2025 Key Stage 2 Mathematics Paper 1: Arithmetic

💡 What are we being asked to do?

The question asks us to break down (partition) the number 7,549 into its place value parts: thousands, hundreds, tens, and ones.

💡 Strategy:

We need to subtract 385 from 456. The column method is the most reliable way to do this. We must align the Ones, Tens, and Hundreds columns carefully.

Details:

• Ones: \( 6 – 5 = 1 \)
• Tens: \( 5 – 8 \) we cannot do. We borrow 1 hundred from the 4 (which becomes 3). The 5 becomes 15.
• Tens (continued): \( 15 – 8 = 7 \)
• Hundreds: \( 3 – 3 = 0 \)

💡 Key Concept:

We are multiplying something by 233 to get an answer of 0.

The “Zero Property of Multiplication” states that if the product of any multiplication is 0, one of the numbers must be 0.

💡 Crucial Rule:

When adding decimal numbers, you MUST align the decimal points vertically. This ensures you add tenths to tenths, hundredths to hundredths, etc.

One number (47.65) has 2 decimal places, and the other (51.783) has 3. It helps to add a placeholder zero to make them the same length: 47.650.

Step-by-step:

• Thousandths: \( 0 + 3 = 3 \)
• Hundredths: \( 5 + 8 = 13 \) (Write 3, carry 1)
• Tenths: \( 6 + 7 + 1(\text{carry}) = 14 \) (Write 4, carry 1, place decimal)
• Ones: \( 7 + 1 + 1(\text{carry}) = 9 \)
• Tens: \( 4 + 5 = 9 \)

💡 Mental or Written?

Since we are only subtracting a small single-digit number (8), we can count back mentally. However, crossing a boundary (from 904 down into the 800s) can be tricky, so let’s verify with a quick column method or number line.

We cannot do \( 4 – 8 \), so we borrow from the tens. The tens column is 0, so we must borrow from the hundreds first.

💡 Strategy:

This asks “How many 12s go into 84?”. Since 12 is a standard times table number, we can count up in 12s.

💡 Strategy:

We are multiplying a 3-digit number by a 1-digit number. The column method (short multiplication) is efficient here. We put the larger number on top.

Step-by-step:

• Ones: \( 2 \times 6 = 12 \) (Write 2, carry 1)
• Tens: \( 3 \times 6 = 18 \). Add the carry: \( 18 + 1 = 19 \) (Write 9, carry 1)
• Hundreds: \( 8 \times 6 = 48 \). Add the carry: \( 48 + 1 = 49 \)

💡 Strategy:

We can multiply these numbers in any order, but usually, it’s easiest to work from left to right, or save the “multiply by 10” for the end because that’s the easiest step.

💡 Strategy:

We start with 326 and add “something” to get to 380. To find that “something”, we can calculate the difference between the two numbers.

Calculation required: \( 380 – 326 \)

Details:

• Ones: \( 0 – 6 \) we can’t do. Borrow from tens (8 becomes 7, 0 becomes 10).
• \( 10 – 6 = 4 \)
• Tens: \( 7 – 2 = 5 \)
• Hundreds: \( 3 – 3 = 0 \)

💡 Strategy:

Division and multiplication are opposites. If \( 72 \div \Box = 9 \), then it is also true that \( 9 \times \Box = 72 \).

We need to think: “How many 9s go into 72?” or “9 times what equals 72?”

💡 Rule:

To multiply fractions, you simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together.

Simplifying (Optional but recommended):

Both numbers end in 0, so we can divide both by 10.

\[ \frac{10}{30} = \frac{1}{3} \]

💡 Strategy:

Instead of doing a long division, look at the numbers. We know a fact about 54 and 6.

💡 Watch Out:

Pay attention to the zero in the middle of 807. When we multiply the tens column, it will just be the carry from the ones column.

Step-by-step:

• Ones: \( 7 \times 8 = 56 \) (Write 6, carry 5)
• Tens: \( 0 \times 8 = 0 \). Add the carry: \( 0 + 5 = 5 \) (Write 5)
• Hundreds: \( 8 \times 8 = 64 \) (Write 64)

💡 Strategy:

We multiply 614 by 2 (the ones), then multiply 614 by 30 (the tens). Finally, we add the two answers together.

