
 x          6 2
 ──────────────
        6 9 3 6
    2 0 8 0 8 0
 ──────────────
    2 1 5 0 1 6
    1 1 1
 

💡 Why we do this:

Dividing a fraction by a whole number makes the pieces smaller. If you have \( \frac{2}{3} \) of a pizza and you share it among 3 people, each person gets a smaller slice.

Mathematically, dividing by 3 is the same as multiplying the denominator (bottom number) by 3.

🔧 Method:

Keep the numerator (top) the same. Multiply the denominator by the whole number.

\[ \frac{2}{3 \times 3} = \frac{2}{9} \]

💡 Strategy:

It is safer to turn mixed numbers into improper fractions before subtracting.

\( 2\frac{1}{2} \): \( 2 \times 2 + 1 = 5 \), so it is \( \frac{5}{2} \).

🔧 Method:

We are subtracting \( \frac{3}{4} \) from \( \frac{5}{2} \).

The common denominator for 2 and 4 is 4.

Convert \( \frac{5}{2} \) to quarters: Multiply top and bottom by 2.

\[ \frac{5 \times 2}{2 \times 2} = \frac{10}{4} \]

\[ \frac{10}{4} – \frac{3}{4} = \frac{7}{4} \]

This answer is acceptable, or you can convert it back to a mixed number: \( 1\frac{3}{4} \).

💡 Strategy:

Break 36% down into smaller chunks: \( 30\% + 5\% + 1\% \).

10%: \( 450 \div 10 = 45 \).

30%: \( 45 \times 3 = 135 \).

5%: Half of 10% (half of 45) = \( 22.5 \).

1%: \( 450 \div 100 = 4.5 \).

💡 Strategy:

Convert \( 1\frac{3}{4} \) to an improper fraction.

\( 1 \times 4 + 3 = 7 \), so it is \( \frac{7}{4} \).

🔧 Method:

Multiply the numerator by the whole number:

\[ \frac{7}{4} \times 10 = \frac{70}{4} \]

We can simplify \( \frac{70}{4} \).

Half of 70 is 35, so \( \frac{35}{2} \).

\( 35 \div 2 = 17.5 \) (or \( 17\frac{1}{2} \)).

💡 Strategy:

When finding a fraction of an amount: Divide by the bottom, Multiply by the top.

🔧 Calculation:

Divide 540 by 6.

We know \( 54 \div 6 = 9 \).

So, \( 540 \div 6 = 90 \).

🔧 Calculation:

Multiply the result (90) by the numerator (5).

\( 90 \times 5 = 450 \).

💡 Strategy:

Dividing by 83 is tricky. Let’s list some multiples or estimate.

83 is close to 80.

• \( 10 \times 83 = 830 \).
• The number 8051 is much larger than 830, so the answer is a 2-digit number.
• Actually, looking at the first three digits (805), we can see how many 83s fit.

🔧 Method:

Part A: 83 into 805.

Since \( 10 \times 83 = 830 \) (too big), let’s try 9.

\( 9 \times 3 = 27 \) (write 7, carry 2)

\( 9 \times 8 = 72 + 2 = 74 \)

So \( 9 \times 83 = 747 \).

Subtract: \( 805 – 747 \).

• \( 15 – 7 = 8 \)
• \( 9 – 4 = 5 \)

Remainder is 58. Bring down the 1 to make 581.

Part B: 83 into 581.

Look at the last digit (1). What times 3 ends in 1? 7.

Let’s test \( 83 \times 7 \).

• \( 7 \times 3 = 21 \)
• \( 7 \times 8 = 56 + 2 = 58 \)

It fits exactly!

          9 7
    ┌────────
 83 │ 8 0 5 1
    - 7 4 7
    ───────
        5 8 1
      - 5 8 1
      ───────
            0