Critical: Don’t forget the placeholder zero when multiplying by the tens digit (3).

Row 1 (×2):

• \( 4 \times 2 = 8 \)
• \( 1 \times 2 = 2 \)
• \( 6 \times 2 = 12 \)
• Result: 1228

Row 2 (×30):

• Place the 0 first.
• \( 4 \times 3 = 12 \) (Write 2, carry 1)
• \( 1 \times 3 = 3 \). Add carry: 4
• \( 6 \times 3 = 18 \)
• Result: 18420

Total: \( 1228 + 18420 = 19648 \)

💡 Strategy:

We can ignore the zeros for a moment and look at the basic division facts.

💡 Strategy:

We can ignore the decimal point for a moment, multiply the whole numbers, and then put the decimal point back in.

Calculation: \( 3 \times 89 \)

Working:

• \( 9 \times 3 = 27 \) (Write 7, carry 2)
• \( 8 \times 3 = 24 \)
• \( 24 + 2 (\text{carry}) = 26 \)

The result is 267.

Now, look at the original question: \( 3 \times 8.9 \). There is one decimal place in the question.

So, we put one decimal place in the answer: 26.7

💡 Rule:

When we divide any number by 10, the digits stay the same, but they shift one place to the right.

This looks like the decimal point moving one place to the left.

💡 Rule:

We cannot subtract fractions unless the bottom numbers (denominators) are the same.

We have denominators 3 and 9. Since 9 is in the 3 times table, we can use 9 as our common denominator.

💡 Strategy:

We are dividing a 3-digit number by 7. We will use the standard short division layout.

Step-by-step:

• 7 into 8: Goes 1 time, with remainder 1. (Carry the 1 to make 19)
• 7 into 19: Goes 2 times (\( 2 \times 7 = 14 \)), remainder 5. (Carry the 5 to make 56)
• 7 into 56: Goes exactly 8 times (\( 8 \times 7 = 56 \)).

💡 Strategy:

Alignment is critical here. We have a number in the millions subtracting a number in the ten-thousands. Be very careful lining up the columns from the right (Ones).

Details:

• Ones: \( 15 – 6 = 9 \)
• Tens: \( 12 – 4 = 8 \) (After borrowing from hundreds)
• Hundreds: \( 11 – 3 = 8 \) (After borrowing from thousands)
• Thousands: \( 3 – 2 = 1 \)
• Ten Thousands: \( 10 – 5 = 5 \) (After borrowing across zeros)
• Hundred Thousands: \( 9 – 0 = 9 \)

💡 Strategy:

We will use the bus stop method to divide 4,104 by 9.

Step-by-step:

• 9 into 4: Doesn’t go. Carry the 4.
• 9 into 41: Goes 4 times (\( 4 \times 9 = 36 \)). Remainder 5.
• 9 into 50: Goes 5 times (\( 5 \times 9 = 45 \)). Remainder 5.
• 9 into 54: Goes 6 times (\( 6 \times 9 = 54 \)).

💡 Strategy:

We need all fractions to have the same bottom number. We have 6, 3, and 12.

Since 12 is a multiple of both 6 and 3, we can use 12 as the common denominator.

The answer \(\frac{13}{12}\) is an improper fraction. We can also write it as a mixed number:

\[ 1 \frac{1}{12} \]

💡 Crucial Step:

The number 17 does not show a decimal point, but it is there at the end: 17.0.

We MUST add the placeholder zero to subtract 3.6 correctly.

Step-by-step:

• Tenths: \( 0 – 6 \) we can’t do. Borrow 1 from the ones (7 becomes 6, 0 becomes 10).
• \( 10 – 6 = 4 \)
• Ones: \( 6 – 3 = 3 \)
• Tens: \( 1 – 0 = 1 \)

💡 Strategy:

We can add the whole numbers first, and then add the fractions.

Whole numbers: \( 2 + 1 = 3 \)

Fractions: \( \frac{3}{5} + \frac{3}{5} \)

💡 Strategy:

To find 5%, it is easiest to find 10% first, and then halve it.

To find 10% of a number, we divide by 10.

💡 Rule:

Dividing a fraction by a whole number is the same as multiplying the denominator (bottom number) by that whole number.

Or, you can use the KFC method: Keep, Flip, Change.

\[ \frac{5}{8} \div \frac{3}{1} \rightarrow \frac{5}{8} \times \frac{1}{3} \]

💡 Strategy:

Calculating 19% directly is hard. It is easier to find 20% (which is \( 2 \times 10\% \)) and then subtract 1%.

\[ 19\% = 20\% – 1\% \]

💡 Coin Card Method:

Dividing by 17 is tricky mentally. It helps to list out some multiples of 17 first (“Coin Card”).

• \( 1 \times 17 = 17 \)
• \( 2 \times 17 = 34 \)
• \( 5 \times 17 = 85 \) (Easy to find: \( 17 \times 10 = 170 \), half of that is 85)
• \( 10 \times 17 = 170 \)

Details:

• 17 into 88: We know \( 5 \times 17 = 85 \). So it goes 5 times with a remainder of \( 88 – 85 = 3 \).
• Bring down the 4 to make 34.
• 17 into 34: We know \( 2 \times 17 = 34 \). It goes exactly 2 times.
• Result: 52

💡 Strategy:

We are adding \( \frac{1}{6} \) and \( \frac{2}{5} \). The whole number (2) can wait.

Multiples of 6: 6, 12, 18, 24, 30.

Multiples of 5: 5, 10, 15, 20, 25, 30.

The lowest common denominator is 30.

Alternatively, convert \( 2\frac{1}{6} \) to \( \frac{13}{6} \):

\( \frac{13 \times 5}{30} + \frac{2 \times 6}{30} = \frac{65}{30} + \frac{12}{30} = \frac{77}{30} \)

💡 Strategy:

This is a 4-digit by 2-digit multiplication. Accuracy is key. We multiply by 4, then by 70 (don’t forget the zero!), then add.

Row 1 (×4):

• \( 9 \times 4 = 36 \) (Write 6, carry 3)
• \( 1 \times 4 = 4 \). Add carry: 7
• \( 4 \times 4 = 16 \) (Write 6, carry 1)
• \( 6 \times 4 = 24 \). Add carry: 25
• Result: 25676

Row 2 (×70):

• Place the 0.
• \( 9 \times 7 = 63 \) (Write 3, carry 6)
• \( 1 \times 7 = 7 \). Add carry: 13 (Write 3, carry 1)
• \( 4 \times 7 = 28 \). Add carry: 29 (Write 9, carry 2)
• \( 6 \times 7 = 42 \). Add carry: 44
• Result: 449330

Total: \( 25676 + 449330 = 475006 \)

💡 Rule:

We must follow the correct order:

1. Brackets
2. Indices (powers/square numbers)
3. Division / Multiplication
4. Addition / Subtraction

💡 Strategy:

We can break 65% down into chunks:

\[ 65\% = 50\% + 10\% + 5\% \]

💡 Strategy:

We can multiply the whole number part and the fraction part separately.

Alternatively, convert to improper fraction:

\[ 3\frac{1}{3} = \frac{10}{3} \]

\[ \frac{10}{3} \times 12 = \frac{120}{3} = 40 \]

💡 Strategy:

Dividing by 45. Let’s write down some multiples of 45 first.

• \( 1 \times 45 = 45 \)
• \( 2 \times 45 = 90 \)
• \( 4 \times 45 = 180 \)
• \( 5 \times 45 = 225 \)
• \( 10 \times 45 = 450 \)

Details:

• 45 into 81: Goes 1 time (\( 1 \times 45 = 45 \)). Remainder \( 81 – 45 = 36 \).
• Bring down 9 → 369.
• 45 into 369: \( 8 \times 45 = 360 \). It fits 8 times. Remainder \( 369 – 360 = 9 \).
• Bring down 0 → 90.
• 45 into 90: Goes exactly 2 times (\( 2 \times 45 = 90 \)).